Theorem omlmod1i2N 29743

Description: Analog of modular law atmod1i2 30341 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 957	. . 3 |- ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X .<_ Z /\ Y C Z ) ) -> K e. OML )
2		simp23 992	. . 3 |- ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X .<_ Z /\ Y C Z ) ) -> Z e. B )
3		simp21 990	. . 3 |- ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X .<_ Z /\ Y C Z ) ) -> X e. B )
4		simp22 991	. . 3 |- ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X .<_ Z /\ Y C Z ) ) -> Y e. B )
5		simp3l 985	. . . . 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 29739	. . . . . 6 |- ( ( K e. OML /\ X e. B /\ Z e. B ) -> ( X .<_ Z -> X C Z ) )
10	1, 3, 2, 9	syl3anc 1184	. . . . 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 29732	. . . . 5 |- ( ( K e. OML /\ X e. B /\ Z e. B ) -> ( X C Z <-> Z C X ) )
13	1, 3, 2, 12	syl3anc 1184	. . . 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 202	. . 3 |- ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X .<_ Z /\ Y C Z ) ) -> Z C X )
15		simp3r 986	. . . 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 29732	. . . . 5 |- ( ( K e. OML /\ Y e. B /\ Z e. B ) -> ( Y C Z <-> Z C Y ) )
17	1, 4, 2, 16	syl3anc 1184	. . . 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 202	. . 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 29741	. . 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 1202	. 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 29725	. . . 4 |- ( K e. OML -> K e. Lat )
24	23	3ad2ant1 978	. . 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 14434	. . . 4 |- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X .\/ Y ) e. B )
26	24, 3, 4, 25	syl3anc 1184	. . 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 14459	. . 3 |- ( ( K e. Lat /\ Z e. B /\ ( X .\/ Y ) e. B ) -> ( Z ./\ ( X .\/ Y ) ) = ( ( X .\/ Y ) ./\ Z ) )
28	24, 2, 26, 27	syl3anc 1184	. 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 14464	. . . . 5 |- ( ( K e. Lat /\ X e. B /\ Z e. B ) -> ( X .<_ Z <-> ( Z ./\ X ) = X ) )
30	24, 3, 2, 29	syl3anc 1184	. . . 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 202	. . 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 14459	. . . 4 |- ( ( K e. Lat /\ Z e. B /\ Y e. B ) -> ( Z ./\ Y ) = ( Y ./\ Z ) )
33	24, 2, 4, 32	syl3anc 1184	. . 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 6058	. 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 2445	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 177   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   Latclat 14429   cmccmtN 29656   OMLcoml 29658

This theorem is referenced by:  omlspjN  29744

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-lat 14430  df-oposet 29659  df-cmtN 29660  df-ol 29661  df-oml 29662