Theorem atmod3i1 32861

Description: Version of modular law that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 4-Jun-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
atmod3i1	|- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) /\ P .<_ X ) -> ( P .\/ ( X ./\ Y ) ) = ( X ./\ ( P .\/ Y ) ) )

Proof of Theorem atmod3i1

Step	Hyp	Ref	Expression
1		simp1 997	. . 3 |- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) /\ P .<_ X ) -> K e. HL )
2		simp21 1030	. . 3 |- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) /\ P .<_ X ) -> P e. A )
3		simp23 1032	. . 3 |- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) /\ P .<_ X ) -> Y e. B )
4		simp22 1031	. . 3 |- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) /\ P .<_ X ) -> X e. B )
5		simp3 999	. . 3 |- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) /\ P .<_ X ) -> P .<_ X )
6		atmod.b	. . . 4 |- B = ( Base ` K )
7		atmod.l	. . . 4 |- .<_ = ( le ` K )
8		atmod.j	. . . 4 |- .\/ = ( join ` K )
9		atmod.m	. . . 4 |- ./\ = ( meet ` K )
10		atmod.a	. . . 4 |- A = ( Atoms ` K )
11	6, 7, 8, 9, 10	atmod1i1 32854	. . 3 |- ( ( K e. HL /\ ( P e. A /\ Y e. B /\ X e. B ) /\ P .<_ X ) -> ( P .\/ ( Y ./\ X ) ) = ( ( P .\/ Y ) ./\ X ) )
12	1, 2, 3, 4, 5, 11	syl131anc 1243	. 2 |- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) /\ P .<_ X ) -> ( P .\/ ( Y ./\ X ) ) = ( ( P .\/ Y ) ./\ X ) )
13		hllat 32361	. . . . 5 |- ( K e. HL -> K e. Lat )
14	13	3ad2ant1 1018	. . . 4 |- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) /\ P .<_ X ) -> K e. Lat )
15	6, 9	latmcom 16027	. . . 4 |- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X ./\ Y ) = ( Y ./\ X ) )
16	14, 4, 3, 15	syl3anc 1230	. . 3 |- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) /\ P .<_ X ) -> ( X ./\ Y ) = ( Y ./\ X ) )
17	16	oveq2d 6293	. 2 |- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) /\ P .<_ X ) -> ( P .\/ ( X ./\ Y ) ) = ( P .\/ ( Y ./\ X ) ) )
18	6, 10	atbase 32287	. . . . 5 |- ( P e. A -> P e. B )
19	2, 18	syl 17	. . . 4 |- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) /\ P .<_ X ) -> P e. B )
20	6, 8	latjcl 16003	. . . 4 |- ( ( K e. Lat /\ P e. B /\ Y e. B ) -> ( P .\/ Y ) e. B )
21	14, 19, 3, 20	syl3anc 1230	. . 3 |- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) /\ P .<_ X ) -> ( P .\/ Y ) e. B )
22	6, 9	latmcom 16027	. . 3 |- ( ( K e. Lat /\ X e. B /\ ( P .\/ Y ) e. B ) -> ( X ./\ ( P .\/ Y ) ) = ( ( P .\/ Y ) ./\ X ) )
23	14, 4, 21, 22	syl3anc 1230	. 2 |- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) /\ P .<_ X ) -> ( X ./\ ( P .\/ Y ) ) = ( ( P .\/ Y ) ./\ X ) )
24	12, 17, 23	3eqtr4d 2453	1 |- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) /\ P .<_ X ) -> ( P .\/ ( X ./\ Y ) ) = ( X ./\ ( P .\/ Y ) ) )

Syntax hints:   -> wi 4   /\ w3a 974   = wceq 1405   e. wcel 1842   class class class wbr 4394   ` cfv 5568  (class class class)co 6277   Basecbs 14839   lecple 14914   joincjn 15895   meetcmee 15896   Latclat 15997   Atomscatm 32261   HLchlt 32348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-preset 15879  df-poset 15897  df-plt 15910  df-lub 15926  df-glb 15927  df-join 15928  df-meet 15929  df-p0 15991  df-lat 15998  df-clat 16060  df-oposet 32174  df-ol 32176  df-oml 32177  df-covers 32264  df-ats 32265  df-atl 32296  df-cvlat 32320  df-hlat 32349  df-psubsp 32500  df-pmap 32501  df-padd 32793

This theorem is referenced by:  dalawlem2  32869  dalawlem3  32870  dalawlem6  32873  lhpmcvr3  33022  cdleme0cp  33212  cdleme0cq  33213  cdleme1  33225  cdleme4  33236  cdleme5  33238  cdleme8  33248  cdleme9  33251  cdleme10  33252  cdleme15b  33273  cdleme22e  33343  cdleme22eALTN  33344  cdleme23c  33350  cdleme35b  33449  cdleme35e  33452  cdleme42a  33470  trlcoabs2N  33721  cdlemi1  33817  cdlemk4  33833  dia2dimlem1  34064  dia2dimlem2  34065  cdlemn10  34206  dihglbcpreN  34300