Theorem pmapglb2xN 35058

Description: The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb2N 35057, where we read S as S ( i ). Extension of Theorem 15.5.2 of [MaedaMaeda] p. 62 that allows I = (/). (Contributed by NM, 21-Jan-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
pmapglb2.b	|- B = ( Base ` K )
pmapglb2.g	|- G = ( glb ` K )
pmapglb2.a	|- A = ( Atoms ` K )
pmapglb2.m	|- M = ( pmap ` K )

Assertion

Ref	Expression
pmapglb2xN	|- ( ( K e. HL /\ A. i e. I S e. B ) -> ( M ` ( G ` { y | E. i e. I y = S } ) ) = ( A i^i |^|_ i e. I ( M ` S ) ) )

Distinct variable groups:   A, i   y, i, B   i, I, y   i, K, y   y, S

Allowed substitution hints:   A( y)   S( i)   G( y, i)   M( y, i)

Proof of Theorem pmapglb2xN

Step	Hyp	Ref	Expression
1		hlop 34649	. . . . 5 |- ( K e. HL -> K e. OP )
2		pmapglb2.g	. . . . . . . 8 |- G = ( glb ` K )
3		eqid 2622	. . . . . . . 8 |- ( 1. ` K ) = ( 1. ` K )
4	2, 3	glb0N 34480	. . . . . . 7 |- ( K e. OP -> ( G ` (/) ) = ( 1. ` K ) )
5	4	fveq2d 6195	. . . . . 6 |- ( K e. OP -> ( M ` ( G ` (/) ) ) = ( M ` ( 1. ` K ) ) )
6		pmapglb2.a	. . . . . . 7 |- A = ( Atoms ` K )
7		pmapglb2.m	. . . . . . 7 |- M = ( pmap ` K )
8	3, 6, 7	pmap1N 35053	. . . . . 6 |- ( K e. OP -> ( M ` ( 1. ` K ) ) = A )
9	5, 8	eqtrd 2656	. . . . 5 |- ( K e. OP -> ( M ` ( G ` (/) ) ) = A )
10	1, 9	syl 17	. . . 4 |- ( K e. HL -> ( M ` ( G ` (/) ) ) = A )
11		rexeq 3139	. . . . . . . . 9 |- ( I = (/) -> ( E. i e. I y = S <-> E. i e. (/) y = S ) )
12	11	abbidv 2741	. . . . . . . 8 |- ( I = (/) -> { y | E. i e. I y = S } = { y | E. i e. (/) y = S } )
13		rex0 3938	. . . . . . . . 9 |- -. E. i e. (/) y = S
14	13	abf 3978	. . . . . . . 8 |- { y | E. i e. (/) y = S } = (/)
15	12, 14	syl6eq 2672	. . . . . . 7 |- ( I = (/) -> { y | E. i e. I y = S } = (/) )
16	15	fveq2d 6195	. . . . . 6 |- ( I = (/) -> ( G ` { y | E. i e. I y = S } ) = ( G ` (/) ) )
17	16	fveq2d 6195	. . . . 5 |- ( I = (/) -> ( M ` ( G ` { y | E. i e. I y = S } ) ) = ( M ` ( G ` (/) ) ) )
18		riin0 4594	. . . . 5 |- ( I = (/) -> ( A i^i |^|_ i e. I ( M ` S ) ) = A )
19	17, 18	eqeq12d 2637	. . . 4 |- ( I = (/) -> ( ( M ` ( G ` { y | E. i e. I y = S } ) ) = ( A i^i |^|_ i e. I ( M ` S ) ) <-> ( M ` ( G ` (/) ) ) = A ) )
20	10, 19	syl5ibrcom 237	. . 3 |- ( K e. HL -> ( I = (/) -> ( M ` ( G ` { y | E. i e. I y = S } ) ) = ( A i^i |^|_ i e. I ( M ` S ) ) ) )
21	20	adantr 481	. 2 |- ( ( K e. HL /\ A. i e. I S e. B ) -> ( I = (/) -> ( M ` ( G ` { y | E. i e. I y = S } ) ) = ( A i^i |^|_ i e. I ( M ` S ) ) ) )
22		pmapglb2.b	. . . . 5 |- B = ( Base ` K )
23	22, 2, 7	pmapglbx 35055	. . . 4 |- ( ( K e. HL /\ A. i e. I S e. B /\ I =/= (/) ) -> ( M ` ( G ` { y | E. i e. I y = S } ) ) = |^|_ i e. I ( M ` S ) )
24		nfv 1843	. . . . . . . . . 10 |- F/ i K e. HL
25		nfra1 2941	. . . . . . . . . 10 |- F/ i A. i e. I S e. B
26	24, 25	nfan 1828	. . . . . . . . 9 |- F/ i ( K e. HL /\ A. i e. I S e. B )
27		simpr 477	. . . . . . . . . . 11 |- ( ( ( K e. HL /\ A. i e. I S e. B ) /\ i e. I ) -> i e. I )
28		simpll 790	. . . . . . . . . . . 12 |- ( ( ( K e. HL /\ A. i e. I S e. B ) /\ i e. I ) -> K e. HL )
29		rspa 2930	. . . . . . . . . . . . 13 |- ( ( A. i e. I S e. B /\ i e. I ) -> S e. B )
30	29	adantll 750	. . . . . . . . . . . 12 |- ( ( ( K e. HL /\ A. i e. I S e. B ) /\ i e. I ) -> S e. B )
31	22, 6, 7	pmapssat 35045	. . . . . . . . . . . 12 |- ( ( K e. HL /\ S e. B ) -> ( M ` S ) C_ A )
32	28, 30, 31	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( K e. HL /\ A. i e. I S e. B ) /\ i e. I ) -> ( M ` S ) C_ A )
33	27, 32	jca 554	. . . . . . . . . 10 |- ( ( ( K e. HL /\ A. i e. I S e. B ) /\ i e. I ) -> ( i e. I /\ ( M ` S ) C_ A ) )
34	33	ex 450	. . . . . . . . 9 |- ( ( K e. HL /\ A. i e. I S e. B ) -> ( i e. I -> ( i e. I /\ ( M ` S ) C_ A ) ) )
35	26, 34	eximd 2085	. . . . . . . 8 |- ( ( K e. HL /\ A. i e. I S e. B ) -> ( E. i i e. I -> E. i ( i e. I /\ ( M ` S ) C_ A ) ) )
36		n0 3931	. . . . . . . 8 |- ( I =/= (/) <-> E. i i e. I )
37		df-rex 2918	. . . . . . . 8 |- ( E. i e. I ( M ` S ) C_ A <-> E. i ( i e. I /\ ( M ` S ) C_ A ) )
38	35, 36, 37	3imtr4g 285	. . . . . . 7 |- ( ( K e. HL /\ A. i e. I S e. B ) -> ( I =/= (/) -> E. i e. I ( M ` S ) C_ A ) )
39	38	3impia 1261	. . . . . 6 |- ( ( K e. HL /\ A. i e. I S e. B /\ I =/= (/) ) -> E. i e. I ( M ` S ) C_ A )
40		iinss 4571	. . . . . 6 |- ( E. i e. I ( M ` S ) C_ A -> |^|_ i e. I ( M ` S ) C_ A )
41	39, 40	syl 17	. . . . 5 |- ( ( K e. HL /\ A. i e. I S e. B /\ I =/= (/) ) -> |^|_ i e. I ( M ` S ) C_ A )
42		sseqin2 3817	. . . . 5 |- ( |^|_ i e. I ( M ` S ) C_ A <-> ( A i^i |^|_ i e. I ( M ` S ) ) = |^|_ i e. I ( M ` S ) )
43	41, 42	sylib 208	. . . 4 |- ( ( K e. HL /\ A. i e. I S e. B /\ I =/= (/) ) -> ( A i^i |^|_ i e. I ( M ` S ) ) = |^|_ i e. I ( M ` S ) )
44	23, 43	eqtr4d 2659	. . 3 |- ( ( K e. HL /\ A. i e. I S e. B /\ I =/= (/) ) -> ( M ` ( G ` { y | E. i e. I y = S } ) ) = ( A i^i |^|_ i e. I ( M ` S ) ) )
45	44	3expia 1267	. 2 |- ( ( K e. HL /\ A. i e. I S e. B ) -> ( I =/= (/) -> ( M ` ( G ` { y | E. i e. I y = S } ) ) = ( A i^i |^|_ i e. I ( M ` S ) ) ) )
46	21, 45	pm2.61dne 2880	1 |- ( ( K e. HL /\ A. i e. I S e. B ) -> ( M ` ( G ` { y | E. i e. I y = S } ) ) = ( A i^i |^|_ i e. I ( M ` S ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   E.wrex 2913   i^i cin 3573   C_ wss 3574   (/)c0 3915   |^|_ciin 4521   ` cfv 5888   Basecbs 15857   glbcglb 16943   1.cp1 17038   OPcops 34459   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-preset 16928  df-poset 16946  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-ats 34554  df-hlat 34638  df-pmap 34790

This theorem is referenced by:  polval2N  35192