Theorem polval2N 37927

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 ∈ HL ∧ 𝑋 ⊆ 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 37926	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑃‘𝑋) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (𝑀‘( ⊥ ‘𝑝))))
6		hlop 37383	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
7	6	ad2antrr 723	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑋) → 𝐾 ∈ OP)
8		ssel2 3917	. . . . . . 7 ⊢ ((𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝑋) → 𝑝 ∈ 𝐴)
9	8	adantll 711	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑋) → 𝑝 ∈ 𝐴)
10		eqid 2739	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
11	10, 2	atbase 37310	. . . . . 6 ⊢ (𝑝 ∈ 𝐴 → 𝑝 ∈ (Base‘𝐾))
12	9, 11	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑋) → 𝑝 ∈ (Base‘𝐾))
13	10, 1	opoccl 37215	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → ( ⊥ ‘𝑝) ∈ (Base‘𝐾))
14	7, 12, 13	syl2anc 584	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑋) → ( ⊥ ‘𝑝) ∈ (Base‘𝐾))
15	14	ralrimiva 3104	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ∀𝑝 ∈ 𝑋 ( ⊥ ‘𝑝) ∈ (Base‘𝐾))
16		eqid 2739	. . . 4 ⊢ (glb‘𝐾) = (glb‘𝐾)
17	10, 16, 2, 3	pmapglb2xN 37793	. . 3 ⊢ ((𝐾 ∈ HL ∧ ∀𝑝 ∈ 𝑋 ( ⊥ ‘𝑝) ∈ (Base‘𝐾)) → (𝑀‘((glb‘𝐾)‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘𝑝)})) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (𝑀‘( ⊥ ‘𝑝))))
18	15, 17	syldan 591	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑀‘((glb‘𝐾)‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘𝑝)})) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (𝑀‘( ⊥ ‘𝑝))))
19		polval2.u	. . . . . 6 ⊢ 𝑈 = (lub‘𝐾)
20	10, 19, 16, 1	glbconxN 37399	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ ∀𝑝 ∈ 𝑋 ( ⊥ ‘𝑝) ∈ (Base‘𝐾)) → ((glb‘𝐾)‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘𝑝)}) = ( ⊥ ‘(𝑈‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥ ‘𝑝))})))
21	15, 20	syldan 591	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((glb‘𝐾)‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘𝑝)}) = ( ⊥ ‘(𝑈‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥ ‘𝑝))})))
22	10, 1	opococ 37216	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → ( ⊥ ‘( ⊥ ‘𝑝)) = 𝑝)
23	7, 12, 22	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑋) → ( ⊥ ‘( ⊥ ‘𝑝)) = 𝑝)
24	23	eqeq2d 2750	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑋) → (𝑥 = ( ⊥ ‘( ⊥ ‘𝑝)) ↔ 𝑥 = 𝑝))
25	24	rexbidva 3226	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥ ‘𝑝)) ↔ ∃𝑝 ∈ 𝑋 𝑥 = 𝑝))
26	25	abbidv 2808	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → {𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥ ‘𝑝))} = {𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = 𝑝})
27		df-rex 3071	. . . . . . . . . 10 ⊢ (∃𝑝 ∈ 𝑋 𝑥 = 𝑝 ↔ ∃𝑝(𝑝 ∈ 𝑋 ∧ 𝑥 = 𝑝))
28		equcom 2022	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑝 ↔ 𝑝 = 𝑥)
29	28	anbi1ci 626	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ 𝑋 ∧ 𝑥 = 𝑝) ↔ (𝑝 = 𝑥 ∧ 𝑝 ∈ 𝑋))
30	29	exbii 1851	. . . . . . . . . 10 ⊢ (∃𝑝(𝑝 ∈ 𝑋 ∧ 𝑥 = 𝑝) ↔ ∃𝑝(𝑝 = 𝑥 ∧ 𝑝 ∈ 𝑋))
31		eleq1w 2822	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑥 → (𝑝 ∈ 𝑋 ↔ 𝑥 ∈ 𝑋))
32	31	equsexvw 2009	. . . . . . . . . 10 ⊢ (∃𝑝(𝑝 = 𝑥 ∧ 𝑝 ∈ 𝑋) ↔ 𝑥 ∈ 𝑋)
33	27, 30, 32	3bitri 297	. . . . . . . . 9 ⊢ (∃𝑝 ∈ 𝑋 𝑥 = 𝑝 ↔ 𝑥 ∈ 𝑋)
34	33	abbii 2809	. . . . . . . 8 ⊢ {𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = 𝑝} = {𝑥 ∣ 𝑥 ∈ 𝑋}
35		abid2 2883	. . . . . . . 8 ⊢ {𝑥 ∣ 𝑥 ∈ 𝑋} = 𝑋
36	34, 35	eqtri 2767	. . . . . . 7 ⊢ {𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = 𝑝} = 𝑋
37	26, 36	eqtrdi 2795	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → {𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥ ‘𝑝))} = 𝑋)
38	37	fveq2d 6787	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑈‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥ ‘𝑝))}) = (𝑈‘𝑋))
39	38	fveq2d 6787	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘(𝑈‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥ ‘𝑝))})) = ( ⊥ ‘(𝑈‘𝑋)))
40	21, 39	eqtrd 2779	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((glb‘𝐾)‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘𝑝)}) = ( ⊥ ‘(𝑈‘𝑋)))
41	40	fveq2d 6787	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑀‘((glb‘𝐾)‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘𝑝)})) = (𝑀‘( ⊥ ‘(𝑈‘𝑋))))
42	5, 18, 41	3eqtr2d 2785	1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑃‘𝑋) = (𝑀‘( ⊥ ‘(𝑈‘𝑋))))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2107  {cab 2716  ∀wral 3065  ∃wrex 3066   ∩ cin 3887   ⊆ wss 3888  ∩ ciin 4926  ‘cfv 6437  Basecbs 16921  occoc 16979  lubclub 18036  glbcglb 18037  OPcops 37193  Atomscatm 37284  HLchlt 37371  pmapcpmap 37518  ⊥_𝑃cpolN 37923

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2710  ax-rep 5210  ax-sep 5224  ax-nul 5231  ax-pow 5289  ax-pr 5353  ax-un 7597  ax-riotaBAD 36974

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-rmo 3072  df-reu 3073  df-rab 3074  df-v 3435  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-iun 4927  df-iin 4928  df-br 5076  df-opab 5138  df-mpt 5159  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6395  df-fun 6439  df-fn 6440  df-f 6441  df-f1 6442  df-fo 6443  df-f1o 6444  df-fv 6445  df-riota 7241  df-ov 7287  df-oprab 7288  df-undef 8098  df-proset 18022  df-poset 18040  df-lub 18073  df-glb 18074  df-join 18075  df-meet 18076  df-p1 18153  df-lat 18159  df-clat 18226  df-oposet 37197  df-ol 37199  df-oml 37200  df-ats 37288  df-hlat 37372  df-pmap 37525  df-polarityN 37924

This theorem is referenced by:  polsubN  37928  pol1N  37931  polpmapN  37933  2polvalN  37935  3polN  37937  poldmj1N  37949  pnonsingN  37954  ispsubcl2N  37968  polsubclN  37973  poml4N  37974