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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 2823 | . . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
5 | lpolv.z | . . . . 5 ⊢ 0 = (0g‘𝑊) | |
6 | eqid 2823 | . . . . 5 ⊢ (LSAtoms‘𝑊) = (LSAtoms‘𝑊) | |
7 | eqid 2823 | . . . . 5 ⊢ (LSHyp‘𝑊) = (LSHyp‘𝑊) | |
8 | lpolv.p | . . . . 5 ⊢ 𝑃 = (LPol‘𝑊) | |
9 | 3, 4, 5, 6, 7, 8 | islpolN 38621 | . . . 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 234 | . 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 1190 | . 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 208 ∧ wa 398 ∧ w3a 1083 ∀wal 1535 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ⊆ wss 3938 𝒫 cpw 4541 {csn 4569 ⟶wf 6353 ‘cfv 6357 Basecbs 16485 0gc0g 16715 LSubSpclss 19705 LSAtomsclsa 36112 LSHypclsh 36113 LPolclpoN 38618 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-map 8410 df-lpolN 38619 |
This theorem is referenced by: (None) |
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