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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 38634 | . . . 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 1535 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ⊆ wss 3936 𝒫 cpw 4539 {csn 4567 ⟶wf 6351 ‘cfv 6355 Basecbs 16483 0gc0g 16713 LSubSpclss 19703 LSAtomsclsa 36125 LSHypclsh 36126 LPolclpoN 38631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-map 8408 df-lpolN 38632 |
This theorem is referenced by: (None) |
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