Theorem omlmod1i2N 34547

Description: Analogue of modular law atmod1i2 35145 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 1061	. . 3 |- ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X .<_ Z /\ Y C Z ) ) -> K e. OML )
2		simp23 1096	. . 3 |- ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X .<_ Z /\ Y C Z ) ) -> Z e. B )
3		simp21 1094	. . 3 |- ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X .<_ Z /\ Y C Z ) ) -> X e. B )
4		simp22 1095	. . 3 |- ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X .<_ Z /\ Y C Z ) ) -> Y e. B )
5		simp3l 1089	. . . . 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 34543	. . . . . 6 |- ( ( K e. OML /\ X e. B /\ Z e. B ) -> ( X .<_ Z -> X C Z ) )
10	1, 3, 2, 9	syl3anc 1326	. . . . 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 34536	. . . . 5 |- ( ( K e. OML /\ X e. B /\ Z e. B ) -> ( X C Z <-> Z C X ) )
13	1, 3, 2, 12	syl3anc 1326	. . . 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 222	. . 3 |- ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X .<_ Z /\ Y C Z ) ) -> Z C X )
15		simp3r 1090	. . . 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 34536	. . . . 5 |- ( ( K e. OML /\ Y e. B /\ Z e. B ) -> ( Y C Z <-> Z C Y ) )
17	1, 4, 2, 16	syl3anc 1326	. . . 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 222	. . 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 34545	. . 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 1344	. 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 34529	. . . 4 |- ( K e. OML -> K e. Lat )
24	23	3ad2ant1 1082	. . 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 17051	. . . 4 |- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X .\/ Y ) e. B )
26	24, 3, 4, 25	syl3anc 1326	. . 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 17075	. . 3 |- ( ( K e. Lat /\ Z e. B /\ ( X .\/ Y ) e. B ) -> ( Z ./\ ( X .\/ Y ) ) = ( ( X .\/ Y ) ./\ Z ) )
28	24, 2, 26, 27	syl3anc 1326	. 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 17080	. . . . 5 |- ( ( K e. Lat /\ X e. B /\ Z e. B ) -> ( X .<_ Z <-> ( Z ./\ X ) = X ) )
30	24, 3, 2, 29	syl3anc 1326	. . . 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 222	. . 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 17075	. . . 4 |- ( ( K e. Lat /\ Z e. B /\ Y e. B ) -> ( Z ./\ Y ) = ( Y ./\ Z ) )
33	24, 2, 4, 32	syl3anc 1326	. . 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 6668	. 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 2665	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 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   meetcmee 16945   Latclat 17045   cmccmtN 34460   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-preset 16928  df-poset 16946  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-lat 17046  df-oposet 34463  df-cmtN 34464  df-ol 34465  df-oml 34466

This theorem is referenced by:  omlspjN  34548