Theorem islpoldN 40658

Description: Properties that determine a polarity. (Contributed by NM, 26-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)
islpold.w	⊢ (𝜑 → 𝑊 ∈ 𝑋)
islpold.1	⊢ (𝜑 → ⊥ :𝒫 𝑉⟶𝑆)
islpold.2	⊢ (𝜑 → ( ⊥ ‘𝑉) = { 0 })
islpold.3	⊢ ((𝜑 ∧ (𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦)) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥))
islpold.4	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ( ⊥ ‘𝑥) ∈ 𝐻)
islpold.5	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)

Assertion

Ref	Expression
islpoldN	⊢ (𝜑 → ⊥ ∈ 𝑃)

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

Step	Hyp	Ref	Expression
1		islpold.1	. 2 ⊢ (𝜑 → ⊥ :𝒫 𝑉⟶𝑆)
2		islpold.2	. . 3 ⊢ (𝜑 → ( ⊥ ‘𝑉) = { 0 })
3		islpold.3	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦)) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥))
4	3	ex 413	. . . 4 ⊢ (𝜑 → ((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)))
5	4	alrimivv 1931	. . 3 ⊢ (𝜑 → ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)))
6		islpold.4	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ( ⊥ ‘𝑥) ∈ 𝐻)
7		islpold.5	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)
8	6, 7	jca 512	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))
9	8	ralrimiva 3146	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))
10	2, 5, 9	3jca 1128	. 2 ⊢ (𝜑 → (( ⊥ ‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)))
11		islpold.w	. . 3 ⊢ (𝜑 → 𝑊 ∈ 𝑋)
12		lpolset.v	. . . 4 ⊢ 𝑉 = (Base‘𝑊)
13		lpolset.s	. . . 4 ⊢ 𝑆 = (LSubSp‘𝑊)
14		lpolset.z	. . . 4 ⊢ 0 = (0g‘𝑊)
15		lpolset.a	. . . 4 ⊢ 𝐴 = (LSAtoms‘𝑊)
16		lpolset.h	. . . 4 ⊢ 𝐻 = (LSHyp‘𝑊)
17		lpolset.p	. . . 4 ⊢ 𝑃 = (LPol‘𝑊)
18	12, 13, 14, 15, 16, 17	islpolN 40657	. . 3 ⊢ (𝑊 ∈ 𝑋 → ( ⊥ ∈ 𝑃 ↔ ( ⊥ :𝒫 𝑉⟶𝑆 ∧ (( ⊥ ‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)))))
19	11, 18	syl 17	. 2 ⊢ (𝜑 → ( ⊥ ∈ 𝑃 ↔ ( ⊥ :𝒫 𝑉⟶𝑆 ∧ (( ⊥ ‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)))))
20	1, 10, 19	mpbir2and 711	1 ⊢ (𝜑 → ⊥ ∈ 𝑃)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087  ∀wal 1539   = wceq 1541   ∈ wcel 2106  ∀wral 3061   ⊆ wss 3948  𝒫 cpw 4602  {csn 4628  ⟶wf 6539  ‘cfv 6543  Basecbs 17148  0_gc0g 17389  LSubSpclss 20686  LSAtomsclsa 38147  LSHypclsh 38148  LPolclpoN 40654

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-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727

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-rab 3433  df-v 3476  df-sbc 3778  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-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-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-map 8824  df-lpolN 40655

This theorem is referenced by:  dochpolN  40664