Theorem omlmod1i2N 35382

Description: Analog of modular law atmod1i2 35980 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 994	. . 3 |- ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X .<_ Z /\ Y C Z ) ) -> K e. OML )
2		simp23 1029	. . 3 |- ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X .<_ Z /\ Y C Z ) ) -> Z e. B )
3		simp21 1027	. . 3 |- ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X .<_ Z /\ Y C Z ) ) -> X e. B )
4		simp22 1028	. . 3 |- ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X .<_ Z /\ Y C Z ) ) -> Y e. B )
5		simp3l 1022	. . . . 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 35378	. . . . . 6 |- ( ( K e. OML /\ X e. B /\ Z e. B ) -> ( X .<_ Z -> X C Z ) )
10	1, 3, 2, 9	syl3anc 1226	. . . . 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 35371	. . . . 5 |- ( ( K e. OML /\ X e. B /\ Z e. B ) -> ( X C Z <-> Z C X ) )
13	1, 3, 2, 12	syl3anc 1226	. . . 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 1023	. . . 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 35371	. . . . 5 |- ( ( K e. OML /\ Y e. B /\ Z e. B ) -> ( Y C Z <-> Z C Y ) )
17	1, 4, 2, 16	syl3anc 1226	. . . 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 35380	. . 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 1244	. 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 35364	. . . 4 |- ( K e. OML -> K e. Lat )
24	23	3ad2ant1 1015	. . 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 15880	. . . 4 |- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X .\/ Y ) e. B )
26	24, 3, 4, 25	syl3anc 1226	. . 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 15904	. . 3 |- ( ( K e. Lat /\ Z e. B /\ ( X .\/ Y ) e. B ) -> ( Z ./\ ( X .\/ Y ) ) = ( ( X .\/ Y ) ./\ Z ) )
28	24, 2, 26, 27	syl3anc 1226	. 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 15909	. . . . 5 |- ( ( K e. Lat /\ X e. B /\ Z e. B ) -> ( X .<_ Z <-> ( Z ./\ X ) = X ) )
30	24, 3, 2, 29	syl3anc 1226	. . . 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 15904	. . . 4 |- ( ( K e. Lat /\ Z e. B /\ Y e. B ) -> ( Z ./\ Y ) = ( Y ./\ Z ) )
33	24, 2, 4, 32	syl3anc 1226	. . 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 6288	. 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 2504	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 367   /\ w3a 971   = wceq 1398   e. wcel 1823   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14716   lecple 14791   joincjn 15772   meetcmee 15773   Latclat 15874   cmccmtN 35295   OMLcoml 35297

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-preset 15756  df-poset 15774  df-lub 15803  df-glb 15804  df-join 15805  df-meet 15806  df-p0 15868  df-lat 15875  df-oposet 35298  df-cmtN 35299  df-ol 35300  df-oml 35301

This theorem is referenced by:  omlspjN  35383