Theorem omlsi 29179

Description: Subspace form of orthomodular law in the Hilbert lattice. Compare the orthomodular law in Theorem 2(ii) of [Kalmbach] p. 22. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)

Hypotheses

Ref	Expression
omls.1	⊢ 𝐴 ∈ Cℋ
omls.2	⊢ 𝐵 ∈ Sℋ

Assertion

Ref	Expression
omlsi	⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ) → 𝐴 = 𝐵)

Proof of Theorem omlsi

Step	Hyp	Ref	Expression
1		eqeq1 2824	. 2 ⊢ (𝐴 = if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ) → (𝐴 = 𝐵 ↔ if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ) = 𝐵))
2		eqeq2 2832	. 2 ⊢ (𝐵 = if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐵, 0ℋ) → (if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ) = 𝐵 ↔ if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ) = if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐵, 0ℋ)))
3		omls.1	. . . 4 ⊢ 𝐴 ∈ Cℋ
4		h0elch 29030	. . . 4 ⊢ 0ℋ ∈ Cℋ
5	3, 4	ifcli 4506	. . 3 ⊢ if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ) ∈ Cℋ
6		omls.2	. . . 4 ⊢ 𝐵 ∈ Sℋ
7		h0elsh 29031	. . . 4 ⊢ 0ℋ ∈ Sℋ
8	6, 7	ifcli 4506	. . 3 ⊢ if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐵, 0ℋ) ∈ Sℋ
9		sseq1 3985	. . . . . 6 ⊢ (𝐴 = if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ) → (𝐴 ⊆ 𝐵 ↔ if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ) ⊆ 𝐵))
10		fveq2 6663	. . . . . . . 8 ⊢ (𝐴 = if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ) → (⊥‘𝐴) = (⊥‘if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ)))
11	10	ineq2d 4182	. . . . . . 7 ⊢ (𝐴 = if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ) → (𝐵 ∩ (⊥‘𝐴)) = (𝐵 ∩ (⊥‘if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ))))
12	11	eqeq1d 2822	. . . . . 6 ⊢ (𝐴 = if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ) → ((𝐵 ∩ (⊥‘𝐴)) = 0ℋ ↔ (𝐵 ∩ (⊥‘if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ))) = 0ℋ))
13	9, 12	anbi12d 632	. . . . 5 ⊢ (𝐴 = if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ) → ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ) ↔ (if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ) ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ))) = 0ℋ)))
14		sseq2 3986	. . . . . 6 ⊢ (𝐵 = if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐵, 0ℋ) → (if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ) ⊆ 𝐵 ↔ if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ) ⊆ if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐵, 0ℋ)))
15		ineq1 4174	. . . . . . 7 ⊢ (𝐵 = if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐵, 0ℋ) → (𝐵 ∩ (⊥‘if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ))) = (if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐵, 0ℋ) ∩ (⊥‘if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ))))
16	15	eqeq1d 2822	. . . . . 6 ⊢ (𝐵 = if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐵, 0ℋ) → ((𝐵 ∩ (⊥‘if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ))) = 0ℋ ↔ (if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐵, 0ℋ) ∩ (⊥‘if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ))) = 0ℋ))
17	14, 16	anbi12d 632	. . . . 5 ⊢ (𝐵 = if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐵, 0ℋ) → ((if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ) ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ))) = 0ℋ) ↔ (if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ) ⊆ if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐵, 0ℋ) ∧ (if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐵, 0ℋ) ∩ (⊥‘if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ))) = 0ℋ)))
18		sseq1 3985	. . . . . 6 ⊢ (0ℋ = if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ) → (0ℋ ⊆ 0ℋ ↔ if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ) ⊆ 0ℋ))
19		fveq2 6663	. . . . . . . 8 ⊢ (0ℋ = if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ) → (⊥‘0ℋ) = (⊥‘if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ)))
20	19	ineq2d 4182	. . . . . . 7 ⊢ (0ℋ = if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ) → (0ℋ ∩ (⊥‘0ℋ)) = (0ℋ ∩ (⊥‘if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ))))
21	20	eqeq1d 2822	. . . . . 6 ⊢ (0ℋ = if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ) → ((0ℋ ∩ (⊥‘0ℋ)) = 0ℋ ↔ (0ℋ ∩ (⊥‘if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ))) = 0ℋ))
22	18, 21	anbi12d 632	. . . . 5 ⊢ (0ℋ = if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ) → ((0ℋ ⊆ 0ℋ ∧ (0ℋ ∩ (⊥‘0ℋ)) = 0ℋ) ↔ (if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ) ⊆ 0ℋ ∧ (0ℋ ∩ (⊥‘if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ))) = 0ℋ)))
23		sseq2 3986	. . . . . 6 ⊢ (0ℋ = if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐵, 0ℋ) → (if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ) ⊆ 0ℋ ↔ if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ) ⊆ if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐵, 0ℋ)))
24		ineq1 4174	. . . . . . 7 ⊢ (0ℋ = if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐵, 0ℋ) → (0ℋ ∩ (⊥‘if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ))) = (if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐵, 0ℋ) ∩ (⊥‘if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ))))
25	24	eqeq1d 2822	. . . . . 6 ⊢ (0ℋ = if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐵, 0ℋ) → ((0ℋ ∩ (⊥‘if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ))) = 0ℋ ↔ (if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐵, 0ℋ) ∩ (⊥‘if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ))) = 0ℋ))
26	23, 25	anbi12d 632	. . . . 5 ⊢ (0ℋ = if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐵, 0ℋ) → ((if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ) ⊆ 0ℋ ∧ (0ℋ ∩ (⊥‘if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ))) = 0ℋ) ↔ (if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ) ⊆ if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐵, 0ℋ) ∧ (if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐵, 0ℋ) ∩ (⊥‘if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ))) = 0ℋ)))
27		ssid 3982	. . . . . 6 ⊢ 0ℋ ⊆ 0ℋ
28		ocin 29071	. . . . . . 7 ⊢ (0ℋ ∈ Sℋ → (0ℋ ∩ (⊥‘0ℋ)) = 0ℋ)
29	7, 28	ax-mp 5	. . . . . 6 ⊢ (0ℋ ∩ (⊥‘0ℋ)) = 0ℋ
30	27, 29	pm3.2i 473	. . . . 5 ⊢ (0ℋ ⊆ 0ℋ ∧ (0ℋ ∩ (⊥‘0ℋ)) = 0ℋ)
31	13, 17, 22, 26, 30	elimhyp2v 4524	. . . 4 ⊢ (if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ) ⊆ if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐵, 0ℋ) ∧ (if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐵, 0ℋ) ∩ (⊥‘if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ))) = 0ℋ)
32	31	simpli 486	. . 3 ⊢ if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ) ⊆ if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐵, 0ℋ)
33	31	simpri 488	. . 3 ⊢ (if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐵, 0ℋ) ∩ (⊥‘if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ))) = 0ℋ
34	5, 8, 32, 33	omlsii 29178	. 2 ⊢ if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐴, 0ℋ) = if((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ), 𝐵, 0ℋ)
35	1, 2, 34	dedth2v 4520	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ) → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1536   ∈ wcel 2113   ∩ cin 3928   ⊆ wss 3929  ifcif 4460  ‘cfv 6348   S_ℋ csh 28703   C_ℋ cch 28704  ⊥cort 28705  0_ℋc0h 28710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5323  ax-un 7454  ax-inf2 9097  ax-cc 9850  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608  ax-addf 10609  ax-mulf 10610  ax-hilex 28774  ax-hfvadd 28775  ax-hvcom 28776  ax-hvass 28777  ax-hv0cl 28778  ax-hvaddid 28779  ax-hfvmul 28780  ax-hvmulid 28781  ax-hvmulass 28782  ax-hvdistr1 28783  ax-hvdistr2 28784  ax-hvmul0 28785  ax-hfi 28854  ax-his1 28857  ax-his2 28858  ax-his3 28859  ax-his4 28860  ax-hcompl 28977

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-nel 3123  df-ral 3142  df-rex 3143  df-reu 3144  df-rmo 3145  df-rab 3146  df-v 3493  df-sbc 3769  df-csb 3877  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-pss 3947  df-nul 4285  df-if 4461  df-pw 4534  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4870  df-iun 4914  df-iin 4915  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7574  df-1st 7682  df-2nd 7683  df-wrecs 7940  df-recs 8001  df-rdg 8039  df-1o 8095  df-oadd 8099  df-omul 8100  df-er 8282  df-map 8401  df-pm 8402  df-en 8503  df-dom 8504  df-sdom 8505  df-fin 8506  df-fi 8868  df-sup 8899  df-inf 8900  df-oi 8967  df-card 9361  df-acn 9364  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11632  df-2 11694  df-3 11695  df-4 11696  df-n0 11892  df-z 11976  df-uz 12238  df-q 12343  df-rp 12384  df-xneg 12501  df-xadd 12502  df-xmul 12503  df-ico 12738  df-icc 12739  df-fz 12890  df-fl 13159  df-seq 13367  df-exp 13427  df-cj 14453  df-re 14454  df-im 14455  df-sqrt 14589  df-abs 14590  df-clim 14840  df-rlim 14841  df-rest 16691  df-topgen 16712  df-psmet 20532  df-xmet 20533  df-met 20534  df-bl 20535  df-mopn 20536  df-fbas 20537  df-fg 20538  df-top 21497  df-topon 21514  df-bases 21549  df-cld 21622  df-ntr 21623  df-cls 21624  df-nei 21701  df-lm 21832  df-haus 21918  df-fil 22449  df-fm 22541  df-flim 22542  df-flf 22543  df-cfil 23853  df-cau 23854  df-cmet 23855  df-grpo 28268  df-gid 28269  df-ginv 28270  df-gdiv 28271  df-ablo 28320  df-vc 28334  df-nv 28367  df-va 28370  df-ba 28371  df-sm 28372  df-0v 28373  df-vs 28374  df-nmcv 28375  df-ims 28376  df-ssp 28497  df-ph 28588  df-cbn 28638  df-hnorm 28743  df-hba 28744  df-hvsub 28746  df-hlim 28747  df-hcau 28748  df-sh 28982  df-ch 28996  df-oc 29027  df-ch0 29028

This theorem is referenced by:  pjomli  29210