Theorem omlsilem 28261

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 e. SH
omlsilem.2	|- H e. SH
omlsilem.3	|- G C_ H
omlsilem.4	|- ( H i^i ( _|_ ` G ) ) = 0H
omlsilem.5	|- A e. H
omlsilem.6	|- B e. G
omlsilem.7	|- C e. ( _|_ ` G )

Assertion

Ref	Expression
omlsilem	|- ( A = ( B +h C ) -> A e. G )

Proof of Theorem omlsilem

Step	Hyp	Ref	Expression
1		omlsilem.2	. . . . . . . . . 10 |- H e. SH
2		omlsilem.5	. . . . . . . . . 10 |- A e. H
3	1, 2	shelii 28072	. . . . . . . . 9 |- A e. ~H
4		omlsilem.1	. . . . . . . . . 10 |- G e. SH
5		omlsilem.6	. . . . . . . . . 10 |- B e. G
6	4, 5	shelii 28072	. . . . . . . . 9 |- B e. ~H
7		shocss 28145	. . . . . . . . . . 11 |- ( G e. SH -> ( _|_ ` G ) C_ ~H )
8	4, 7	ax-mp 5	. . . . . . . . . 10 |- ( _|_ ` G ) C_ ~H
9		omlsilem.7	. . . . . . . . . 10 |- C e. ( _|_ ` G )
10	8, 9	sselii 3600	. . . . . . . . 9 |- C e. ~H
11	3, 6, 10	hvsubaddi 27923	. . . . . . . 8 |- ( ( A -h B ) = C <-> ( B +h C ) = A )
12		eqcom 2629	. . . . . . . 8 |- ( ( B +h C ) = A <-> A = ( B +h C ) )
13	11, 12	bitri 264	. . . . . . 7 |- ( ( A -h B ) = C <-> A = ( B +h C ) )
14		omlsilem.3	. . . . . . . . . 10 |- G C_ H
15	14, 5	sselii 3600	. . . . . . . . 9 |- B e. H
16		shsubcl 28077	. . . . . . . . 9 |- ( ( H e. SH /\ A e. H /\ B e. H ) -> ( A -h B ) e. H )
17	1, 2, 15, 16	mp3an 1424	. . . . . . . 8 |- ( A -h B ) e. H
18		eleq1 2689	. . . . . . . 8 |- ( ( A -h B ) = C -> ( ( A -h B ) e. H <-> C e. H ) )
19	17, 18	mpbii 223	. . . . . . 7 |- ( ( A -h B ) = C -> C e. H )
20	13, 19	sylbir 225	. . . . . 6 |- ( A = ( B +h C ) -> C e. H )
21		omlsilem.4	. . . . . . . 8 |- ( H i^i ( _|_ ` G ) ) = 0H
22	21	eleq2i 2693	. . . . . . 7 |- ( C e. ( H i^i ( _|_ ` G ) ) <-> C e. 0H )
23		elin 3796	. . . . . . 7 |- ( C e. ( H i^i ( _|_ ` G ) ) <-> ( C e. H /\ C e. ( _|_ ` G ) ) )
24		elch0 28111	. . . . . . 7 |- ( C e. 0H <-> C = 0h )
25	22, 23, 24	3bitr3i 290	. . . . . 6 |- ( ( C e. H /\ C e. ( _|_ ` G ) ) <-> C = 0h )
26	20, 9, 25	sylanblc 696	. . . . 5 |- ( A = ( B +h C ) -> C = 0h )
27	26	oveq2d 6666	. . . 4 |- ( A = ( B +h C ) -> ( B +h C ) = ( B +h 0h ) )
28		ax-hvaddid 27861	. . . . 5 |- ( B e. ~H -> ( B +h 0h ) = B )
29	6, 28	ax-mp 5	. . . 4 |- ( B +h 0h ) = B
30	27, 29	syl6eq 2672	. . 3 |- ( A = ( B +h C ) -> ( B +h C ) = B )
31	30, 5	syl6eqel 2709	. 2 |- ( A = ( B +h C ) -> ( B +h C ) e. G )
32		eleq1 2689	. 2 |- ( A = ( B +h C ) -> ( A e. G <-> ( B +h C ) e. G ) )
33	31, 32	mpbird 247	1 |- ( A = ( B +h C ) -> A e. G )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   ` cfv 5888  (class class class)co 6650   ~Hchil 27776   +h cva 27777   0hc0v 27781   -h cmv 27782   SHcsh 27785   _|_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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  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-hilex 27856  ax-hfvadd 27857  ax-hvcom 27858  ax-hvass 27859  ax-hv0cl 27860  ax-hvaddid 27861  ax-hfvmul 27862  ax-hvmulid 27863  ax-hvdistr2 27866  ax-hvmul0 27867  ax-hfi 27936  ax-his2 27940  ax-his3 27941

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-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-ltxr 10079  df-sub 10268  df-neg 10269  df-hvsub 27828  df-sh 28064  df-oc 28109  df-ch0 28110

This theorem is referenced by:  omlsii  28262