Theorem llnmod1i2 35146

Description: Version of modular law pmod1i 35134 that holds in a Hilbert lattice, when one element is a lattice line (expressed as the join P .\/ Q). (Contributed by NM, 16-Sep-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
llnmod1i2	|- ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( P e. A /\ Q e. A ) /\ X .<_ Y ) -> ( X .\/ ( ( P .\/ Q ) ./\ Y ) ) = ( ( X .\/ ( P .\/ Q ) ) ./\ Y ) )

Proof of Theorem llnmod1i2

Step	Hyp	Ref	Expression
1		simpl1 1064	. . . . 5 |- ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( P e. A /\ Q e. A ) ) -> K e. HL )
2		simpl2 1065	. . . . 5 |- ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( P e. A /\ Q e. A ) ) -> X e. B )
3		simprl 794	. . . . 5 |- ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( P e. A /\ Q e. A ) ) -> P e. A )
4		simprr 796	. . . . 5 |- ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( P e. A /\ Q e. A ) ) -> Q e. A )
5		atmod.b	. . . . . 6 |- B = ( Base ` K )
6		atmod.j	. . . . . 6 |- .\/ = ( join ` K )
7		atmod.a	. . . . . 6 |- A = ( Atoms ` K )
8		eqid 2622	. . . . . 6 |- ( pmap ` K ) = ( pmap ` K )
9		eqid 2622	. . . . . 6 |- ( +P ` K ) = ( +P ` K )
10	5, 6, 7, 8, 9	pmapjlln1 35141	. . . . 5 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( ( pmap ` K ) ` ( X .\/ ( P .\/ Q ) ) ) = ( ( ( pmap ` K ) ` X ) ( +P ` K ) ( ( pmap ` K ) ` ( P .\/ Q ) ) ) )
11	1, 2, 3, 4, 10	syl13anc 1328	. . . 4 |- ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( P e. A /\ Q e. A ) ) -> ( ( pmap ` K ) ` ( X .\/ ( P .\/ Q ) ) ) = ( ( ( pmap ` K ) ` X ) ( +P ` K ) ( ( pmap ` K ) ` ( P .\/ Q ) ) ) )
12		hllat 34650	. . . . . . 7 |- ( K e. HL -> K e. Lat )
13	1, 12	syl 17	. . . . . 6 |- ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( P e. A /\ Q e. A ) ) -> K e. Lat )
14	5, 7	atbase 34576	. . . . . . 7 |- ( P e. A -> P e. B )
15	3, 14	syl 17	. . . . . 6 |- ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( P e. A /\ Q e. A ) ) -> P e. B )
16	5, 7	atbase 34576	. . . . . . 7 |- ( Q e. A -> Q e. B )
17	4, 16	syl 17	. . . . . 6 |- ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( P e. A /\ Q e. A ) ) -> Q e. B )
18	5, 6	latjcl 17051	. . . . . 6 |- ( ( K e. Lat /\ P e. B /\ Q e. B ) -> ( P .\/ Q ) e. B )
19	13, 15, 17, 18	syl3anc 1326	. . . . 5 |- ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( P e. A /\ Q e. A ) ) -> ( P .\/ Q ) e. B )
20		simpl3 1066	. . . . 5 |- ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( P e. A /\ Q e. A ) ) -> Y e. B )
21		atmod.l	. . . . . 6 |- .<_ = ( le ` K )
22		atmod.m	. . . . . 6 |- ./\ = ( meet ` K )
23	5, 21, 6, 22, 8, 9	hlmod1i 35142	. . . . 5 |- ( ( K e. HL /\ ( X e. B /\ ( P .\/ Q ) e. B /\ Y e. B ) ) -> ( ( X .<_ Y /\ ( ( pmap ` K ) ` ( X .\/ ( P .\/ Q ) ) ) = ( ( ( pmap ` K ) ` X ) ( +P ` K ) ( ( pmap ` K ) ` ( P .\/ Q ) ) ) ) -> ( ( X .\/ ( P .\/ Q ) ) ./\ Y ) = ( X .\/ ( ( P .\/ Q ) ./\ Y ) ) ) )
24	1, 2, 19, 20, 23	syl13anc 1328	. . . 4 |- ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( P e. A /\ Q e. A ) ) -> ( ( X .<_ Y /\ ( ( pmap ` K ) ` ( X .\/ ( P .\/ Q ) ) ) = ( ( ( pmap ` K ) ` X ) ( +P ` K ) ( ( pmap ` K ) ` ( P .\/ Q ) ) ) ) -> ( ( X .\/ ( P .\/ Q ) ) ./\ Y ) = ( X .\/ ( ( P .\/ Q ) ./\ Y ) ) ) )
25	11, 24	mpan2d 710	. . 3 |- ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( P e. A /\ Q e. A ) ) -> ( X .<_ Y -> ( ( X .\/ ( P .\/ Q ) ) ./\ Y ) = ( X .\/ ( ( P .\/ Q ) ./\ Y ) ) ) )
26	25	3impia 1261	. 2 |- ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( P e. A /\ Q e. A ) /\ X .<_ Y ) -> ( ( X .\/ ( P .\/ Q ) ) ./\ Y ) = ( X .\/ ( ( P .\/ Q ) ./\ Y ) ) )
27	26	eqcomd 2628	1 |- ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( P e. A /\ Q e. A ) /\ X .<_ Y ) -> ( X .\/ ( ( P .\/ Q ) ./\ Y ) ) = ( ( X .\/ ( P .\/ Q ) ) ./\ Y ) )

Syntax hints:   -> wi 4   /\ 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   Atomscatm 34550   HLchlt 34637   pmapcpmap 34783   +Pcpadd 35081

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:  llnmod2i2  35149  dalawlem12  35168