Theorem omllaw5N 34534

Description: The orthomodular law. Remark in [Kalmbach] p. 22. (pjoml5 28472 analog.) (Contributed by NM, 14-Nov-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
omllaw5.b	|- B = ( Base ` K )
omllaw5.j	|- .\/ = ( join ` K )
omllaw5.m	|- ./\ = ( meet ` K )
omllaw5.o	|- ._|_ = ( oc ` K )

Assertion

Ref	Expression
omllaw5N	|- ( ( K e. OML /\ X e. B /\ Y e. B ) -> ( X .\/ ( ( ._|_ ` X ) ./\ ( X .\/ Y ) ) ) = ( X .\/ Y ) )

Proof of Theorem omllaw5N

Step	Hyp	Ref	Expression
1		simp1 1061	. . 3 |- ( ( K e. OML /\ X e. B /\ Y e. B ) -> K e. OML )
2		simp2 1062	. . 3 |- ( ( K e. OML /\ X e. B /\ Y e. B ) -> X e. B )
3		omllat 34529	. . . 4 |- ( K e. OML -> K e. Lat )
4		omllaw5.b	. . . . 5 |- B = ( Base ` K )
5		omllaw5.j	. . . . 5 |- .\/ = ( join ` K )
6	4, 5	latjcl 17051	. . . 4 |- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X .\/ Y ) e. B )
7	3, 6	syl3an1 1359	. . 3 |- ( ( K e. OML /\ X e. B /\ Y e. B ) -> ( X .\/ Y ) e. B )
8	1, 2, 7	3jca 1242	. 2 |- ( ( K e. OML /\ X e. B /\ Y e. B ) -> ( K e. OML /\ X e. B /\ ( X .\/ Y ) e. B ) )
9		eqid 2622	. . . 4 |- ( le ` K ) = ( le ` K )
10	4, 9, 5	latlej1 17060	. . 3 |- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> X ( le ` K ) ( X .\/ Y ) )
11	3, 10	syl3an1 1359	. 2 |- ( ( K e. OML /\ X e. B /\ Y e. B ) -> X ( le ` K ) ( X .\/ Y ) )
12		omllaw5.m	. . 3 |- ./\ = ( meet ` K )
13		omllaw5.o	. . 3 |- ._|_ = ( oc ` K )
14	4, 9, 5, 12, 13	omllaw2N 34531	. 2 |- ( ( K e. OML /\ X e. B /\ ( X .\/ Y ) e. B ) -> ( X ( le ` K ) ( X .\/ Y ) -> ( X .\/ ( ( ._|_ ` X ) ./\ ( X .\/ Y ) ) ) = ( X .\/ Y ) ) )
15	8, 11, 14	sylc 65	1 |- ( ( K e. OML /\ X e. B /\ Y e. B ) -> ( X .\/ ( ( ._|_ ` X ) ./\ ( X .\/ Y ) ) ) = ( X .\/ Y ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   occoc 15949   joincjn 16944   meetcmee 16945   Latclat 17045   OMLcoml 34462

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-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-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-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-lat 17046  df-oposet 34463  df-ol 34465  df-oml 34466

This theorem is referenced by: (None)