Theorem polval2N 40527

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 40526	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑃‘𝑋) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (𝑀‘( ⊥ ‘𝑝))))
6		hlop 39983	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
7	6	ad2antrr 736	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑋) → 𝐾 ∈ OP)
8		ssel2 3931	. . . . . . 7 ⊢ ((𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝑋) → 𝑝 ∈ 𝐴)
9	8	adantll 724	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑋) → 𝑝 ∈ 𝐴)
10		eqid 2762	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
11	10, 2	atbase 39910	. . . . . 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 39815	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → ( ⊥ ‘𝑝) ∈ (Base‘𝐾))
14	7, 12, 13	syl2anc 593	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑋) → ( ⊥ ‘𝑝) ∈ (Base‘𝐾))
15	14	ralrimiva 3154	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ∀𝑝 ∈ 𝑋 ( ⊥ ‘𝑝) ∈ (Base‘𝐾))
16		eqid 2762	. . . 4 ⊢ (glb‘𝐾) = (glb‘𝐾)
17	10, 16, 2, 3	pmapglb2xN 40393	. . 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 600	. 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 39999	. . . . 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 600	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((glb‘𝐾)‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘𝑝)}) = ( ⊥ ‘(𝑈‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥ ‘𝑝))})))
22	10, 1	opococ 39816	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → ( ⊥ ‘( ⊥ ‘𝑝)) = 𝑝)
23	7, 12, 22	syl2anc 593	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑋) → ( ⊥ ‘( ⊥ ‘𝑝)) = 𝑝)
24	23	eqeq2d 2773	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑋) → (𝑥 = ( ⊥ ‘( ⊥ ‘𝑝)) ↔ 𝑥 = 𝑝))
25	24	rexbidva 3184	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥ ‘𝑝)) ↔ ∃𝑝 ∈ 𝑋 𝑥 = 𝑝))
26	25	abbidv 2828	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → {𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥ ‘𝑝))} = {𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = 𝑝})
27		df-rex 3087	. . . . . . . . . 10 ⊢ (∃𝑝 ∈ 𝑋 𝑥 = 𝑝 ↔ ∃𝑝(𝑝 ∈ 𝑋 ∧ 𝑥 = 𝑝))
28		equcom 2038	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑝 ↔ 𝑝 = 𝑥)
29	28	anbi1ci 635	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ 𝑋 ∧ 𝑥 = 𝑝) ↔ (𝑝 = 𝑥 ∧ 𝑝 ∈ 𝑋))
30	29	exbii 1868	. . . . . . . . . 10 ⊢ (∃𝑝(𝑝 ∈ 𝑋 ∧ 𝑥 = 𝑝) ↔ ∃𝑝(𝑝 = 𝑥 ∧ 𝑝 ∈ 𝑋))
31		eleq1w 2845	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑥 → (𝑝 ∈ 𝑋 ↔ 𝑥 ∈ 𝑋))
32	31	equsexvw 2025	. . . . . . . . . 10 ⊢ (∃𝑝(𝑝 = 𝑥 ∧ 𝑝 ∈ 𝑋) ↔ 𝑥 ∈ 𝑋)
33	27, 30, 32	3bitri 299	. . . . . . . . 9 ⊢ (∃𝑝 ∈ 𝑋 𝑥 = 𝑝 ↔ 𝑥 ∈ 𝑋)
34	33	abbii 2829	. . . . . . . 8 ⊢ {𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = 𝑝} = {𝑥 ∣ 𝑥 ∈ 𝑋}
35		abid2 2899	. . . . . . . 8 ⊢ {𝑥 ∣ 𝑥 ∈ 𝑋} = 𝑋
36	34, 35	eqtri 2785	. . . . . . 7 ⊢ {𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = 𝑝} = 𝑋
37	26, 36	eqtrdi 2813	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → {𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥ ‘𝑝))} = 𝑋)
38	37	fveq2d 6871	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑈‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥ ‘𝑝))}) = (𝑈‘𝑋))
39	38	fveq2d 6871	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘(𝑈‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥ ‘𝑝))})) = ( ⊥ ‘(𝑈‘𝑋)))
40	21, 39	eqtrd 2797	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((glb‘𝐾)‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘𝑝)}) = ( ⊥ ‘(𝑈‘𝑋)))
41	40	fveq2d 6871	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑀‘((glb‘𝐾)‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘𝑝)})) = (𝑀‘( ⊥ ‘(𝑈‘𝑋))))
42	5, 18, 41	3eqtr2d 2803	1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑃‘𝑋) = (𝑀‘( ⊥ ‘(𝑈‘𝑋))))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560  ∃wex 1799   ∈ wcel 2142  {cab 2740  ∀wral 3076  ∃wrex 3086   ∩ cin 3903   ⊆ wss 3904  ∩ ciin 4950  ‘cfv 6521  Basecbs 17245  occoc 17294  lubclub 18341  glbcglb 18342  OPcops 39793  Atomscatm 39884  HLchlt 39971  pmapcpmap 40118  ⊥_𝑃cpolN 40523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-iin 4952  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-proset 18326  df-poset 18345  df-lub 18376  df-glb 18377  df-join 18378  df-meet 18379  df-p1 18456  df-lat 18464  df-clat 18531  df-oposet 39797  df-ol 39799  df-oml 39800  df-ats 39888  df-hlat 39972  df-pmap 40125  df-polarityN 40524

This theorem is referenced by:  polsubN  40528  pol1N  40531  polpmapN  40533  2polvalN  40535  3polN  40537  poldmj1N  40549  pnonsingN  40554  ispsubcl2N  40568  polsubclN  40573  poml4N  40574