Theorem pmapglb2N 35057

Description: The projective map of the GLB of a set of lattice elements S. Variant of Theorem 15.5.2 of [MaedaMaeda] p. 62. Allows S = (/). (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
pmapglb2N	|- ( ( K e. HL /\ S C_ B ) -> ( M ` ( G ` S ) ) = ( A i^i |^|_ x e. S ( M ` x ) ) )

Distinct variable groups:   x, A   x, B   x, K   x, S

Allowed substitution hints:   G( x)   M( x)

Proof of Theorem pmapglb2N

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		fveq2 6191	. . . . . 6 |- ( S = (/) -> ( G ` S ) = ( G ` (/) ) )
12	11	fveq2d 6195	. . . . 5 |- ( S = (/) -> ( M ` ( G ` S ) ) = ( M ` ( G ` (/) ) ) )
13		riin0 4594	. . . . 5 |- ( S = (/) -> ( A i^i |^|_ x e. S ( M ` x ) ) = A )
14	12, 13	eqeq12d 2637	. . . 4 |- ( S = (/) -> ( ( M ` ( G ` S ) ) = ( A i^i |^|_ x e. S ( M ` x ) ) <-> ( M ` ( G ` (/) ) ) = A ) )
15	10, 14	syl5ibrcom 237	. . 3 |- ( K e. HL -> ( S = (/) -> ( M ` ( G ` S ) ) = ( A i^i |^|_ x e. S ( M ` x ) ) ) )
16	15	adantr 481	. 2 |- ( ( K e. HL /\ S C_ B ) -> ( S = (/) -> ( M ` ( G ` S ) ) = ( A i^i |^|_ x e. S ( M ` x ) ) ) )
17		pmapglb2.b	. . . . 5 |- B = ( Base ` K )
18	17, 2, 7	pmapglb 35056	. . . 4 |- ( ( K e. HL /\ S C_ B /\ S =/= (/) ) -> ( M ` ( G ` S ) ) = |^|_ x e. S ( M ` x ) )
19		simpr 477	. . . . . . . . . . 11 |- ( ( ( K e. HL /\ S C_ B ) /\ x e. S ) -> x e. S )
20		simpll 790	. . . . . . . . . . . 12 |- ( ( ( K e. HL /\ S C_ B ) /\ x e. S ) -> K e. HL )
21		ssel2 3598	. . . . . . . . . . . . 13 |- ( ( S C_ B /\ x e. S ) -> x e. B )
22	21	adantll 750	. . . . . . . . . . . 12 |- ( ( ( K e. HL /\ S C_ B ) /\ x e. S ) -> x e. B )
23	17, 6, 7	pmapssat 35045	. . . . . . . . . . . 12 |- ( ( K e. HL /\ x e. B ) -> ( M ` x ) C_ A )
24	20, 22, 23	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( K e. HL /\ S C_ B ) /\ x e. S ) -> ( M ` x ) C_ A )
25	19, 24	jca 554	. . . . . . . . . 10 |- ( ( ( K e. HL /\ S C_ B ) /\ x e. S ) -> ( x e. S /\ ( M ` x ) C_ A ) )
26	25	ex 450	. . . . . . . . 9 |- ( ( K e. HL /\ S C_ B ) -> ( x e. S -> ( x e. S /\ ( M ` x ) C_ A ) ) )
27	26	eximdv 1846	. . . . . . . 8 |- ( ( K e. HL /\ S C_ B ) -> ( E. x x e. S -> E. x ( x e. S /\ ( M ` x ) C_ A ) ) )
28		n0 3931	. . . . . . . 8 |- ( S =/= (/) <-> E. x x e. S )
29		df-rex 2918	. . . . . . . 8 |- ( E. x e. S ( M ` x ) C_ A <-> E. x ( x e. S /\ ( M ` x ) C_ A ) )
30	27, 28, 29	3imtr4g 285	. . . . . . 7 |- ( ( K e. HL /\ S C_ B ) -> ( S =/= (/) -> E. x e. S ( M ` x ) C_ A ) )
31	30	3impia 1261	. . . . . 6 |- ( ( K e. HL /\ S C_ B /\ S =/= (/) ) -> E. x e. S ( M ` x ) C_ A )
32		iinss 4571	. . . . . 6 |- ( E. x e. S ( M ` x ) C_ A -> |^|_ x e. S ( M ` x ) C_ A )
33	31, 32	syl 17	. . . . 5 |- ( ( K e. HL /\ S C_ B /\ S =/= (/) ) -> |^|_ x e. S ( M ` x ) C_ A )
34		sseqin2 3817	. . . . 5 |- ( |^|_ x e. S ( M ` x ) C_ A <-> ( A i^i |^|_ x e. S ( M ` x ) ) = |^|_ x e. S ( M ` x ) )
35	33, 34	sylib 208	. . . 4 |- ( ( K e. HL /\ S C_ B /\ S =/= (/) ) -> ( A i^i |^|_ x e. S ( M ` x ) ) = |^|_ x e. S ( M ` x ) )
36	18, 35	eqtr4d 2659	. . 3 |- ( ( K e. HL /\ S C_ B /\ S =/= (/) ) -> ( M ` ( G ` S ) ) = ( A i^i |^|_ x e. S ( M ` x ) ) )
37	36	3expia 1267	. 2 |- ( ( K e. HL /\ S C_ B ) -> ( S =/= (/) -> ( M ` ( G ` S ) ) = ( A i^i |^|_ x e. S ( M ` x ) ) ) )
38	16, 37	pm2.61dne 2880	1 |- ( ( K e. HL /\ S C_ B ) -> ( M ` ( G ` S ) ) = ( A i^i |^|_ x e. S ( M ` x ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   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: (None)