Theorem islpolN 42359

Description: The predicate "is a polarity". (Contributed by NM, 24-Nov-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
lpolset.v	⊢ 𝑉 = (Base‘𝑊)
lpolset.s	⊢ 𝑆 = (LSubSp‘𝑊)
lpolset.z	⊢ 0 = (0g‘𝑊)
lpolset.a	⊢ 𝐴 = (LSAtoms‘𝑊)
lpolset.h	⊢ 𝐻 = (LSHyp‘𝑊)
lpolset.p	⊢ 𝑃 = (LPol‘𝑊)

Assertion

Ref	Expression
islpolN	⊢ (𝑊 ∈ 𝑋 → ( ⊥ ∈ 𝑃 ↔ ( ⊥ :𝒫 𝑉⟶𝑆 ∧ (( ⊥ ‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)))))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝑊   𝑥, ⊥ ,𝑦

Allowed substitution hints:   𝐴(𝑦)   𝑃(𝑥, 𝑦)   𝑆(𝑥, 𝑦)   𝐻(𝑥, 𝑦)   𝑉(𝑥, 𝑦)   𝑋(𝑥, 𝑦)   0 (𝑥, 𝑦)

Proof of Theorem islpolN

Dummy variable 𝑜 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lpolset.v	. . . 4 ⊢ 𝑉 = (Base‘𝑊)
2		lpolset.s	. . . 4 ⊢ 𝑆 = (LSubSp‘𝑊)
3		lpolset.z	. . . 4 ⊢ 0 = (0g‘𝑊)
4		lpolset.a	. . . 4 ⊢ 𝐴 = (LSAtoms‘𝑊)
5		lpolset.h	. . . 4 ⊢ 𝐻 = (LSHyp‘𝑊)
6		lpolset.p	. . . 4 ⊢ 𝑃 = (LPol‘𝑊)
7	1, 2, 3, 4, 5, 6	lpolsetN 42358	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝑃 = {𝑜 ∈ (𝑆 ↑m 𝒫 𝑉) ∣ ((𝑜‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 ((𝑜‘𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥))})
8	7	eleq2d 2846	. 2 ⊢ (𝑊 ∈ 𝑋 → ( ⊥ ∈ 𝑃 ↔ ⊥ ∈ {𝑜 ∈ (𝑆 ↑m 𝒫 𝑉) ∣ ((𝑜‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 ((𝑜‘𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥))}))
9		fveq1 6878	. . . . . 6 ⊢ (𝑜 = ⊥ → (𝑜‘𝑉) = ( ⊥ ‘𝑉))
10	9	eqeq1d 2762	. . . . 5 ⊢ (𝑜 = ⊥ → ((𝑜‘𝑉) = { 0 } ↔ ( ⊥ ‘𝑉) = { 0 }))
11		fveq1 6878	. . . . . . . 8 ⊢ (𝑜 = ⊥ → (𝑜‘𝑦) = ( ⊥ ‘𝑦))
12		fveq1 6878	. . . . . . . 8 ⊢ (𝑜 = ⊥ → (𝑜‘𝑥) = ( ⊥ ‘𝑥))
13	11, 12	sseq12d 3964	. . . . . . 7 ⊢ (𝑜 = ⊥ → ((𝑜‘𝑦) ⊆ (𝑜‘𝑥) ↔ ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)))
14	13	imbi2d 343	. . . . . 6 ⊢ (𝑜 = ⊥ → (((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥)) ↔ ((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥))))
15	14	2albidv 1956	. . . . 5 ⊢ (𝑜 = ⊥ → (∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥)) ↔ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥))))
16	12	eleq1d 2845	. . . . . . 7 ⊢ (𝑜 = ⊥ → ((𝑜‘𝑥) ∈ 𝐻 ↔ ( ⊥ ‘𝑥) ∈ 𝐻))
17		id 23	. . . . . . . . 9 ⊢ (𝑜 = ⊥ → 𝑜 = ⊥ )
18	17, 12	fveq12d 6886	. . . . . . . 8 ⊢ (𝑜 = ⊥ → (𝑜‘(𝑜‘𝑥)) = ( ⊥ ‘( ⊥ ‘𝑥)))
19	18	eqeq1d 2762	. . . . . . 7 ⊢ (𝑜 = ⊥ → ((𝑜‘(𝑜‘𝑥)) = 𝑥 ↔ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))
20	16, 19	anbi12d 644	. . . . . 6 ⊢ (𝑜 = ⊥ → (((𝑜‘𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥) ↔ (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)))
21	20	ralbidv 3185	. . . . 5 ⊢ (𝑜 = ⊥ → (∀𝑥 ∈ 𝐴 ((𝑜‘𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥) ↔ ∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)))
22	10, 15, 21	3anbi123d 1464	. . . 4 ⊢ (𝑜 = ⊥ → (((𝑜‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 ((𝑜‘𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥)) ↔ (( ⊥ ‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))))
23	22	elrab 3645	. . 3 ⊢ ( ⊥ ∈ {𝑜 ∈ (𝑆 ↑m 𝒫 𝑉) ∣ ((𝑜‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 ((𝑜‘𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥))} ↔ ( ⊥ ∈ (𝑆 ↑m 𝒫 𝑉) ∧ (( ⊥ ‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))))
24	2	fvexi 6893	. . . . 5 ⊢ 𝑆 ∈ V
25	1	fvexi 6893	. . . . . 6 ⊢ 𝑉 ∈ V
26	25	pwex 5345	. . . . 5 ⊢ 𝒫 𝑉 ∈ V
27	24, 26	elmap 8881	. . . 4 ⊢ ( ⊥ ∈ (𝑆 ↑m 𝒫 𝑉) ↔ ⊥ :𝒫 𝑉⟶𝑆)
28	27	anbi1i 636	. . 3 ⊢ (( ⊥ ∈ (𝑆 ↑m 𝒫 𝑉) ∧ (( ⊥ ‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))) ↔ ( ⊥ :𝒫 𝑉⟶𝑆 ∧ (( ⊥ ‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))))
29	23, 28	bitri 278	. 2 ⊢ ( ⊥ ∈ {𝑜 ∈ (𝑆 ↑m 𝒫 𝑉) ∣ ((𝑜‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 ((𝑜‘𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥))} ↔ ( ⊥ :𝒫 𝑉⟶𝑆 ∧ (( ⊥ ‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))))
30	8, 29	bitrdi 290	1 ⊢ (𝑊 ∈ 𝑋 → ( ⊥ ∈ 𝑃 ↔ ( ⊥ :𝒫 𝑉⟶𝑆 ∧ (( ⊥ ‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2145  ∀wral 3076  {crab 3412   ⊆ wss 3899  𝒫 cpw 4557  {csn 4584  ⟶wf 6529  ‘cfv 6533  (class class class)co 7414   ↑_m cmap 8829  Basecbs 17304  0_gc0g 17527  LSubSpclss 21118  LSAtomsclsa 39850  LSHypclsh 39851  LPolclpoN 42356

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-map 8831  df-lpolN 42357

This theorem is used by:  islpoldN  42360  lpolfN  42361  lpolvN  42362  lpolconN  42363  lpolsatN  42364  lpolpolsatN  42365