Theorem islpoldN 35791

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 449	. . . 4 ⊢ (𝜑 → ((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)))
5	4	alrimivv 1843	. . 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 553	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))
9	8	ralrimiva 2949	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))
10	2, 5, 9	3jca 1235	. 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 35790	. . 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 959	1 ⊢ (𝜑 → ⊥ ∈ 𝑃)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125  ⟶wf 5800  ‘cfv 5804  Basecbs 15695  0_gc0g 15923  LSubSpclss 18753  LSAtomsclsa 33279  LSHypclsh 33280  LPolclpoN 35787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-lpolN 35788

This theorem is referenced by:  dochpolN  35797