Theorem polval2N 40105

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 40104	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑃‘𝑋) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (𝑀‘( ⊥ ‘𝑝))))
6		hlop 39561	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
7	6	ad2antrr 726	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑋) → 𝐾 ∈ OP)
8		ssel2 3926	. . . . . . 7 ⊢ ((𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝑋) → 𝑝 ∈ 𝐴)
9	8	adantll 714	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑋) → 𝑝 ∈ 𝐴)
10		eqid 2734	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
11	10, 2	atbase 39488	. . . . . 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 39393	. . . . 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 3126	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ∀𝑝 ∈ 𝑋 ( ⊥ ‘𝑝) ∈ (Base‘𝐾))
16		eqid 2734	. . . 4 ⊢ (glb‘𝐾) = (glb‘𝐾)
17	10, 16, 2, 3	pmapglb2xN 39971	. . 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 39577	. . . . 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 39394	. . . . . . . . . . 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 2745	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑋) → (𝑥 = ( ⊥ ‘( ⊥ ‘𝑝)) ↔ 𝑥 = 𝑝))
25	24	rexbidva 3156	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥ ‘𝑝)) ↔ ∃𝑝 ∈ 𝑋 𝑥 = 𝑝))
26	25	abbidv 2800	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → {𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥ ‘𝑝))} = {𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = 𝑝})
27		df-rex 3059	. . . . . . . . . 10 ⊢ (∃𝑝 ∈ 𝑋 𝑥 = 𝑝 ↔ ∃𝑝(𝑝 ∈ 𝑋 ∧ 𝑥 = 𝑝))
28		equcom 2019	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑝 ↔ 𝑝 = 𝑥)
29	28	anbi1ci 626	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ 𝑋 ∧ 𝑥 = 𝑝) ↔ (𝑝 = 𝑥 ∧ 𝑝 ∈ 𝑋))
30	29	exbii 1849	. . . . . . . . . 10 ⊢ (∃𝑝(𝑝 ∈ 𝑋 ∧ 𝑥 = 𝑝) ↔ ∃𝑝(𝑝 = 𝑥 ∧ 𝑝 ∈ 𝑋))
31		eleq1w 2817	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑥 → (𝑝 ∈ 𝑋 ↔ 𝑥 ∈ 𝑋))
32	31	equsexvw 2006	. . . . . . . . . 10 ⊢ (∃𝑝(𝑝 = 𝑥 ∧ 𝑝 ∈ 𝑋) ↔ 𝑥 ∈ 𝑋)
33	27, 30, 32	3bitri 297	. . . . . . . . 9 ⊢ (∃𝑝 ∈ 𝑋 𝑥 = 𝑝 ↔ 𝑥 ∈ 𝑋)
34	33	abbii 2801	. . . . . . . 8 ⊢ {𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = 𝑝} = {𝑥 ∣ 𝑥 ∈ 𝑋}
35		abid2 2871	. . . . . . . 8 ⊢ {𝑥 ∣ 𝑥 ∈ 𝑋} = 𝑋
36	34, 35	eqtri 2757	. . . . . . 7 ⊢ {𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = 𝑝} = 𝑋
37	26, 36	eqtrdi 2785	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → {𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥ ‘𝑝))} = 𝑋)
38	37	fveq2d 6836	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑈‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥ ‘𝑝))}) = (𝑈‘𝑋))
39	38	fveq2d 6836	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘(𝑈‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥ ‘𝑝))})) = ( ⊥ ‘(𝑈‘𝑋)))
40	21, 39	eqtrd 2769	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((glb‘𝐾)‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘𝑝)}) = ( ⊥ ‘(𝑈‘𝑋)))
41	40	fveq2d 6836	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑀‘((glb‘𝐾)‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘𝑝)})) = (𝑀‘( ⊥ ‘(𝑈‘𝑋))))
42	5, 18, 41	3eqtr2d 2775	1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑃‘𝑋) = (𝑀‘( ⊥ ‘(𝑈‘𝑋))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  {cab 2712  ∀wral 3049  ∃wrex 3058   ∩ cin 3898   ⊆ wss 3899  ∩ ciin 4945  ‘cfv 6490  Basecbs 17134  occoc 17183  lubclub 18230  glbcglb 18231  OPcops 39371  Atomscatm 39462  HLchlt 39549  pmapcpmap 39696  ⊥_𝑃cpolN 40101

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-iin 4947  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-proset 18215  df-poset 18234  df-lub 18265  df-glb 18266  df-join 18267  df-meet 18268  df-p1 18345  df-lat 18353  df-clat 18420  df-oposet 39375  df-ol 39377  df-oml 39378  df-ats 39466  df-hlat 39550  df-pmap 39703  df-polarityN 40102

This theorem is referenced by:  polsubN  40106  pol1N  40109  polpmapN  40111  2polvalN  40113  3polN  40115  poldmj1N  40127  pnonsingN  40132  ispsubcl2N  40146  polsubclN  40151  poml4N  40152