Theorem polval2N 38765

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 38764	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑃‘𝑋) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (𝑀‘( ⊥ ‘𝑝))))
6		hlop 38220	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
7	6	ad2antrr 724	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑋) → 𝐾 ∈ OP)
8		ssel2 3976	. . . . . . 7 ⊢ ((𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝑋) → 𝑝 ∈ 𝐴)
9	8	adantll 712	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑋) → 𝑝 ∈ 𝐴)
10		eqid 2732	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
11	10, 2	atbase 38147	. . . . . 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 38052	. . . . 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 3146	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ∀𝑝 ∈ 𝑋 ( ⊥ ‘𝑝) ∈ (Base‘𝐾))
16		eqid 2732	. . . 4 ⊢ (glb‘𝐾) = (glb‘𝐾)
17	10, 16, 2, 3	pmapglb2xN 38631	. . 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 38237	. . . . 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 38053	. . . . . . . . . . 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 2743	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑋) → (𝑥 = ( ⊥ ‘( ⊥ ‘𝑝)) ↔ 𝑥 = 𝑝))
25	24	rexbidva 3176	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥ ‘𝑝)) ↔ ∃𝑝 ∈ 𝑋 𝑥 = 𝑝))
26	25	abbidv 2801	. . . . . . 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 2021	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑝 ↔ 𝑝 = 𝑥)
29	28	anbi1ci 626	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ 𝑋 ∧ 𝑥 = 𝑝) ↔ (𝑝 = 𝑥 ∧ 𝑝 ∈ 𝑋))
30	29	exbii 1850	. . . . . . . . . 10 ⊢ (∃𝑝(𝑝 ∈ 𝑋 ∧ 𝑥 = 𝑝) ↔ ∃𝑝(𝑝 = 𝑥 ∧ 𝑝 ∈ 𝑋))
31		eleq1w 2816	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑥 → (𝑝 ∈ 𝑋 ↔ 𝑥 ∈ 𝑋))
32	31	equsexvw 2008	. . . . . . . . . 10 ⊢ (∃𝑝(𝑝 = 𝑥 ∧ 𝑝 ∈ 𝑋) ↔ 𝑥 ∈ 𝑋)
33	27, 30, 32	3bitri 296	. . . . . . . . 9 ⊢ (∃𝑝 ∈ 𝑋 𝑥 = 𝑝 ↔ 𝑥 ∈ 𝑋)
34	33	abbii 2802	. . . . . . . 8 ⊢ {𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = 𝑝} = {𝑥 ∣ 𝑥 ∈ 𝑋}
35		abid2 2871	. . . . . . . 8 ⊢ {𝑥 ∣ 𝑥 ∈ 𝑋} = 𝑋
36	34, 35	eqtri 2760	. . . . . . 7 ⊢ {𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = 𝑝} = 𝑋
37	26, 36	eqtrdi 2788	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → {𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥ ‘𝑝))} = 𝑋)
38	37	fveq2d 6892	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑈‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥ ‘𝑝))}) = (𝑈‘𝑋))
39	38	fveq2d 6892	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘(𝑈‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘( ⊥ ‘𝑝))})) = ( ⊥ ‘(𝑈‘𝑋)))
40	21, 39	eqtrd 2772	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((glb‘𝐾)‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘𝑝)}) = ( ⊥ ‘(𝑈‘𝑋)))
41	40	fveq2d 6892	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑀‘((glb‘𝐾)‘{𝑥 ∣ ∃𝑝 ∈ 𝑋 𝑥 = ( ⊥ ‘𝑝)})) = (𝑀‘( ⊥ ‘(𝑈‘𝑋))))
42	5, 18, 41	3eqtr2d 2778	1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑃‘𝑋) = (𝑀‘( ⊥ ‘(𝑈‘𝑋))))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  {cab 2709  ∀wral 3061  ∃wrex 3070   ∩ cin 3946   ⊆ wss 3947  ∩ ciin 4997  ‘cfv 6540  Basecbs 17140  occoc 17201  lubclub 18258  glbcglb 18259  OPcops 38030  Atomscatm 38121  HLchlt 38208  pmapcpmap 38356  ⊥_𝑃cpolN 38761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-proset 18244  df-poset 18262  df-lub 18295  df-glb 18296  df-join 18297  df-meet 18298  df-p1 18375  df-lat 18381  df-clat 18448  df-oposet 38034  df-ol 38036  df-oml 38037  df-ats 38125  df-hlat 38209  df-pmap 38363  df-polarityN 38762

This theorem is referenced by:  polsubN  38766  pol1N  38769  polpmapN  38771  2polvalN  38773  3polN  38775  poldmj1N  38787  pnonsingN  38792  ispsubcl2N  38806  polsubclN  38811  poml4N  38812