Theorem polval2N 35192

Description: Alternate expression for value of the projective subspace polarity function. Equation for polarity in [Holland95] p. 223. (Contributed by NM, 22-Jan-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
polval2.u	|- U = ( lub ` K )
polval2.o	|- ._|_ = ( oc ` K )
polval2.a	|- A = ( Atoms ` K )
polval2.m	|- M = ( pmap ` K )
polval2.p	|- P = ( _|_P ` K )

Assertion

Ref	Expression
polval2N	|- ( ( K e. HL /\ X C_ A ) -> ( P ` X ) = ( M ` ( ._|_ ` ( U ` X ) ) ) )

Proof of Theorem polval2N

Dummy variables x p are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		polval2.o	. . 3 |- ._|_ = ( oc ` K )
2		polval2.a	. . 3 |- A = ( Atoms ` K )
3		polval2.m	. . 3 |- M = ( pmap ` K )
4		polval2.p	. . 3 |- P = ( _|_P ` K )
5	1, 2, 3, 4	polvalN 35191	. 2 |- ( ( K e. HL /\ X C_ A ) -> ( P ` X ) = ( A i^i |^|_ p e. X ( M ` ( ._|_ ` p ) ) ) )
6		hlop 34649	. . . . . 6 |- ( K e. HL -> K e. OP )
7	6	ad2antrr 762	. . . . 5 |- ( ( ( K e. HL /\ X C_ A ) /\ p e. X ) -> K e. OP )
8		ssel2 3598	. . . . . . 7 |- ( ( X C_ A /\ p e. X ) -> p e. A )
9	8	adantll 750	. . . . . 6 |- ( ( ( K e. HL /\ X C_ A ) /\ p e. X ) -> p e. A )
10		eqid 2622	. . . . . . 7 |- ( Base ` K ) = ( Base ` K )
11	10, 2	atbase 34576	. . . . . 6 |- ( p e. A -> p e. ( Base ` K ) )
12	9, 11	syl 17	. . . . 5 |- ( ( ( K e. HL /\ X C_ A ) /\ p e. X ) -> p e. ( Base ` K ) )
13	10, 1	opoccl 34481	. . . . 5 |- ( ( K e. OP /\ p e. ( Base ` K ) ) -> ( ._|_ ` p ) e. ( Base ` K ) )
14	7, 12, 13	syl2anc 693	. . . 4 |- ( ( ( K e. HL /\ X C_ A ) /\ p e. X ) -> ( ._|_ ` p ) e. ( Base ` K ) )
15	14	ralrimiva 2966	. . 3 |- ( ( K e. HL /\ X C_ A ) -> A. p e. X ( ._|_ ` p ) e. ( Base ` K ) )
16		eqid 2622	. . . 4 |- ( glb ` K ) = ( glb ` K )
17	10, 16, 2, 3	pmapglb2xN 35058	. . 3 |- ( ( K e. HL /\ A. p e. X ( ._|_ ` p ) e. ( Base ` K ) ) -> ( M ` ( ( glb ` K ) ` { x | E. p e. X x = ( ._|_ ` p ) } ) ) = ( A i^i |^|_ p e. X ( M ` ( ._|_ ` p ) ) ) )
18	15, 17	syldan 487	. 2 |- ( ( K e. HL /\ X C_ A ) -> ( M ` ( ( glb ` K ) ` { x | E. p e. X x = ( ._|_ ` p ) } ) ) = ( A i^i |^|_ p e. X ( M ` ( ._|_ ` p ) ) ) )
19		polval2.u	. . . . . 6 |- U = ( lub ` K )
20	10, 19, 16, 1	glbconxN 34664	. . . . 5 |- ( ( K e. HL /\ A. p e. X ( ._|_ ` p ) e. ( Base ` K ) ) -> ( ( glb ` K ) ` { x | E. p e. X x = ( ._|_ ` p ) } ) = ( ._|_ ` ( U ` { x | E. p e. X x = ( ._|_ ` ( ._|_ ` p ) ) } ) ) )
21	15, 20	syldan 487	. . . 4 |- ( ( K e. HL /\ X C_ A ) -> ( ( glb ` K ) ` { x | E. p e. X x = ( ._|_ ` p ) } ) = ( ._|_ ` ( U ` { x | E. p e. X x = ( ._|_ ` ( ._|_ ` p ) ) } ) ) )
22	10, 1	opococ 34482	. . . . . . . . . . 11 |- ( ( K e. OP /\ p e. ( Base ` K ) ) -> ( ._|_ ` ( ._|_ ` p ) ) = p )
23	7, 12, 22	syl2anc 693	. . . . . . . . . 10 |- ( ( ( K e. HL /\ X C_ A ) /\ p e. X ) -> ( ._|_ ` ( ._|_ ` p ) ) = p )
24	23	eqeq2d 2632	. . . . . . . . 9 |- ( ( ( K e. HL /\ X C_ A ) /\ p e. X ) -> ( x = ( ._|_ ` ( ._|_ ` p ) ) <-> x = p ) )
25	24	rexbidva 3049	. . . . . . . 8 |- ( ( K e. HL /\ X C_ A ) -> ( E. p e. X x = ( ._|_ ` ( ._|_ ` p ) ) <-> E. p e. X x = p ) )
26	25	abbidv 2741	. . . . . . 7 |- ( ( K e. HL /\ X C_ A ) -> { x | E. p e. X x = ( ._|_ ` ( ._|_ ` p ) ) } = { x | E. p e. X x = p } )
27		df-rex 2918	. . . . . . . . . 10 |- ( E. p e. X x = p <-> E. p ( p e. X /\ x = p ) )
28		equcom 1945	. . . . . . . . . . . . 13 |- ( x = p <-> p = x )
29	28	anbi2i 730	. . . . . . . . . . . 12 |- ( ( p e. X /\ x = p ) <-> ( p e. X /\ p = x ) )
30		ancom 466	. . . . . . . . . . . 12 |- ( ( p e. X /\ p = x ) <-> ( p = x /\ p e. X ) )
31	29, 30	bitri 264	. . . . . . . . . . 11 |- ( ( p e. X /\ x = p ) <-> ( p = x /\ p e. X ) )
32	31	exbii 1774	. . . . . . . . . 10 |- ( E. p ( p e. X /\ x = p ) <-> E. p ( p = x /\ p e. X ) )
33		vex 3203	. . . . . . . . . . 11 |- x e. _V
34		eleq1 2689	. . . . . . . . . . 11 |- ( p = x -> ( p e. X <-> x e. X ) )
35	33, 34	ceqsexv 3242	. . . . . . . . . 10 |- ( E. p ( p = x /\ p e. X ) <-> x e. X )
36	27, 32, 35	3bitri 286	. . . . . . . . 9 |- ( E. p e. X x = p <-> x e. X )
37	36	abbii 2739	. . . . . . . 8 |- { x | E. p e. X x = p } = { x | x e. X }
38		abid2 2745	. . . . . . . 8 |- { x | x e. X } = X
39	37, 38	eqtri 2644	. . . . . . 7 |- { x | E. p e. X x = p } = X
40	26, 39	syl6eq 2672	. . . . . 6 |- ( ( K e. HL /\ X C_ A ) -> { x | E. p e. X x = ( ._|_ ` ( ._|_ ` p ) ) } = X )
41	40	fveq2d 6195	. . . . 5 |- ( ( K e. HL /\ X C_ A ) -> ( U ` { x | E. p e. X x = ( ._|_ ` ( ._|_ ` p ) ) } ) = ( U ` X ) )
42	41	fveq2d 6195	. . . 4 |- ( ( K e. HL /\ X C_ A ) -> ( ._|_ ` ( U ` { x | E. p e. X x = ( ._|_ ` ( ._|_ ` p ) ) } ) ) = ( ._|_ ` ( U ` X ) ) )
43	21, 42	eqtrd 2656	. . 3 |- ( ( K e. HL /\ X C_ A ) -> ( ( glb ` K ) ` { x | E. p e. X x = ( ._|_ ` p ) } ) = ( ._|_ ` ( U ` X ) ) )
44	43	fveq2d 6195	. 2 |- ( ( K e. HL /\ X C_ A ) -> ( M ` ( ( glb ` K ) ` { x | E. p e. X x = ( ._|_ ` p ) } ) ) = ( M ` ( ._|_ ` ( U ` X ) ) ) )
45	5, 18, 44	3eqtr2d 2662	1 |- ( ( K e. HL /\ X C_ A ) -> ( P ` X ) = ( M ` ( ._|_ ` ( U ` X ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   i^i cin 3573   C_ wss 3574   |^|_ciin 4521   ` cfv 5888   Basecbs 15857   occoc 15949   lubclub 16942   glbcglb 16943   OPcops 34459   Atomscatm 34550   HLchlt 34637   pmapcpmap 34783   _|_PcpolN 35188

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  ax-riotaBAD 34239

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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-undef 7399  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  df-polarityN 35189

This theorem is referenced by:  polsubN  35193  pol1N  35196  polpmapN  35198  2polvalN  35200  3polN  35202  poldmj1N  35214  pnonsingN  35219  ispsubcl2N  35233  polsubclN  35238  poml4N  35239