Theorem islpoldN 41473

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 412	. . . 4 ⊢ (𝜑 → ((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)))
5	4	alrimivv 1928	. . 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 511	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))
9	8	ralrimiva 3121	. . 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 41472	. . 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 713	1 ⊢ (𝜑 → ⊥ ∈ 𝑃)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1538   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ⊆ wss 3903  𝒫 cpw 4551  {csn 4577  ⟶wf 6478  ‘cfv 6482  Basecbs 17120  0_gc0g 17343  LSubSpclss 20834  LSAtomsclsa 38963  LSHypclsh 38964  LPolclpoN 41469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-map 8755  df-lpolN 41470

This theorem is referenced by:  dochpolN  41479