Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lpolvN | Structured version Visualization version GIF version |
Description: The polarity of the whole space is the zero subspace. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lpolv.v | ⊢ 𝑉 = (Base‘𝑊) |
lpolv.z | ⊢ 0 = (0g‘𝑊) |
lpolv.p | ⊢ 𝑃 = (LPol‘𝑊) |
lpolv.w | ⊢ (𝜑 → 𝑊 ∈ 𝑋) |
lpolv.o | ⊢ (𝜑 → ⊥ ∈ 𝑃) |
Ref | Expression |
---|---|
lpolvN | ⊢ (𝜑 → ( ⊥ ‘𝑉) = { 0 }) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lpolv.o | . . 3 ⊢ (𝜑 → ⊥ ∈ 𝑃) | |
2 | lpolv.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝑋) | |
3 | lpolv.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
4 | eqid 2818 | . . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
5 | lpolv.z | . . . . 5 ⊢ 0 = (0g‘𝑊) | |
6 | eqid 2818 | . . . . 5 ⊢ (LSAtoms‘𝑊) = (LSAtoms‘𝑊) | |
7 | eqid 2818 | . . . . 5 ⊢ (LSHyp‘𝑊) = (LSHyp‘𝑊) | |
8 | lpolv.p | . . . . 5 ⊢ 𝑃 = (LPol‘𝑊) | |
9 | 3, 4, 5, 6, 7, 8 | islpolN 38499 | . . . 4 ⊢ (𝑊 ∈ 𝑋 → ( ⊥ ∈ 𝑃 ↔ ( ⊥ :𝒫 𝑉⟶(LSubSp‘𝑊) ∧ (( ⊥ ‘𝑉) = { 0 } ∧ ∀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⟶(LSubSp‘𝑊) ∧ (( ⊥ ‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑊)(( ⊥ ‘𝑥) ∈ (LSHyp‘𝑊) ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))))) |
11 | 1, 10 | mpbid 233 | . 2 ⊢ (𝜑 → ( ⊥ :𝒫 𝑉⟶(LSubSp‘𝑊) ∧ (( ⊥ ‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑊)(( ⊥ ‘𝑥) ∈ (LSHyp‘𝑊) ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)))) |
12 | simpr1 1186 | . 2 ⊢ (( ⊥ :𝒫 𝑉⟶(LSubSp‘𝑊) ∧ (( ⊥ ‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑊)(( ⊥ ‘𝑥) ∈ (LSHyp‘𝑊) ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))) → ( ⊥ ‘𝑉) = { 0 }) | |
13 | 11, 12 | syl 17 | 1 ⊢ (𝜑 → ( ⊥ ‘𝑉) = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1079 ∀wal 1526 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ⊆ wss 3933 𝒫 cpw 4535 {csn 4557 ⟶wf 6344 ‘cfv 6348 Basecbs 16471 0gc0g 16701 LSubSpclss 19632 LSAtomsclsa 35990 LSHypclsh 35991 LPolclpoN 38496 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-map 8397 df-lpolN 38497 |
This theorem is referenced by: (None) |
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