Theorem islpoldN 42342

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 418	. . . 4 ⊢ (𝜑 → ((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)))
5	4	alrimivv 1961	. . 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 521	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))
9	8	ralrimiva 3156	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))
10	2, 5, 9	3jca 1146	. 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 42341	. . 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 18	. 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 726	1 ⊢ (𝜑 → ⊥ ∈ 𝑃)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2145  ∀wral 3078   ⊆ wss 3902  𝒫 cpw 4560  {csn 4587  ⟶wf 6533  ‘cfv 6537  Basecbs 17303  0_gc0g 17526  LSubSpclss 21114  LSAtomsclsa 39832  LSHypclsh 39833  LPolclpoN 42338

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 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-map 8831  df-lpolN 42339

This theorem is used by:  dochpolN  42348