5oalem1 - Hilbert Space Explorer

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Theorem 5oalem1 31457

Description: Lemma for orthoarguesian law 5OA. (Contributed by NM, 1-Apr-2000.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
5oalem1.1	⊢ 𝐴 ∈ S_ℋ
5oalem1.2	⊢ 𝐵 ∈ S_ℋ
5oalem1.3	⊢ 𝐶 ∈ S_ℋ
5oalem1.4	⊢ 𝑅 ∈ S_ℋ

**Assertion**
Ref	Expression
5oalem1	⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = (𝑥 +_ℎ 𝑦)) ∧ (𝑧 ∈ 𝐶 ∧ (𝑥 −_ℎ 𝑧) ∈ 𝑅)) → 𝑣 ∈ (𝐵 +_ℋ (𝐴 ∩ (𝐶 +_ℋ 𝑅))))

**Proof of Theorem 5oalem1**
Step	Hyp	Ref	Expression
1		simplll 774	. . . 4 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = (𝑥 +_ℎ 𝑦)) ∧ (𝑧 ∈ 𝐶 ∧ (𝑥 −_ℎ 𝑧) ∈ 𝑅)) → 𝑥 ∈ 𝐴)
2		5oalem1.1	. . . . . . . 8 ⊢ 𝐴 ∈ S_ℋ
3	2	sheli 31017	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℋ)
4	3	ad2antrr 725	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = (𝑥 +_ℎ 𝑦)) → 𝑥 ∈ ℋ)
5		5oalem1.3	. . . . . . . 8 ⊢ 𝐶 ∈ S_ℋ
6	5	sheli 31017	. . . . . . 7 ⊢ (𝑧 ∈ 𝐶 → 𝑧 ∈ ℋ)
7	6	adantr 480	. . . . . 6 ⊢ ((𝑧 ∈ 𝐶 ∧ (𝑥 −_ℎ 𝑧) ∈ 𝑅) → 𝑧 ∈ ℋ)
8		hvaddsub12 30841	. . . . . . . 8 ⊢ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑥 +_ℎ (𝑧 −_ℎ 𝑧)) = (𝑧 +_ℎ (𝑥 −_ℎ 𝑧)))
9	8	3anidm23 1419	. . . . . . 7 ⊢ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑥 +_ℎ (𝑧 −_ℎ 𝑧)) = (𝑧 +_ℎ (𝑥 −_ℎ 𝑧)))
10		hvsubid 30829	. . . . . . . . 9 ⊢ (𝑧 ∈ ℋ → (𝑧 −_ℎ 𝑧) = 0_ℎ)
11	10	oveq2d 7430	. . . . . . . 8 ⊢ (𝑧 ∈ ℋ → (𝑥 +_ℎ (𝑧 −_ℎ 𝑧)) = (𝑥 +_ℎ 0_ℎ))
12		ax-hvaddid 30807	. . . . . . . 8 ⊢ (𝑥 ∈ ℋ → (𝑥 +_ℎ 0_ℎ) = 𝑥)
13	11, 12	sylan9eqr 2790	. . . . . . 7 ⊢ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑥 +_ℎ (𝑧 −_ℎ 𝑧)) = 𝑥)
14	9, 13	eqtr3d 2770	. . . . . 6 ⊢ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑧 +_ℎ (𝑥 −_ℎ 𝑧)) = 𝑥)
15	4, 7, 14	syl2an 595	. . . . 5 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = (𝑥 +_ℎ 𝑦)) ∧ (𝑧 ∈ 𝐶 ∧ (𝑥 −_ℎ 𝑧) ∈ 𝑅)) → (𝑧 +_ℎ (𝑥 −_ℎ 𝑧)) = 𝑥)
16		5oalem1.4	. . . . . . 7 ⊢ 𝑅 ∈ S_ℋ
17	5, 16	shsvai 31167	. . . . . 6 ⊢ ((𝑧 ∈ 𝐶 ∧ (𝑥 −_ℎ 𝑧) ∈ 𝑅) → (𝑧 +_ℎ (𝑥 −_ℎ 𝑧)) ∈ (𝐶 +_ℋ 𝑅))
18	17	adantl 481	. . . . 5 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = (𝑥 +_ℎ 𝑦)) ∧ (𝑧 ∈ 𝐶 ∧ (𝑥 −_ℎ 𝑧) ∈ 𝑅)) → (𝑧 +_ℎ (𝑥 −_ℎ 𝑧)) ∈ (𝐶 +_ℋ 𝑅))
19	15, 18	eqeltrrd 2830	. . . 4 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = (𝑥 +_ℎ 𝑦)) ∧ (𝑧 ∈ 𝐶 ∧ (𝑥 −_ℎ 𝑧) ∈ 𝑅)) → 𝑥 ∈ (𝐶 +_ℋ 𝑅))
20	1, 19	elind 4190	. . 3 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = (𝑥 +_ℎ 𝑦)) ∧ (𝑧 ∈ 𝐶 ∧ (𝑥 −_ℎ 𝑧) ∈ 𝑅)) → 𝑥 ∈ (𝐴 ∩ (𝐶 +_ℋ 𝑅)))
21		simpllr 775	. . 3 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = (𝑥 +_ℎ 𝑦)) ∧ (𝑧 ∈ 𝐶 ∧ (𝑥 −_ℎ 𝑧) ∈ 𝑅)) → 𝑦 ∈ 𝐵)
22	5, 16	shscli 31120	. . . . . 6 ⊢ (𝐶 +_ℋ 𝑅) ∈ S_ℋ
23	2, 22	shincli 31165	. . . . 5 ⊢ (𝐴 ∩ (𝐶 +_ℋ 𝑅)) ∈ S_ℋ
24		5oalem1.2	. . . . 5 ⊢ 𝐵 ∈ S_ℋ
25	23, 24	shsvai 31167	. . . 4 ⊢ ((𝑥 ∈ (𝐴 ∩ (𝐶 +_ℋ 𝑅)) ∧ 𝑦 ∈ 𝐵) → (𝑥 +_ℎ 𝑦) ∈ ((𝐴 ∩ (𝐶 +_ℋ 𝑅)) +_ℋ 𝐵))
26	23, 24	shscomi 31166	. . . 4 ⊢ ((𝐴 ∩ (𝐶 +_ℋ 𝑅)) +_ℋ 𝐵) = (𝐵 +_ℋ (𝐴 ∩ (𝐶 +_ℋ 𝑅)))
27	25, 26	eleqtrdi 2839	. . 3 ⊢ ((𝑥 ∈ (𝐴 ∩ (𝐶 +_ℋ 𝑅)) ∧ 𝑦 ∈ 𝐵) → (𝑥 +_ℎ 𝑦) ∈ (𝐵 +_ℋ (𝐴 ∩ (𝐶 +_ℋ 𝑅))))
28	20, 21, 27	syl2anc 583	. 2 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = (𝑥 +_ℎ 𝑦)) ∧ (𝑧 ∈ 𝐶 ∧ (𝑥 −_ℎ 𝑧) ∈ 𝑅)) → (𝑥 +_ℎ 𝑦) ∈ (𝐵 +_ℋ (𝐴 ∩ (𝐶 +_ℋ 𝑅))))
29		eleq1 2817	. . 3 ⊢ (𝑣 = (𝑥 +_ℎ 𝑦) → (𝑣 ∈ (𝐵 +_ℋ (𝐴 ∩ (𝐶 +_ℋ 𝑅))) ↔ (𝑥 +_ℎ 𝑦) ∈ (𝐵 +_ℋ (𝐴 ∩ (𝐶 +_ℋ 𝑅)))))
30	29	ad2antlr 726	. 2 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = (𝑥 +_ℎ 𝑦)) ∧ (𝑧 ∈ 𝐶 ∧ (𝑥 −_ℎ 𝑧) ∈ 𝑅)) → (𝑣 ∈ (𝐵 +_ℋ (𝐴 ∩ (𝐶 +_ℋ 𝑅))) ↔ (𝑥 +_ℎ 𝑦) ∈ (𝐵 +_ℋ (𝐴 ∩ (𝐶 +_ℋ 𝑅)))))
31	28, 30	mpbird 257	1 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = (𝑥 +_ℎ 𝑦)) ∧ (𝑧 ∈ 𝐶 ∧ (𝑥 −_ℎ 𝑧) ∈ 𝑅)) → 𝑣 ∈ (𝐵 +_ℋ (𝐴 ∩ (𝐶 +_ℋ 𝑅))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∩ cin 3944 (class class class)co 7414 ℋchba 30722 +_ℎ cva 30723 0_ℎc0v 30727 −_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-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-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-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-id 5570 df-po 5584 df-so 5585 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-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-er 8718 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-grpo 30296 df-ablo 30348 df-hvsub 30774 df-sh 31010 df-shs 31111

This theorem is referenced by: 5oalem6 31462

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