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Theorem omlsilem 31205

Description: Lemma for orthomodular law in the Hilbert lattice. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
omlsilem.1	⊢ 𝐺 ∈ S_ℋ
omlsilem.2	⊢ 𝐻 ∈ S_ℋ
omlsilem.3	⊢ 𝐺 ⊆ 𝐻
omlsilem.4	⊢ (𝐻 ∩ (⊥‘𝐺)) = 0_ℋ
omlsilem.5	⊢ 𝐴 ∈ 𝐻
omlsilem.6	⊢ 𝐵 ∈ 𝐺
omlsilem.7	⊢ 𝐶 ∈ (⊥‘𝐺)

**Assertion**
Ref	Expression
omlsilem	⊢ (𝐴 = (𝐵 +_ℎ 𝐶) → 𝐴 ∈ 𝐺)

**Proof of Theorem omlsilem**
Step	Hyp	Ref	Expression
1		omlsilem.2	. . . . . . . . . 10 ⊢ 𝐻 ∈ S_ℋ
2		omlsilem.5	. . . . . . . . . 10 ⊢ 𝐴 ∈ 𝐻
3	1, 2	shelii 31018	. . . . . . . . 9 ⊢ 𝐴 ∈ ℋ
4		omlsilem.1	. . . . . . . . . 10 ⊢ 𝐺 ∈ S_ℋ
5		omlsilem.6	. . . . . . . . . 10 ⊢ 𝐵 ∈ 𝐺
6	4, 5	shelii 31018	. . . . . . . . 9 ⊢ 𝐵 ∈ ℋ
7		shocss 31089	. . . . . . . . . . 11 ⊢ (𝐺 ∈ S_ℋ → (⊥‘𝐺) ⊆ ℋ)
8	4, 7	ax-mp 5	. . . . . . . . . 10 ⊢ (⊥‘𝐺) ⊆ ℋ
9		omlsilem.7	. . . . . . . . . 10 ⊢ 𝐶 ∈ (⊥‘𝐺)
10	8, 9	sselii 3975	. . . . . . . . 9 ⊢ 𝐶 ∈ ℋ
11	3, 6, 10	hvsubaddi 30869	. . . . . . . 8 ⊢ ((𝐴 −_ℎ 𝐵) = 𝐶 ↔ (𝐵 +_ℎ 𝐶) = 𝐴)
12		eqcom 2735	. . . . . . . 8 ⊢ ((𝐵 +_ℎ 𝐶) = 𝐴 ↔ 𝐴 = (𝐵 +_ℎ 𝐶))
13	11, 12	bitri 275	. . . . . . 7 ⊢ ((𝐴 −_ℎ 𝐵) = 𝐶 ↔ 𝐴 = (𝐵 +_ℎ 𝐶))
14		omlsilem.3	. . . . . . . . . 10 ⊢ 𝐺 ⊆ 𝐻
15	14, 5	sselii 3975	. . . . . . . . 9 ⊢ 𝐵 ∈ 𝐻
16		shsubcl 31023	. . . . . . . . 9 ⊢ ((𝐻 ∈ S_ℋ ∧ 𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) → (𝐴 −_ℎ 𝐵) ∈ 𝐻)
17	1, 2, 15, 16	mp3an 1458	. . . . . . . 8 ⊢ (𝐴 −_ℎ 𝐵) ∈ 𝐻
18		eleq1 2817	. . . . . . . 8 ⊢ ((𝐴 −_ℎ 𝐵) = 𝐶 → ((𝐴 −_ℎ 𝐵) ∈ 𝐻 ↔ 𝐶 ∈ 𝐻))
19	17, 18	mpbii 232	. . . . . . 7 ⊢ ((𝐴 −_ℎ 𝐵) = 𝐶 → 𝐶 ∈ 𝐻)
20	13, 19	sylbir 234	. . . . . 6 ⊢ (𝐴 = (𝐵 +_ℎ 𝐶) → 𝐶 ∈ 𝐻)
21		omlsilem.4	. . . . . . . 8 ⊢ (𝐻 ∩ (⊥‘𝐺)) = 0_ℋ
22	21	eleq2i 2821	. . . . . . 7 ⊢ (𝐶 ∈ (𝐻 ∩ (⊥‘𝐺)) ↔ 𝐶 ∈ 0_ℋ)
23		elin 3961	. . . . . . 7 ⊢ (𝐶 ∈ (𝐻 ∩ (⊥‘𝐺)) ↔ (𝐶 ∈ 𝐻 ∧ 𝐶 ∈ (⊥‘𝐺)))
24		elch0 31057	. . . . . . 7 ⊢ (𝐶 ∈ 0_ℋ ↔ 𝐶 = 0_ℎ)
25	22, 23, 24	3bitr3i 301	. . . . . 6 ⊢ ((𝐶 ∈ 𝐻 ∧ 𝐶 ∈ (⊥‘𝐺)) ↔ 𝐶 = 0_ℎ)
26	20, 9, 25	sylanblc 588	. . . . 5 ⊢ (𝐴 = (𝐵 +_ℎ 𝐶) → 𝐶 = 0_ℎ)
27	26	oveq2d 7430	. . . 4 ⊢ (𝐴 = (𝐵 +_ℎ 𝐶) → (𝐵 +_ℎ 𝐶) = (𝐵 +_ℎ 0_ℎ))
28		ax-hvaddid 30807	. . . . 5 ⊢ (𝐵 ∈ ℋ → (𝐵 +_ℎ 0_ℎ) = 𝐵)
29	6, 28	ax-mp 5	. . . 4 ⊢ (𝐵 +_ℎ 0_ℎ) = 𝐵
30	27, 29	eqtrdi 2784	. . 3 ⊢ (𝐴 = (𝐵 +_ℎ 𝐶) → (𝐵 +_ℎ 𝐶) = 𝐵)
31	30, 5	eqeltrdi 2837	. 2 ⊢ (𝐴 = (𝐵 +_ℎ 𝐶) → (𝐵 +_ℎ 𝐶) ∈ 𝐺)
32		eleq1 2817	. 2 ⊢ (𝐴 = (𝐵 +_ℎ 𝐶) → (𝐴 ∈ 𝐺 ↔ (𝐵 +_ℎ 𝐶) ∈ 𝐺))
33	31, 32	mpbird 257	1 ⊢ (𝐴 = (𝐵 +_ℎ 𝐶) → 𝐴 ∈ 𝐺)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∩ cin 3944 ⊆ wss 3945 ‘cfv 6542 (class class class)co 7414 ℋchba 30722 +_ℎ cva 30723 0_ℎc0v 30727 −_ℎ cmv 30728 S_ℋ csh 30731 ⊥cort 30733 0_ℋc0h 30738

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-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-hvdistr2 30812 ax-hvmul0 30813 ax-hfi 30882 ax-his2 30886 ax-his3 30887

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-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-hvsub 30774 df-sh 31010 df-oc 31055 df-ch0 31056

This theorem is referenced by: omlsii 31206

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