Theorem atmod2i1 35147

Description: Version of modular law pmod2iN 35135 that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 14-May-2012.) (Revised by Mario Carneiro, 10-May-2013.)

Hypotheses

Ref	Expression
atmod.b	|- B = ( Base ` K )
atmod.l	|- .<_ = ( le ` K )
atmod.j	|- .\/ = ( join ` K )
atmod.m	|- ./\ = ( meet ` K )
atmod.a	|- A = ( Atoms ` K )

Assertion

Ref	Expression
atmod2i1	|- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) /\ P .<_ X ) -> ( ( X ./\ Y ) .\/ P ) = ( X ./\ ( Y .\/ P ) ) )

Proof of Theorem atmod2i1

Step	Hyp	Ref	Expression
1		hllat 34650	. . . . 5 |- ( K e. HL -> K e. Lat )
2	1	3ad2ant1 1082	. . . 4 |- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) /\ P .<_ X ) -> K e. Lat )
3		simp22 1095	. . . 4 |- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) /\ P .<_ X ) -> X e. B )
4		simp23 1096	. . . 4 |- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) /\ P .<_ X ) -> Y e. B )
5		atmod.b	. . . . 5 |- B = ( Base ` K )
6		atmod.m	. . . . 5 |- ./\ = ( meet ` K )
7	5, 6	latmcom 17075	. . . 4 |- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X ./\ Y ) = ( Y ./\ X ) )
8	2, 3, 4, 7	syl3anc 1326	. . 3 |- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) /\ P .<_ X ) -> ( X ./\ Y ) = ( Y ./\ X ) )
9	8	oveq2d 6666	. 2 |- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) /\ P .<_ X ) -> ( P .\/ ( X ./\ Y ) ) = ( P .\/ ( Y ./\ X ) ) )
10		simp21 1094	. . . 4 |- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) /\ P .<_ X ) -> P e. A )
11		atmod.a	. . . . 5 |- A = ( Atoms ` K )
12	5, 11	atbase 34576	. . . 4 |- ( P e. A -> P e. B )
13	10, 12	syl 17	. . 3 |- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) /\ P .<_ X ) -> P e. B )
14	5, 6	latmcl 17052	. . . 4 |- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X ./\ Y ) e. B )
15	2, 3, 4, 14	syl3anc 1326	. . 3 |- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) /\ P .<_ X ) -> ( X ./\ Y ) e. B )
16		atmod.j	. . . 4 |- .\/ = ( join ` K )
17	5, 16	latjcom 17059	. . 3 |- ( ( K e. Lat /\ P e. B /\ ( X ./\ Y ) e. B ) -> ( P .\/ ( X ./\ Y ) ) = ( ( X ./\ Y ) .\/ P ) )
18	2, 13, 15, 17	syl3anc 1326	. 2 |- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) /\ P .<_ X ) -> ( P .\/ ( X ./\ Y ) ) = ( ( X ./\ Y ) .\/ P ) )
19	5, 16	latjcl 17051	. . . . 5 |- ( ( K e. Lat /\ P e. B /\ Y e. B ) -> ( P .\/ Y ) e. B )
20	2, 13, 4, 19	syl3anc 1326	. . . 4 |- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) /\ P .<_ X ) -> ( P .\/ Y ) e. B )
21	5, 6	latmcom 17075	. . . 4 |- ( ( K e. Lat /\ ( P .\/ Y ) e. B /\ X e. B ) -> ( ( P .\/ Y ) ./\ X ) = ( X ./\ ( P .\/ Y ) ) )
22	2, 20, 3, 21	syl3anc 1326	. . 3 |- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) /\ P .<_ X ) -> ( ( P .\/ Y ) ./\ X ) = ( X ./\ ( P .\/ Y ) ) )
23		simp1 1061	. . . 4 |- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) /\ P .<_ X ) -> K e. HL )
24		simp3 1063	. . . 4 |- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) /\ P .<_ X ) -> P .<_ X )
25		atmod.l	. . . . 5 |- .<_ = ( le ` K )
26	5, 25, 16, 6, 11	atmod1i1 35143	. . . 4 |- ( ( K e. HL /\ ( P e. A /\ Y e. B /\ X e. B ) /\ P .<_ X ) -> ( P .\/ ( Y ./\ X ) ) = ( ( P .\/ Y ) ./\ X ) )
27	23, 10, 4, 3, 24, 26	syl131anc 1339	. . 3 |- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) /\ P .<_ X ) -> ( P .\/ ( Y ./\ X ) ) = ( ( P .\/ Y ) ./\ X ) )
28	5, 16	latjcom 17059	. . . . 5 |- ( ( K e. Lat /\ Y e. B /\ P e. B ) -> ( Y .\/ P ) = ( P .\/ Y ) )
29	2, 4, 13, 28	syl3anc 1326	. . . 4 |- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) /\ P .<_ X ) -> ( Y .\/ P ) = ( P .\/ Y ) )
30	29	oveq2d 6666	. . 3 |- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) /\ P .<_ X ) -> ( X ./\ ( Y .\/ P ) ) = ( X ./\ ( P .\/ Y ) ) )
31	22, 27, 30	3eqtr4d 2666	. 2 |- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) /\ P .<_ X ) -> ( P .\/ ( Y ./\ X ) ) = ( X ./\ ( Y .\/ P ) ) )
32	9, 18, 31	3eqtr3d 2664	1 |- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) /\ P .<_ X ) -> ( ( X ./\ Y ) .\/ P ) = ( X ./\ ( Y .\/ P ) ) )

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   joincjn 16944   meetcmee 16945   Latclat 17045   Atomscatm 34550   HLchlt 34637

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-iin 4523  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-mpt2 6655  df-1st 7168  df-2nd 7169  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-psubsp 34789  df-pmap 34790  df-padd 35082

This theorem is referenced by:  lhpmod6i1  35325  trljat1  35453  trljat2  35454  cdlemc1  35478  cdlemc6  35483  cdleme16b  35566  cdleme20c  35599  cdleme20j  35606  cdleme22e  35632  cdleme22eALTN  35633  cdlemkid1  36210  dihmeetlem5  36597