Theorem polval2N 39081

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 39080	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑃‘𝑋) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (𝑀‘( ⊥ ‘𝑝))))
6		hlop 38536	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
7	6	ad2antrr 723	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑋) → 𝐾 ∈ OP)
8		ssel2 3977	. . . . . . 7 ⊢ ((𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝑋) → 𝑝 ∈ 𝐴)
9	8	adantll 711	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑋) → 𝑝 ∈ 𝐴)
10		eqid 2731	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
11	10, 2	atbase 38463	. . . . . 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 38368	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → ( ⊥ ‘𝑝) ∈ (Base‘𝐾))
14	7, 12, 13	syl2anc 583	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑋) → ( ⊥ ‘𝑝) ∈ (Base‘𝐾))
15	14	ralrimiva 3145	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ∀𝑝 ∈ 𝑋 ( ⊥ ‘𝑝) ∈ (Base‘𝐾))
16		eqid 2731	. . . 4 ⊢ (glb‘𝐾) = (glb‘𝐾)
17	10, 16, 2, 3	pmapglb2xN 38947	. . 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 590	. 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 38553	. . . . 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 590	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((glb‘𝐾)‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘𝑝)}) = ( ⊥ ‘(𝑈‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥ ‘𝑝))})))
22	10, 1	opococ 38369	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → ( ⊥ ‘( ⊥ ‘𝑝)) = 𝑝)
23	7, 12, 22	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑋) → ( ⊥ ‘( ⊥ ‘𝑝)) = 𝑝)
24	23	eqeq2d 2742	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑋) → (𝑥 = ( ⊥ ‘( ⊥ ‘𝑝)) ↔ 𝑥 = 𝑝))
25	24	rexbidva 3175	. . . . . . . 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 3070	. . . . . . . . . 10 ⊢ (∃𝑝 ∈ 𝑋 𝑥 = 𝑝 ↔ ∃𝑝(𝑝 ∈ 𝑋 ∧ 𝑥 = 𝑝))
28		equcom 2020	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑝 ↔ 𝑝 = 𝑥)
29	28	anbi1ci 625	. . . . . . . . . . 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 2815	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑥 → (𝑝 ∈ 𝑋 ↔ 𝑥 ∈ 𝑋))
32	31	equsexvw 2007	. . . . . . . . . 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 2870	. . . . . . . 8 ⊢ {𝑥 ∣ 𝑥 ∈ 𝑋} = 𝑋
36	34, 35	eqtri 2759	. . . . . . 7 ⊢ {𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = 𝑝} = 𝑋
37	26, 36	eqtrdi 2787	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → {𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥ ‘𝑝))} = 𝑋)
38	37	fveq2d 6895	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑈‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥ ‘𝑝))}) = (𝑈‘𝑋))
39	38	fveq2d 6895	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘(𝑈‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥ ‘𝑝))})) = ( ⊥ ‘(𝑈‘𝑋)))
40	21, 39	eqtrd 2771	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((glb‘𝐾)‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘𝑝)}) = ( ⊥ ‘(𝑈‘𝑋)))
41	40	fveq2d 6895	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑀‘((glb‘𝐾)‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘𝑝)})) = (𝑀‘( ⊥ ‘(𝑈‘𝑋))))
42	5, 18, 41	3eqtr2d 2777	1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑃‘𝑋) = (𝑀‘( ⊥ ‘(𝑈‘𝑋))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1780   ∈ wcel 2105  {cab 2708  ∀wral 3060  ∃wrex 3069   ∩ cin 3947   ⊆ wss 3948  ∩ ciin 4998  ‘cfv 6543  Basecbs 17149  occoc 17210  lubclub 18267  glbcglb 18268  OPcops 38346  Atomscatm 38437  HLchlt 38524  pmapcpmap 38672  ⊥_𝑃cpolN 39077

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7728

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-proset 18253  df-poset 18271  df-lub 18304  df-glb 18305  df-join 18306  df-meet 18307  df-p1 18384  df-lat 18390  df-clat 18457  df-oposet 38350  df-ol 38352  df-oml 38353  df-ats 38441  df-hlat 38525  df-pmap 38679  df-polarityN 39078

This theorem is referenced by:  polsubN  39082  pol1N  39085  polpmapN  39087  2polvalN  39089  3polN  39091  poldmj1N  39103  pnonsingN  39108  ispsubcl2N  39122  polsubclN  39127  poml4N  39128