Theorem omlmod1i2N 33211

Description: Analog of modular law atmod1i2 33809 that holds in any OML. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
omlmod.b	|- B = ( Base ` K )
omlmod.l	|- .<_ = ( le ` K )
omlmod.j	|- .\/ = ( join ` K )
omlmod.m	|- ./\ = ( meet ` K )
omlmod.c	|- C = ( cm ` K )

Assertion

Ref	Expression
omlmod1i2N	|- ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X .<_ Z /\ Y C Z ) ) -> ( X .\/ ( Y ./\ Z ) ) = ( ( X .\/ Y ) ./\ Z ) )

Proof of Theorem omlmod1i2N

Step	Hyp	Ref	Expression
1		simp1 988	. . 3 |- ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X .<_ Z /\ Y C Z ) ) -> K e. OML )
2		simp23 1023	. . 3 |- ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X .<_ Z /\ Y C Z ) ) -> Z e. B )
3		simp21 1021	. . 3 |- ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X .<_ Z /\ Y C Z ) ) -> X e. B )
4		simp22 1022	. . 3 |- ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X .<_ Z /\ Y C Z ) ) -> Y e. B )
5		simp3l 1016	. . . . 5 |- ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X .<_ Z /\ Y C Z ) ) -> X .<_ Z )
6		omlmod.b	. . . . . . 7 |- B = ( Base ` K )
7		omlmod.l	. . . . . . 7 |- .<_ = ( le ` K )
8		omlmod.c	. . . . . . 7 |- C = ( cm ` K )
9	6, 7, 8	lecmtN 33207	. . . . . 6 |- ( ( K e. OML /\ X e. B /\ Z e. B ) -> ( X .<_ Z -> X C Z ) )
10	1, 3, 2, 9	syl3anc 1219	. . . . 5 |- ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X .<_ Z /\ Y C Z ) ) -> ( X .<_ Z -> X C Z ) )
11	5, 10	mpd 15	. . . 4 |- ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X .<_ Z /\ Y C Z ) ) -> X C Z )
12	6, 8	cmtcomN 33200	. . . . 5 |- ( ( K e. OML /\ X e. B /\ Z e. B ) -> ( X C Z <-> Z C X ) )
13	1, 3, 2, 12	syl3anc 1219	. . . 4 |- ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X .<_ Z /\ Y C Z ) ) -> ( X C Z <-> Z C X ) )
14	11, 13	mpbid 210	. . 3 |- ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X .<_ Z /\ Y C Z ) ) -> Z C X )
15		simp3r 1017	. . . 4 |- ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X .<_ Z /\ Y C Z ) ) -> Y C Z )
16	6, 8	cmtcomN 33200	. . . . 5 |- ( ( K e. OML /\ Y e. B /\ Z e. B ) -> ( Y C Z <-> Z C Y ) )
17	1, 4, 2, 16	syl3anc 1219	. . . 4 |- ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X .<_ Z /\ Y C Z ) ) -> ( Y C Z <-> Z C Y ) )
18	15, 17	mpbid 210	. . 3 |- ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X .<_ Z /\ Y C Z ) ) -> Z C Y )
19		omlmod.j	. . . 4 |- .\/ = ( join ` K )
20		omlmod.m	. . . 4 |- ./\ = ( meet ` K )
21	6, 19, 20, 8	omlfh1N 33209	. . 3 |- ( ( K e. OML /\ ( Z e. B /\ X e. B /\ Y e. B ) /\ ( Z C X /\ Z C Y ) ) -> ( Z ./\ ( X .\/ Y ) ) = ( ( Z ./\ X ) .\/ ( Z ./\ Y ) ) )
22	1, 2, 3, 4, 14, 18, 21	syl132anc 1237	. 2 |- ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X .<_ Z /\ Y C Z ) ) -> ( Z ./\ ( X .\/ Y ) ) = ( ( Z ./\ X ) .\/ ( Z ./\ Y ) ) )
23		omllat 33193	. . . 4 |- ( K e. OML -> K e. Lat )
24	23	3ad2ant1 1009	. . 3 |- ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X .<_ Z /\ Y C Z ) ) -> K e. Lat )
25	6, 19	latjcl 15323	. . . 4 |- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X .\/ Y ) e. B )
26	24, 3, 4, 25	syl3anc 1219	. . 3 |- ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X .<_ Z /\ Y C Z ) ) -> ( X .\/ Y ) e. B )
27	6, 20	latmcom 15347	. . 3 |- ( ( K e. Lat /\ Z e. B /\ ( X .\/ Y ) e. B ) -> ( Z ./\ ( X .\/ Y ) ) = ( ( X .\/ Y ) ./\ Z ) )
28	24, 2, 26, 27	syl3anc 1219	. 2 |- ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X .<_ Z /\ Y C Z ) ) -> ( Z ./\ ( X .\/ Y ) ) = ( ( X .\/ Y ) ./\ Z ) )
29	6, 7, 20	latleeqm2 15352	. . . . 5 |- ( ( K e. Lat /\ X e. B /\ Z e. B ) -> ( X .<_ Z <-> ( Z ./\ X ) = X ) )
30	24, 3, 2, 29	syl3anc 1219	. . . 4 |- ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X .<_ Z /\ Y C Z ) ) -> ( X .<_ Z <-> ( Z ./\ X ) = X ) )
31	5, 30	mpbid 210	. . 3 |- ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X .<_ Z /\ Y C Z ) ) -> ( Z ./\ X ) = X )
32	6, 20	latmcom 15347	. . . 4 |- ( ( K e. Lat /\ Z e. B /\ Y e. B ) -> ( Z ./\ Y ) = ( Y ./\ Z ) )
33	24, 2, 4, 32	syl3anc 1219	. . 3 |- ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X .<_ Z /\ Y C Z ) ) -> ( Z ./\ Y ) = ( Y ./\ Z ) )
34	31, 33	oveq12d 6208	. 2 |- ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X .<_ Z /\ Y C Z ) ) -> ( ( Z ./\ X ) .\/ ( Z ./\ Y ) ) = ( X .\/ ( Y ./\ Z ) ) )
35	22, 28, 34	3eqtr3rd 2501	1 |- ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X .<_ Z /\ Y C Z ) ) -> ( X .\/ ( Y ./\ Z ) ) = ( ( X .\/ Y ) ./\ Z ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   class class class wbr 4390   ` cfv 5516  (class class class)co 6190   Basecbs 14276   lecple 14347   joincjn 15216   meetcmee 15217   Latclat 15317   cmccmtN 33124   OMLcoml 33126

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-poset 15218  df-lub 15246  df-glb 15247  df-join 15248  df-meet 15249  df-p0 15311  df-lat 15318  df-oposet 33127  df-cmtN 33128  df-ol 33129  df-oml 33130

This theorem is referenced by:  omlspjN  33212