Theorem pmapmeet 35059

Description: The projective map of a meet. (Contributed by NM, 25-Jan-2012.)

Hypotheses

Ref	Expression
pmapmeet.b	|- B = ( Base ` K )
pmapmeet.m	|- ./\ = ( meet ` K )
pmapmeet.a	|- A = ( Atoms ` K )
pmapmeet.p	|- P = ( pmap ` K )

Assertion

Ref	Expression
pmapmeet	|- ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( P ` ( X ./\ Y ) ) = ( ( P ` X ) i^i ( P ` Y ) ) )

Proof of Theorem pmapmeet

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 |- ( glb ` K ) = ( glb ` K )
2		pmapmeet.m	. . . 4 |- ./\ = ( meet ` K )
3		simp1 1061	. . . 4 |- ( ( K e. HL /\ X e. B /\ Y e. B ) -> K e. HL )
4		simp2 1062	. . . 4 |- ( ( K e. HL /\ X e. B /\ Y e. B ) -> X e. B )
5		simp3 1063	. . . 4 |- ( ( K e. HL /\ X e. B /\ Y e. B ) -> Y e. B )
6	1, 2, 3, 4, 5	meetval 17019	. . 3 |- ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( X ./\ Y ) = ( ( glb ` K ) ` { X , Y } ) )
7	6	fveq2d 6195	. 2 |- ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( P ` ( X ./\ Y ) ) = ( P ` ( ( glb ` K ) ` { X , Y } ) ) )
8		prssi 4353	. . . 4 |- ( ( X e. B /\ Y e. B ) -> { X , Y } C_ B )
9	8	3adant1 1079	. . 3 |- ( ( K e. HL /\ X e. B /\ Y e. B ) -> { X , Y } C_ B )
10		prnzg 4311	. . . 4 |- ( X e. B -> { X , Y } =/= (/) )
11	10	3ad2ant2 1083	. . 3 |- ( ( K e. HL /\ X e. B /\ Y e. B ) -> { X , Y } =/= (/) )
12		pmapmeet.b	. . . 4 |- B = ( Base ` K )
13		pmapmeet.p	. . . 4 |- P = ( pmap ` K )
14	12, 1, 13	pmapglb 35056	. . 3 |- ( ( K e. HL /\ { X , Y } C_ B /\ { X , Y } =/= (/) ) -> ( P ` ( ( glb ` K ) ` { X , Y } ) ) = |^|_ x e. { X , Y } ( P ` x ) )
15	3, 9, 11, 14	syl3anc 1326	. 2 |- ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( P ` ( ( glb ` K ) ` { X , Y } ) ) = |^|_ x e. { X , Y } ( P ` x ) )
16		fveq2 6191	. . . 4 |- ( x = X -> ( P ` x ) = ( P ` X ) )
17		fveq2 6191	. . . 4 |- ( x = Y -> ( P ` x ) = ( P ` Y ) )
18	16, 17	iinxprg 4601	. . 3 |- ( ( X e. B /\ Y e. B ) -> |^|_ x e. { X , Y } ( P ` x ) = ( ( P ` X ) i^i ( P ` Y ) ) )
19	18	3adant1 1079	. 2 |- ( ( K e. HL /\ X e. B /\ Y e. B ) -> |^|_ x e. { X , Y } ( P ` x ) = ( ( P ` X ) i^i ( P ` Y ) ) )
20	7, 15, 19	3eqtrd 2660	1 |- ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( P ` ( X ./\ Y ) ) = ( ( P ` X ) i^i ( P ` Y ) ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   i^i cin 3573   C_ wss 3574   (/)c0 3915   {cpr 4179   |^|_ciin 4521   ` cfv 5888  (class class class)co 6650   Basecbs 15857   glbcglb 16943   meetcmee 16945   Atomscatm 34550   HLchlt 34637   pmapcpmap 34783

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-poset 16946  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-lat 17046  df-clat 17108  df-ats 34554  df-hlat 34638  df-pmap 34790

This theorem is referenced by:  hlmod1i  35142  poldmj1N  35214  pmapj2N  35215  pnonsingN  35219  psubclinN  35234  poml4N  35239  pl42lem1N  35265  pl42lem2N  35266