Theorem omlsi 28263

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 e. CH
omls.2	|- B e. SH

Assertion

Ref	Expression
omlsi	|- ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) -> A = B )

Proof of Theorem omlsi

Step	Hyp	Ref	Expression
1		eqeq1 2626	. 2 |- ( A = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) -> ( A = B <-> if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) = B ) )
2		eqeq2 2633	. 2 |- ( B = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) -> ( if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) = B <-> if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) ) )
3		omls.1	. . . 4 |- A e. CH
4		h0elch 28112	. . . 4 |- 0H e. CH
5	3, 4	keepel 4155	. . 3 |- if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) e. CH
6		omls.2	. . . 4 |- B e. SH
7		h0elsh 28113	. . . 4 |- 0H e. SH
8	6, 7	keepel 4155	. . 3 |- if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) e. SH
9		sseq1 3626	. . . . . 6 |- ( A = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) -> ( A C_ B <-> if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) C_ B ) )
10		fveq2 6191	. . . . . . . 8 |- ( A = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) -> ( _|_ ` A ) = ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) )
11	10	ineq2d 3814	. . . . . . 7 |- ( A = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) -> ( B i^i ( _|_ ` A ) ) = ( B i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) )
12	11	eqeq1d 2624	. . . . . 6 |- ( A = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) -> ( ( B i^i ( _|_ ` A ) ) = 0H <-> ( B i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) = 0H ) )
13	9, 12	anbi12d 747	. . . . 5 |- ( A = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) -> ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) <-> ( if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) C_ B /\ ( B i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) = 0H ) ) )
14		sseq2 3627	. . . . . 6 |- ( B = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) -> ( if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) C_ B <-> if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) C_ if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) ) )
15		ineq1 3807	. . . . . . 7 |- ( B = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) -> ( B i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) = ( if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) )
16	15	eqeq1d 2624	. . . . . 6 |- ( B = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) -> ( ( B i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) = 0H <-> ( if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) = 0H ) )
17	14, 16	anbi12d 747	. . . . 5 |- ( B = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) -> ( ( if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) C_ B /\ ( B i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) = 0H ) <-> ( if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) C_ if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) /\ ( if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) = 0H ) ) )
18		sseq1 3626	. . . . . 6 |- ( 0H = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) -> ( 0H C_ 0H <-> if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) C_ 0H ) )
19		fveq2 6191	. . . . . . . 8 |- ( 0H = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) -> ( _|_ ` 0H ) = ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) )
20	19	ineq2d 3814	. . . . . . 7 |- ( 0H = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) -> ( 0H i^i ( _|_ ` 0H ) ) = ( 0H i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) )
21	20	eqeq1d 2624	. . . . . 6 |- ( 0H = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) -> ( ( 0H i^i ( _|_ ` 0H ) ) = 0H <-> ( 0H i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) = 0H ) )
22	18, 21	anbi12d 747	. . . . 5 |- ( 0H = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) -> ( ( 0H C_ 0H /\ ( 0H i^i ( _|_ ` 0H ) ) = 0H ) <-> ( if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) C_ 0H /\ ( 0H i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) = 0H ) ) )
23		sseq2 3627	. . . . . 6 |- ( 0H = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) -> ( if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) C_ 0H <-> if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) C_ if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) ) )
24		ineq1 3807	. . . . . . 7 |- ( 0H = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) -> ( 0H i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) = ( if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) )
25	24	eqeq1d 2624	. . . . . 6 |- ( 0H = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) -> ( ( 0H i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) = 0H <-> ( if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) = 0H ) )
26	23, 25	anbi12d 747	. . . . 5 |- ( 0H = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) -> ( ( if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) C_ 0H /\ ( 0H i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) = 0H ) <-> ( if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) C_ if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) /\ ( if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) = 0H ) ) )
27		ssid 3624	. . . . . 6 |- 0H C_ 0H
28		ocin 28155	. . . . . . 7 |- ( 0H e. SH -> ( 0H i^i ( _|_ ` 0H ) ) = 0H )
29	7, 28	ax-mp 5	. . . . . 6 |- ( 0H i^i ( _|_ ` 0H ) ) = 0H
30	27, 29	pm3.2i 471	. . . . 5 |- ( 0H C_ 0H /\ ( 0H i^i ( _|_ ` 0H ) ) = 0H )
31	13, 17, 22, 26, 30	elimhyp2v 4147	. . . 4 |- ( if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) C_ if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) /\ ( if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) = 0H )
32	31	simpli 474	. . 3 |- if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) C_ if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H )
33	31	simpri 478	. . 3 |- ( if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) = 0H
34	5, 8, 32, 33	omlsii 28262	. 2 |- if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H )
35	1, 2, 34	dedth2v 4143	1 |- ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) -> A = B )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   ifcif 4086   ` cfv 5888   SHcsh 27785   CHcch 27786   _|_cort 27787   0Hc0h 27792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cc 9257  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016  ax-hilex 27856  ax-hfvadd 27857  ax-hvcom 27858  ax-hvass 27859  ax-hv0cl 27860  ax-hvaddid 27861  ax-hfvmul 27862  ax-hvmulid 27863  ax-hvmulass 27864  ax-hvdistr1 27865  ax-hvdistr2 27866  ax-hvmul0 27867  ax-hfi 27936  ax-his1 27939  ax-his2 27940  ax-his3 27941  ax-his4 27942  ax-hcompl 28059

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ico 12181  df-icc 12182  df-fz 12327  df-fl 12593  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-rest 16083  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-top 20699  df-topon 20716  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lm 21033  df-haus 21119  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-cfil 23053  df-cau 23054  df-cmet 23055  df-grpo 27347  df-gid 27348  df-ginv 27349  df-gdiv 27350  df-ablo 27399  df-vc 27414  df-nv 27447  df-va 27450  df-ba 27451  df-sm 27452  df-0v 27453  df-vs 27454  df-nmcv 27455  df-ims 27456  df-ssp 27577  df-ph 27668  df-cbn 27719  df-hnorm 27825  df-hba 27826  df-hvsub 27828  df-hlim 27829  df-hcau 27830  df-sh 28064  df-ch 28078  df-oc 28109  df-ch0 28110

This theorem is referenced by:  pjomli  28294