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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 2821 | . . . . 5 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
6 | eqid 2821 | . . . . 5 ⊢ (LSAtoms‘𝑊) = (LSAtoms‘𝑊) | |
7 | eqid 2821 | . . . . 5 ⊢ (LSHyp‘𝑊) = (LSHyp‘𝑊) | |
8 | lpolf.p | . . . . 5 ⊢ 𝑃 = (LPol‘𝑊) | |
9 | 3, 4, 5, 6, 7, 8 | islpolN 38613 | . . . 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 234 | . 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 497 | 1 ⊢ (𝜑 → ⊥ :𝒫 𝑉⟶𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 ∀wal 1531 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ⊆ wss 3935 𝒫 cpw 4538 {csn 4560 ⟶wf 6345 ‘cfv 6349 Basecbs 16477 0gc0g 16707 LSubSpclss 19697 LSAtomsclsa 36104 LSHypclsh 36105 LPolclpoN 38610 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-map 8402 df-lpolN 38611 |
This theorem is referenced by: (None) |
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