5oalem6 - Hilbert Space Explorer

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Theorem 5oalem6 31462

Description: Lemma for orthoarguesian law 5OA. (Contributed by NM, 4-May-2000.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
5oalem5.1	⊢ 𝐴 ∈ S_ℋ
5oalem5.2	⊢ 𝐵 ∈ S_ℋ
5oalem5.3	⊢ 𝐶 ∈ S_ℋ
5oalem5.4	⊢ 𝐷 ∈ S_ℋ
5oalem5.5	⊢ 𝐹 ∈ S_ℋ
5oalem5.6	⊢ 𝐺 ∈ S_ℋ
5oalem5.7	⊢ 𝑅 ∈ S_ℋ
5oalem5.8	⊢ 𝑆 ∈ S_ℋ

**Assertion**
Ref	Expression
5oalem6	⊢ (((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ℎ = (𝑥 +_ℎ 𝑦)) ∧ ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷) ∧ ℎ = (𝑧 +_ℎ 𝑤))) ∧ (((𝑓 ∈ 𝐹 ∧ 𝑔 ∈ 𝐺) ∧ ℎ = (𝑓 +_ℎ 𝑔)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑢 ∈ 𝑆) ∧ ℎ = (𝑣 +_ℎ 𝑢)))) → ℎ ∈ (𝐵 +_ℋ (𝐴 ∩ (𝐶 +_ℋ ((((𝐴 +_ℋ 𝐶) ∩ (𝐵 +_ℋ 𝐷)) ∩ (((𝐴 +_ℋ 𝑅) ∩ (𝐵 +_ℋ 𝑆)) +_ℋ ((𝐶 +_ℋ 𝑅) ∩ (𝐷 +_ℋ 𝑆)))) ∩ ((((𝐴 +_ℋ 𝐹) ∩ (𝐵 +_ℋ 𝐺)) ∩ (((𝐴 +_ℋ 𝑅) ∩ (𝐵 +_ℋ 𝑆)) +_ℋ ((𝐹 +_ℋ 𝑅) ∩ (𝐺 +_ℋ 𝑆)))) +_ℋ (((𝐶 +_ℋ 𝐹) ∩ (𝐷 +_ℋ 𝐺)) ∩ (((𝐶 +_ℋ 𝑅) ∩ (𝐷 +_ℋ 𝑆)) +_ℋ ((𝐹 +_ℋ 𝑅) ∩ (𝐺 +_ℋ 𝑆))))))))))

**Proof of Theorem 5oalem6**
Step	Hyp	Ref	Expression
1		an4 655	. . . 4 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ℎ = (𝑥 +_ℎ 𝑦)) ∧ ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷) ∧ ℎ = (𝑧 +_ℎ 𝑤))) ↔ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷)) ∧ (ℎ = (𝑥 +_ℎ 𝑦) ∧ ℎ = (𝑧 +_ℎ 𝑤))))
2		an4 655	. . . 4 ⊢ ((((𝑓 ∈ 𝐹 ∧ 𝑔 ∈ 𝐺) ∧ ℎ = (𝑓 +_ℎ 𝑔)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑢 ∈ 𝑆) ∧ ℎ = (𝑣 +_ℎ 𝑢))) ↔ (((𝑓 ∈ 𝐹 ∧ 𝑔 ∈ 𝐺) ∧ (𝑣 ∈ 𝑅 ∧ 𝑢 ∈ 𝑆)) ∧ (ℎ = (𝑓 +_ℎ 𝑔) ∧ ℎ = (𝑣 +_ℎ 𝑢))))
3		eqeq1 2732	. . . . . . . . . . 11 ⊢ (ℎ = (𝑥 +_ℎ 𝑦) → (ℎ = (𝑣 +_ℎ 𝑢) ↔ (𝑥 +_ℎ 𝑦) = (𝑣 +_ℎ 𝑢)))
4	3	biimpcd 248	. . . . . . . . . 10 ⊢ (ℎ = (𝑣 +_ℎ 𝑢) → (ℎ = (𝑥 +_ℎ 𝑦) → (𝑥 +_ℎ 𝑦) = (𝑣 +_ℎ 𝑢)))
5		eqeq1 2732	. . . . . . . . . . 11 ⊢ (ℎ = (𝑧 +_ℎ 𝑤) → (ℎ = (𝑣 +_ℎ 𝑢) ↔ (𝑧 +_ℎ 𝑤) = (𝑣 +_ℎ 𝑢)))
6	5	biimpcd 248	. . . . . . . . . 10 ⊢ (ℎ = (𝑣 +_ℎ 𝑢) → (ℎ = (𝑧 +_ℎ 𝑤) → (𝑧 +_ℎ 𝑤) = (𝑣 +_ℎ 𝑢)))
7	4, 6	anim12d 608	. . . . . . . . 9 ⊢ (ℎ = (𝑣 +_ℎ 𝑢) → ((ℎ = (𝑥 +_ℎ 𝑦) ∧ ℎ = (𝑧 +_ℎ 𝑤)) → ((𝑥 +_ℎ 𝑦) = (𝑣 +_ℎ 𝑢) ∧ (𝑧 +_ℎ 𝑤) = (𝑣 +_ℎ 𝑢))))
8		eqeq1 2732	. . . . . . . . . 10 ⊢ (ℎ = (𝑓 +_ℎ 𝑔) → (ℎ = (𝑣 +_ℎ 𝑢) ↔ (𝑓 +_ℎ 𝑔) = (𝑣 +_ℎ 𝑢)))
9	8	biimpcd 248	. . . . . . . . 9 ⊢ (ℎ = (𝑣 +_ℎ 𝑢) → (ℎ = (𝑓 +_ℎ 𝑔) → (𝑓 +_ℎ 𝑔) = (𝑣 +_ℎ 𝑢)))
10	7, 9	anim12d 608	. . . . . . . 8 ⊢ (ℎ = (𝑣 +_ℎ 𝑢) → (((ℎ = (𝑥 +_ℎ 𝑦) ∧ ℎ = (𝑧 +_ℎ 𝑤)) ∧ ℎ = (𝑓 +_ℎ 𝑔)) → (((𝑥 +_ℎ 𝑦) = (𝑣 +_ℎ 𝑢) ∧ (𝑧 +_ℎ 𝑤) = (𝑣 +_ℎ 𝑢)) ∧ (𝑓 +_ℎ 𝑔) = (𝑣 +_ℎ 𝑢))))
11	10	expdcom 414	. . . . . . 7 ⊢ ((ℎ = (𝑥 +_ℎ 𝑦) ∧ ℎ = (𝑧 +_ℎ 𝑤)) → (ℎ = (𝑓 +_ℎ 𝑔) → (ℎ = (𝑣 +_ℎ 𝑢) → (((𝑥 +_ℎ 𝑦) = (𝑣 +_ℎ 𝑢) ∧ (𝑧 +_ℎ 𝑤) = (𝑣 +_ℎ 𝑢)) ∧ (𝑓 +_ℎ 𝑔) = (𝑣 +_ℎ 𝑢)))))
12	11	imp32 418	. . . . . 6 ⊢ (((ℎ = (𝑥 +_ℎ 𝑦) ∧ ℎ = (𝑧 +_ℎ 𝑤)) ∧ (ℎ = (𝑓 +_ℎ 𝑔) ∧ ℎ = (𝑣 +_ℎ 𝑢))) → (((𝑥 +_ℎ 𝑦) = (𝑣 +_ℎ 𝑢) ∧ (𝑧 +_ℎ 𝑤) = (𝑣 +_ℎ 𝑢)) ∧ (𝑓 +_ℎ 𝑔) = (𝑣 +_ℎ 𝑢)))
13	12	anim2i 616	. . . . 5 ⊢ (((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷)) ∧ ((𝑓 ∈ 𝐹 ∧ 𝑔 ∈ 𝐺) ∧ (𝑣 ∈ 𝑅 ∧ 𝑢 ∈ 𝑆))) ∧ ((ℎ = (𝑥 +_ℎ 𝑦) ∧ ℎ = (𝑧 +_ℎ 𝑤)) ∧ (ℎ = (𝑓 +_ℎ 𝑔) ∧ ℎ = (𝑣 +_ℎ 𝑢)))) → ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷)) ∧ ((𝑓 ∈ 𝐹 ∧ 𝑔 ∈ 𝐺) ∧ (𝑣 ∈ 𝑅 ∧ 𝑢 ∈ 𝑆))) ∧ (((𝑥 +_ℎ 𝑦) = (𝑣 +_ℎ 𝑢) ∧ (𝑧 +_ℎ 𝑤) = (𝑣 +_ℎ 𝑢)) ∧ (𝑓 +_ℎ 𝑔) = (𝑣 +_ℎ 𝑢))))
14	13	an4s 659	. . . 4 ⊢ (((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷)) ∧ (ℎ = (𝑥 +_ℎ 𝑦) ∧ ℎ = (𝑧 +_ℎ 𝑤))) ∧ (((𝑓 ∈ 𝐹 ∧ 𝑔 ∈ 𝐺) ∧ (𝑣 ∈ 𝑅 ∧ 𝑢 ∈ 𝑆)) ∧ (ℎ = (𝑓 +_ℎ 𝑔) ∧ ℎ = (𝑣 +_ℎ 𝑢)))) → ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷)) ∧ ((𝑓 ∈ 𝐹 ∧ 𝑔 ∈ 𝐺) ∧ (𝑣 ∈ 𝑅 ∧ 𝑢 ∈ 𝑆))) ∧ (((𝑥 +_ℎ 𝑦) = (𝑣 +_ℎ 𝑢) ∧ (𝑧 +_ℎ 𝑤) = (𝑣 +_ℎ 𝑢)) ∧ (𝑓 +_ℎ 𝑔) = (𝑣 +_ℎ 𝑢))))
15	1, 2, 14	syl2anb 597	. . 3 ⊢ (((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ℎ = (𝑥 +_ℎ 𝑦)) ∧ ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷) ∧ ℎ = (𝑧 +_ℎ 𝑤))) ∧ (((𝑓 ∈ 𝐹 ∧ 𝑔 ∈ 𝐺) ∧ ℎ = (𝑓 +_ℎ 𝑔)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑢 ∈ 𝑆) ∧ ℎ = (𝑣 +_ℎ 𝑢)))) → ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷)) ∧ ((𝑓 ∈ 𝐹 ∧ 𝑔 ∈ 𝐺) ∧ (𝑣 ∈ 𝑅 ∧ 𝑢 ∈ 𝑆))) ∧ (((𝑥 +_ℎ 𝑦) = (𝑣 +_ℎ 𝑢) ∧ (𝑧 +_ℎ 𝑤) = (𝑣 +_ℎ 𝑢)) ∧ (𝑓 +_ℎ 𝑔) = (𝑣 +_ℎ 𝑢))))
16		5oalem5.1	. . . 4 ⊢ 𝐴 ∈ S_ℋ
17		5oalem5.2	. . . 4 ⊢ 𝐵 ∈ S_ℋ
18		5oalem5.3	. . . 4 ⊢ 𝐶 ∈ S_ℋ
19		5oalem5.4	. . . 4 ⊢ 𝐷 ∈ S_ℋ
20		5oalem5.5	. . . 4 ⊢ 𝐹 ∈ S_ℋ
21		5oalem5.6	. . . 4 ⊢ 𝐺 ∈ S_ℋ
22		5oalem5.7	. . . 4 ⊢ 𝑅 ∈ S_ℋ
23		5oalem5.8	. . . 4 ⊢ 𝑆 ∈ S_ℋ
24	16, 17, 18, 19, 20, 21, 22, 23	5oalem5 31461	. . 3 ⊢ (((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷)) ∧ ((𝑓 ∈ 𝐹 ∧ 𝑔 ∈ 𝐺) ∧ (𝑣 ∈ 𝑅 ∧ 𝑢 ∈ 𝑆))) ∧ (((𝑥 +_ℎ 𝑦) = (𝑣 +_ℎ 𝑢) ∧ (𝑧 +_ℎ 𝑤) = (𝑣 +_ℎ 𝑢)) ∧ (𝑓 +_ℎ 𝑔) = (𝑣 +_ℎ 𝑢))) → (𝑥 −_ℎ 𝑧) ∈ ((((𝐴 +_ℋ 𝐶) ∩ (𝐵 +_ℋ 𝐷)) ∩ (((𝐴 +_ℋ 𝑅) ∩ (𝐵 +_ℋ 𝑆)) +_ℋ ((𝐶 +_ℋ 𝑅) ∩ (𝐷 +_ℋ 𝑆)))) ∩ ((((𝐴 +_ℋ 𝐹) ∩ (𝐵 +_ℋ 𝐺)) ∩ (((𝐴 +_ℋ 𝑅) ∩ (𝐵 +_ℋ 𝑆)) +_ℋ ((𝐹 +_ℋ 𝑅) ∩ (𝐺 +_ℋ 𝑆)))) +_ℋ (((𝐶 +_ℋ 𝐹) ∩ (𝐷 +_ℋ 𝐺)) ∩ (((𝐶 +_ℋ 𝑅) ∩ (𝐷 +_ℋ 𝑆)) +_ℋ ((𝐹 +_ℋ 𝑅) ∩ (𝐺 +_ℋ 𝑆)))))))
25	15, 24	syl 17	. 2 ⊢ (((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ℎ = (𝑥 +_ℎ 𝑦)) ∧ ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷) ∧ ℎ = (𝑧 +_ℎ 𝑤))) ∧ (((𝑓 ∈ 𝐹 ∧ 𝑔 ∈ 𝐺) ∧ ℎ = (𝑓 +_ℎ 𝑔)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑢 ∈ 𝑆) ∧ ℎ = (𝑣 +_ℎ 𝑢)))) → (𝑥 −_ℎ 𝑧) ∈ ((((𝐴 +_ℋ 𝐶) ∩ (𝐵 +_ℋ 𝐷)) ∩ (((𝐴 +_ℋ 𝑅) ∩ (𝐵 +_ℋ 𝑆)) +_ℋ ((𝐶 +_ℋ 𝑅) ∩ (𝐷 +_ℋ 𝑆)))) ∩ ((((𝐴 +_ℋ 𝐹) ∩ (𝐵 +_ℋ 𝐺)) ∩ (((𝐴 +_ℋ 𝑅) ∩ (𝐵 +_ℋ 𝑆)) +_ℋ ((𝐹 +_ℋ 𝑅) ∩ (𝐺 +_ℋ 𝑆)))) +_ℋ (((𝐶 +_ℋ 𝐹) ∩ (𝐷 +_ℋ 𝐺)) ∩ (((𝐶 +_ℋ 𝑅) ∩ (𝐷 +_ℋ 𝑆)) +_ℋ ((𝐹 +_ℋ 𝑅) ∩ (𝐺 +_ℋ 𝑆)))))))
26	16, 18	shscli 31120	. . . . . . . . . 10 ⊢ (𝐴 +_ℋ 𝐶) ∈ S_ℋ
27	17, 19	shscli 31120	. . . . . . . . . 10 ⊢ (𝐵 +_ℋ 𝐷) ∈ S_ℋ
28	26, 27	shincli 31165	. . . . . . . . 9 ⊢ ((𝐴 +_ℋ 𝐶) ∩ (𝐵 +_ℋ 𝐷)) ∈ S_ℋ
29	16, 22	shscli 31120	. . . . . . . . . . 11 ⊢ (𝐴 +_ℋ 𝑅) ∈ S_ℋ
30	17, 23	shscli 31120	. . . . . . . . . . 11 ⊢ (𝐵 +_ℋ 𝑆) ∈ S_ℋ
31	29, 30	shincli 31165	. . . . . . . . . 10 ⊢ ((𝐴 +_ℋ 𝑅) ∩ (𝐵 +_ℋ 𝑆)) ∈ S_ℋ
32	18, 22	shscli 31120	. . . . . . . . . . 11 ⊢ (𝐶 +_ℋ 𝑅) ∈ S_ℋ
33	19, 23	shscli 31120	. . . . . . . . . . 11 ⊢ (𝐷 +_ℋ 𝑆) ∈ S_ℋ
34	32, 33	shincli 31165	. . . . . . . . . 10 ⊢ ((𝐶 +_ℋ 𝑅) ∩ (𝐷 +_ℋ 𝑆)) ∈ S_ℋ
35	31, 34	shscli 31120	. . . . . . . . 9 ⊢ (((𝐴 +_ℋ 𝑅) ∩ (𝐵 +_ℋ 𝑆)) +_ℋ ((𝐶 +_ℋ 𝑅) ∩ (𝐷 +_ℋ 𝑆))) ∈ S_ℋ
36	28, 35	shincli 31165	. . . . . . . 8 ⊢ (((𝐴 +_ℋ 𝐶) ∩ (𝐵 +_ℋ 𝐷)) ∩ (((𝐴 +_ℋ 𝑅) ∩ (𝐵 +_ℋ 𝑆)) +_ℋ ((𝐶 +_ℋ 𝑅) ∩ (𝐷 +_ℋ 𝑆)))) ∈ S_ℋ
37	16, 20	shscli 31120	. . . . . . . . . . 11 ⊢ (𝐴 +_ℋ 𝐹) ∈ S_ℋ
38	17, 21	shscli 31120	. . . . . . . . . . 11 ⊢ (𝐵 +_ℋ 𝐺) ∈ S_ℋ
39	37, 38	shincli 31165	. . . . . . . . . 10 ⊢ ((𝐴 +_ℋ 𝐹) ∩ (𝐵 +_ℋ 𝐺)) ∈ S_ℋ
40	20, 22	shscli 31120	. . . . . . . . . . . 12 ⊢ (𝐹 +_ℋ 𝑅) ∈ S_ℋ
41	21, 23	shscli 31120	. . . . . . . . . . . 12 ⊢ (𝐺 +_ℋ 𝑆) ∈ S_ℋ
42	40, 41	shincli 31165	. . . . . . . . . . 11 ⊢ ((𝐹 +_ℋ 𝑅) ∩ (𝐺 +_ℋ 𝑆)) ∈ S_ℋ
43	31, 42	shscli 31120	. . . . . . . . . 10 ⊢ (((𝐴 +_ℋ 𝑅) ∩ (𝐵 +_ℋ 𝑆)) +_ℋ ((𝐹 +_ℋ 𝑅) ∩ (𝐺 +_ℋ 𝑆))) ∈ S_ℋ
44	39, 43	shincli 31165	. . . . . . . . 9 ⊢ (((𝐴 +_ℋ 𝐹) ∩ (𝐵 +_ℋ 𝐺)) ∩ (((𝐴 +_ℋ 𝑅) ∩ (𝐵 +_ℋ 𝑆)) +_ℋ ((𝐹 +_ℋ 𝑅) ∩ (𝐺 +_ℋ 𝑆)))) ∈ S_ℋ
45	18, 20	shscli 31120	. . . . . . . . . . 11 ⊢ (𝐶 +_ℋ 𝐹) ∈ S_ℋ
46	19, 21	shscli 31120	. . . . . . . . . . 11 ⊢ (𝐷 +_ℋ 𝐺) ∈ S_ℋ
47	45, 46	shincli 31165	. . . . . . . . . 10 ⊢ ((𝐶 +_ℋ 𝐹) ∩ (𝐷 +_ℋ 𝐺)) ∈ S_ℋ
48	34, 42	shscli 31120	. . . . . . . . . 10 ⊢ (((𝐶 +_ℋ 𝑅) ∩ (𝐷 +_ℋ 𝑆)) +_ℋ ((𝐹 +_ℋ 𝑅) ∩ (𝐺 +_ℋ 𝑆))) ∈ S_ℋ
49	47, 48	shincli 31165	. . . . . . . . 9 ⊢ (((𝐶 +_ℋ 𝐹) ∩ (𝐷 +_ℋ 𝐺)) ∩ (((𝐶 +_ℋ 𝑅) ∩ (𝐷 +_ℋ 𝑆)) +_ℋ ((𝐹 +_ℋ 𝑅) ∩ (𝐺 +_ℋ 𝑆)))) ∈ S_ℋ
50	44, 49	shscli 31120	. . . . . . . 8 ⊢ ((((𝐴 +_ℋ 𝐹) ∩ (𝐵 +_ℋ 𝐺)) ∩ (((𝐴 +_ℋ 𝑅) ∩ (𝐵 +_ℋ 𝑆)) +_ℋ ((𝐹 +_ℋ 𝑅) ∩ (𝐺 +_ℋ 𝑆)))) +_ℋ (((𝐶 +_ℋ 𝐹) ∩ (𝐷 +_ℋ 𝐺)) ∩ (((𝐶 +_ℋ 𝑅) ∩ (𝐷 +_ℋ 𝑆)) +_ℋ ((𝐹 +_ℋ 𝑅) ∩ (𝐺 +_ℋ 𝑆))))) ∈ S_ℋ
51	36, 50	shincli 31165	. . . . . . 7 ⊢ ((((𝐴 +_ℋ 𝐶) ∩ (𝐵 +_ℋ 𝐷)) ∩ (((𝐴 +_ℋ 𝑅) ∩ (𝐵 +_ℋ 𝑆)) +_ℋ ((𝐶 +_ℋ 𝑅) ∩ (𝐷 +_ℋ 𝑆)))) ∩ ((((𝐴 +_ℋ 𝐹) ∩ (𝐵 +_ℋ 𝐺)) ∩ (((𝐴 +_ℋ 𝑅) ∩ (𝐵 +_ℋ 𝑆)) +_ℋ ((𝐹 +_ℋ 𝑅) ∩ (𝐺 +_ℋ 𝑆)))) +_ℋ (((𝐶 +_ℋ 𝐹) ∩ (𝐷 +_ℋ 𝐺)) ∩ (((𝐶 +_ℋ 𝑅) ∩ (𝐷 +_ℋ 𝑆)) +_ℋ ((𝐹 +_ℋ 𝑅) ∩ (𝐺 +_ℋ 𝑆)))))) ∈ S_ℋ
52	16, 17, 18, 51	5oalem1 31457	. . . . . 6 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ℎ = (𝑥 +_ℎ 𝑦)) ∧ (𝑧 ∈ 𝐶 ∧ (𝑥 −_ℎ 𝑧) ∈ ((((𝐴 +_ℋ 𝐶) ∩ (𝐵 +_ℋ 𝐷)) ∩ (((𝐴 +_ℋ 𝑅) ∩ (𝐵 +_ℋ 𝑆)) +_ℋ ((𝐶 +_ℋ 𝑅) ∩ (𝐷 +_ℋ 𝑆)))) ∩ ((((𝐴 +_ℋ 𝐹) ∩ (𝐵 +_ℋ 𝐺)) ∩ (((𝐴 +_ℋ 𝑅) ∩ (𝐵 +_ℋ 𝑆)) +_ℋ ((𝐹 +_ℋ 𝑅) ∩ (𝐺 +_ℋ 𝑆)))) +_ℋ (((𝐶 +_ℋ 𝐹) ∩ (𝐷 +_ℋ 𝐺)) ∩ (((𝐶 +_ℋ 𝑅) ∩ (𝐷 +_ℋ 𝑆)) +_ℋ ((𝐹 +_ℋ 𝑅) ∩ (𝐺 +_ℋ 𝑆)))))))) → ℎ ∈ (𝐵 +_ℋ (𝐴 ∩ (𝐶 +_ℋ ((((𝐴 +_ℋ 𝐶) ∩ (𝐵 +_ℋ 𝐷)) ∩ (((𝐴 +_ℋ 𝑅) ∩ (𝐵 +_ℋ 𝑆)) +_ℋ ((𝐶 +_ℋ 𝑅) ∩ (𝐷 +_ℋ 𝑆)))) ∩ ((((𝐴 +_ℋ 𝐹) ∩ (𝐵 +_ℋ 𝐺)) ∩ (((𝐴 +_ℋ 𝑅) ∩ (𝐵 +_ℋ 𝑆)) +_ℋ ((𝐹 +_ℋ 𝑅) ∩ (𝐺 +_ℋ 𝑆)))) +_ℋ (((𝐶 +_ℋ 𝐹) ∩ (𝐷 +_ℋ 𝐺)) ∩ (((𝐶 +_ℋ 𝑅) ∩ (𝐷 +_ℋ 𝑆)) +_ℋ ((𝐹 +_ℋ 𝑅) ∩ (𝐺 +_ℋ 𝑆))))))))))
53	52	expr 456	. . . . 5 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ℎ = (𝑥 +_ℎ 𝑦)) ∧ 𝑧 ∈ 𝐶) → ((𝑥 −_ℎ 𝑧) ∈ ((((𝐴 +_ℋ 𝐶) ∩ (𝐵 +_ℋ 𝐷)) ∩ (((𝐴 +_ℋ 𝑅) ∩ (𝐵 +_ℋ 𝑆)) +_ℋ ((𝐶 +_ℋ 𝑅) ∩ (𝐷 +_ℋ 𝑆)))) ∩ ((((𝐴 +_ℋ 𝐹) ∩ (𝐵 +_ℋ 𝐺)) ∩ (((𝐴 +_ℋ 𝑅) ∩ (𝐵 +_ℋ 𝑆)) +_ℋ ((𝐹 +_ℋ 𝑅) ∩ (𝐺 +_ℋ 𝑆)))) +_ℋ (((𝐶 +_ℋ 𝐹) ∩ (𝐷 +_ℋ 𝐺)) ∩ (((𝐶 +_ℋ 𝑅) ∩ (𝐷 +_ℋ 𝑆)) +_ℋ ((𝐹 +_ℋ 𝑅) ∩ (𝐺 +_ℋ 𝑆)))))) → ℎ ∈ (𝐵 +_ℋ (𝐴 ∩ (𝐶 +_ℋ ((((𝐴 +_ℋ 𝐶) ∩ (𝐵 +_ℋ 𝐷)) ∩ (((𝐴 +_ℋ 𝑅) ∩ (𝐵 +_ℋ 𝑆)) +_ℋ ((𝐶 +_ℋ 𝑅) ∩ (𝐷 +_ℋ 𝑆)))) ∩ ((((𝐴 +_ℋ 𝐹) ∩ (𝐵 +_ℋ 𝐺)) ∩ (((𝐴 +_ℋ 𝑅) ∩ (𝐵 +_ℋ 𝑆)) +_ℋ ((𝐹 +_ℋ 𝑅) ∩ (𝐺 +_ℋ 𝑆)))) +_ℋ (((𝐶 +_ℋ 𝐹) ∩ (𝐷 +_ℋ 𝐺)) ∩ (((𝐶 +_ℋ 𝑅) ∩ (𝐷 +_ℋ 𝑆)) +_ℋ ((𝐹 +_ℋ 𝑅) ∩ (𝐺 +_ℋ 𝑆)))))))))))
54	53	adantrr 716	. . . 4 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ℎ = (𝑥 +_ℎ 𝑦)) ∧ (𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷)) → ((𝑥 −_ℎ 𝑧) ∈ ((((𝐴 +_ℋ 𝐶) ∩ (𝐵 +_ℋ 𝐷)) ∩ (((𝐴 +_ℋ 𝑅) ∩ (𝐵 +_ℋ 𝑆)) +_ℋ ((𝐶 +_ℋ 𝑅) ∩ (𝐷 +_ℋ 𝑆)))) ∩ ((((𝐴 +_ℋ 𝐹) ∩ (𝐵 +_ℋ 𝐺)) ∩ (((𝐴 +_ℋ 𝑅) ∩ (𝐵 +_ℋ 𝑆)) +_ℋ ((𝐹 +_ℋ 𝑅) ∩ (𝐺 +_ℋ 𝑆)))) +_ℋ (((𝐶 +_ℋ 𝐹) ∩ (𝐷 +_ℋ 𝐺)) ∩ (((𝐶 +_ℋ 𝑅) ∩ (𝐷 +_ℋ 𝑆)) +_ℋ ((𝐹 +_ℋ 𝑅) ∩ (𝐺 +_ℋ 𝑆)))))) → ℎ ∈ (𝐵 +_ℋ (𝐴 ∩ (𝐶 +_ℋ ((((𝐴 +_ℋ 𝐶) ∩ (𝐵 +_ℋ 𝐷)) ∩ (((𝐴 +_ℋ 𝑅) ∩ (𝐵 +_ℋ 𝑆)) +_ℋ ((𝐶 +_ℋ 𝑅) ∩ (𝐷 +_ℋ 𝑆)))) ∩ ((((𝐴 +_ℋ 𝐹) ∩ (𝐵 +_ℋ 𝐺)) ∩ (((𝐴 +_ℋ 𝑅) ∩ (𝐵 +_ℋ 𝑆)) +_ℋ ((𝐹 +_ℋ 𝑅) ∩ (𝐺 +_ℋ 𝑆)))) +_ℋ (((𝐶 +_ℋ 𝐹) ∩ (𝐷 +_ℋ 𝐺)) ∩ (((𝐶 +_ℋ 𝑅) ∩ (𝐷 +_ℋ 𝑆)) +_ℋ ((𝐹 +_ℋ 𝑅) ∩ (𝐺 +_ℋ 𝑆)))))))))))
55	54	adantrr 716	. . 3 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ℎ = (𝑥 +_ℎ 𝑦)) ∧ ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷) ∧ ℎ = (𝑧 +_ℎ 𝑤))) → ((𝑥 −_ℎ 𝑧) ∈ ((((𝐴 +_ℋ 𝐶) ∩ (𝐵 +_ℋ 𝐷)) ∩ (((𝐴 +_ℋ 𝑅) ∩ (𝐵 +_ℋ 𝑆)) +_ℋ ((𝐶 +_ℋ 𝑅) ∩ (𝐷 +_ℋ 𝑆)))) ∩ ((((𝐴 +_ℋ 𝐹) ∩ (𝐵 +_ℋ 𝐺)) ∩ (((𝐴 +_ℋ 𝑅) ∩ (𝐵 +_ℋ 𝑆)) +_ℋ ((𝐹 +_ℋ 𝑅) ∩ (𝐺 +_ℋ 𝑆)))) +_ℋ (((𝐶 +_ℋ 𝐹) ∩ (𝐷 +_ℋ 𝐺)) ∩ (((𝐶 +_ℋ 𝑅) ∩ (𝐷 +_ℋ 𝑆)) +_ℋ ((𝐹 +_ℋ 𝑅) ∩ (𝐺 +_ℋ 𝑆)))))) → ℎ ∈ (𝐵 +_ℋ (𝐴 ∩ (𝐶 +_ℋ ((((𝐴 +_ℋ 𝐶) ∩ (𝐵 +_ℋ 𝐷)) ∩ (((𝐴 +_ℋ 𝑅) ∩ (𝐵 +_ℋ 𝑆)) +_ℋ ((𝐶 +_ℋ 𝑅) ∩ (𝐷 +_ℋ 𝑆)))) ∩ ((((𝐴 +_ℋ 𝐹) ∩ (𝐵 +_ℋ 𝐺)) ∩ (((𝐴 +_ℋ 𝑅) ∩ (𝐵 +_ℋ 𝑆)) +_ℋ ((𝐹 +_ℋ 𝑅) ∩ (𝐺 +_ℋ 𝑆)))) +_ℋ (((𝐶 +_ℋ 𝐹) ∩ (𝐷 +_ℋ 𝐺)) ∩ (((𝐶 +_ℋ 𝑅) ∩ (𝐷 +_ℋ 𝑆)) +_ℋ ((𝐹 +_ℋ 𝑅) ∩ (𝐺 +_ℋ 𝑆)))))))))))
56	55	adantr 480	. 2 ⊢ (((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ℎ = (𝑥 +_ℎ 𝑦)) ∧ ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷) ∧ ℎ = (𝑧 +_ℎ 𝑤))) ∧ (((𝑓 ∈ 𝐹 ∧ 𝑔 ∈ 𝐺) ∧ ℎ = (𝑓 +_ℎ 𝑔)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑢 ∈ 𝑆) ∧ ℎ = (𝑣 +_ℎ 𝑢)))) → ((𝑥 −_ℎ 𝑧) ∈ ((((𝐴 +_ℋ 𝐶) ∩ (𝐵 +_ℋ 𝐷)) ∩ (((𝐴 +_ℋ 𝑅) ∩ (𝐵 +_ℋ 𝑆)) +_ℋ ((𝐶 +_ℋ 𝑅) ∩ (𝐷 +_ℋ 𝑆)))) ∩ ((((𝐴 +_ℋ 𝐹) ∩ (𝐵 +_ℋ 𝐺)) ∩ (((𝐴 +_ℋ 𝑅) ∩ (𝐵 +_ℋ 𝑆)) +_ℋ ((𝐹 +_ℋ 𝑅) ∩ (𝐺 +_ℋ 𝑆)))) +_ℋ (((𝐶 +_ℋ 𝐹) ∩ (𝐷 +_ℋ 𝐺)) ∩ (((𝐶 +_ℋ 𝑅) ∩ (𝐷 +_ℋ 𝑆)) +_ℋ ((𝐹 +_ℋ 𝑅) ∩ (𝐺 +_ℋ 𝑆)))))) → ℎ ∈ (𝐵 +_ℋ (𝐴 ∩ (𝐶 +_ℋ ((((𝐴 +_ℋ 𝐶) ∩ (𝐵 +_ℋ 𝐷)) ∩ (((𝐴 +_ℋ 𝑅) ∩ (𝐵 +_ℋ 𝑆)) +_ℋ ((𝐶 +_ℋ 𝑅) ∩ (𝐷 +_ℋ 𝑆)))) ∩ ((((𝐴 +_ℋ 𝐹) ∩ (𝐵 +_ℋ 𝐺)) ∩ (((𝐴 +_ℋ 𝑅) ∩ (𝐵 +_ℋ 𝑆)) +_ℋ ((𝐹 +_ℋ 𝑅) ∩ (𝐺 +_ℋ 𝑆)))) +_ℋ (((𝐶 +_ℋ 𝐹) ∩ (𝐷 +_ℋ 𝐺)) ∩ (((𝐶 +_ℋ 𝑅) ∩ (𝐷 +_ℋ 𝑆)) +_ℋ ((𝐹 +_ℋ 𝑅) ∩ (𝐺 +_ℋ 𝑆)))))))))))
57	25, 56	mpd 15	1 ⊢ (((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ℎ = (𝑥 +_ℎ 𝑦)) ∧ ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷) ∧ ℎ = (𝑧 +_ℎ 𝑤))) ∧ (((𝑓 ∈ 𝐹 ∧ 𝑔 ∈ 𝐺) ∧ ℎ = (𝑓 +_ℎ 𝑔)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑢 ∈ 𝑆) ∧ ℎ = (𝑣 +_ℎ 𝑢)))) → ℎ ∈ (𝐵 +_ℋ (𝐴 ∩ (𝐶 +_ℋ ((((𝐴 +_ℋ 𝐶) ∩ (𝐵 +_ℋ 𝐷)) ∩ (((𝐴 +_ℋ 𝑅) ∩ (𝐵 +_ℋ 𝑆)) +_ℋ ((𝐶 +_ℋ 𝑅) ∩ (𝐷 +_ℋ 𝑆)))) ∩ ((((𝐴 +_ℋ 𝐹) ∩ (𝐵 +_ℋ 𝐺)) ∩ (((𝐴 +_ℋ 𝑅) ∩ (𝐵 +_ℋ 𝑆)) +_ℋ ((𝐹 +_ℋ 𝑅) ∩ (𝐺 +_ℋ 𝑆)))) +_ℋ (((𝐶 +_ℋ 𝐹) ∩ (𝐷 +_ℋ 𝐺)) ∩ (((𝐶 +_ℋ 𝑅) ∩ (𝐷 +_ℋ 𝑆)) +_ℋ ((𝐹 +_ℋ 𝑅) ∩ (𝐺 +_ℋ 𝑆))))))))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∩ cin 3944 (class class class)co 7414 +_ℎ cva 30723 −_ℎ cmv 30728 S_ℋ csh 30731 +_ℋ cph 30734

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-hilex 30802 ax-hfvadd 30803 ax-hvcom 30804 ax-hvass 30805 ax-hv0cl 30806 ax-hvaddid 30807 ax-hfvmul 30808 ax-hvmulid 30809 ax-hvmulass 30810 ax-hvdistr1 30811 ax-hvdistr2 30812 ax-hvmul0 30813

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-ltxr 11277 df-sub 11470 df-neg 11471 df-nn 12237 df-grpo 30296 df-ablo 30348 df-hvsub 30774 df-hlim 30775 df-sh 31010 df-ch 31024 df-shs 31111

This theorem is referenced by: 5oalem7 31463

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