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Mirrors > Home > MPE Home > Th. List > Mathboxes > lpolfN | Structured version Visualization version GIF version |
Description: Functionality of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lpolf.v | ⊢ 𝑉 = (Base‘𝑊) |
lpolf.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lpolf.p | ⊢ 𝑃 = (LPol‘𝑊) |
lpolf.w | ⊢ (𝜑 → 𝑊 ∈ 𝑋) |
lpolf.o | ⊢ (𝜑 → ⊥ ∈ 𝑃) |
Ref | Expression |
---|---|
lpolfN | ⊢ (𝜑 → ⊥ :𝒫 𝑉⟶𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lpolf.o | . . 3 ⊢ (𝜑 → ⊥ ∈ 𝑃) | |
2 | lpolf.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝑋) | |
3 | lpolf.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
4 | lpolf.s | . . . . 5 ⊢ 𝑆 = (LSubSp‘𝑊) | |
5 | eqid 2738 | . . . . 5 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
6 | eqid 2738 | . . . . 5 ⊢ (LSAtoms‘𝑊) = (LSAtoms‘𝑊) | |
7 | eqid 2738 | . . . . 5 ⊢ (LSHyp‘𝑊) = (LSHyp‘𝑊) | |
8 | lpolf.p | . . . . 5 ⊢ 𝑃 = (LPol‘𝑊) | |
9 | 3, 4, 5, 6, 7, 8 | islpolN 39109 | . . . 4 ⊢ (𝑊 ∈ 𝑋 → ( ⊥ ∈ 𝑃 ↔ ( ⊥ :𝒫 𝑉⟶𝑆 ∧ (( ⊥ ‘𝑉) = {(0g‘𝑊)} ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑊)(( ⊥ ‘𝑥) ∈ (LSHyp‘𝑊) ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))))) |
10 | 2, 9 | syl 17 | . . 3 ⊢ (𝜑 → ( ⊥ ∈ 𝑃 ↔ ( ⊥ :𝒫 𝑉⟶𝑆 ∧ (( ⊥ ‘𝑉) = {(0g‘𝑊)} ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑊)(( ⊥ ‘𝑥) ∈ (LSHyp‘𝑊) ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))))) |
11 | 1, 10 | mpbid 235 | . 2 ⊢ (𝜑 → ( ⊥ :𝒫 𝑉⟶𝑆 ∧ (( ⊥ ‘𝑉) = {(0g‘𝑊)} ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑊)(( ⊥ ‘𝑥) ∈ (LSHyp‘𝑊) ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)))) |
12 | 11 | simpld 498 | 1 ⊢ (𝜑 → ⊥ :𝒫 𝑉⟶𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1088 ∀wal 1540 = wceq 1542 ∈ wcel 2113 ∀wral 3053 ⊆ wss 3841 𝒫 cpw 4485 {csn 4513 ⟶wf 6329 ‘cfv 6333 Basecbs 16579 0gc0g 16809 LSubSpclss 19815 LSAtomsclsa 36600 LSHypclsh 36601 LPolclpoN 39106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-sbc 3680 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-fv 6341 df-ov 7167 df-oprab 7168 df-mpo 7169 df-map 8432 df-lpolN 39107 |
This theorem is referenced by: (None) |
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