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Mirrors > Home > MPE Home > Th. List > Mathboxes > islpoldN | Structured version Visualization version GIF version |
Description: Properties that determine a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lpolset.v | ⊢ 𝑉 = (Base‘𝑊) |
lpolset.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lpolset.z | ⊢ 0 = (0g‘𝑊) |
lpolset.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
lpolset.h | ⊢ 𝐻 = (LSHyp‘𝑊) |
lpolset.p | ⊢ 𝑃 = (LPol‘𝑊) |
islpold.w | ⊢ (𝜑 → 𝑊 ∈ 𝑋) |
islpold.1 | ⊢ (𝜑 → ⊥ :𝒫 𝑉⟶𝑆) |
islpold.2 | ⊢ (𝜑 → ( ⊥ ‘𝑉) = { 0 }) |
islpold.3 | ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦)) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) |
islpold.4 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ( ⊥ ‘𝑥) ∈ 𝐻) |
islpold.5 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥) |
Ref | Expression |
---|---|
islpoldN | ⊢ (𝜑 → ⊥ ∈ 𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | islpold.1 | . 2 ⊢ (𝜑 → ⊥ :𝒫 𝑉⟶𝑆) | |
2 | islpold.2 | . . 3 ⊢ (𝜑 → ( ⊥ ‘𝑉) = { 0 }) | |
3 | islpold.3 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦)) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) | |
4 | 3 | ex 450 | . . . 4 ⊢ (𝜑 → ((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥))) |
5 | 4 | alrimivv 1856 | . . 3 ⊢ (𝜑 → ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥))) |
6 | islpold.4 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ( ⊥ ‘𝑥) ∈ 𝐻) | |
7 | islpold.5 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥) | |
8 | 6, 7 | jca 554 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)) |
9 | 8 | ralrimiva 2966 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)) |
10 | 2, 5, 9 | 3jca 1242 | . 2 ⊢ (𝜑 → (( ⊥ ‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))) |
11 | islpold.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ 𝑋) | |
12 | lpolset.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
13 | lpolset.s | . . . 4 ⊢ 𝑆 = (LSubSp‘𝑊) | |
14 | lpolset.z | . . . 4 ⊢ 0 = (0g‘𝑊) | |
15 | lpolset.a | . . . 4 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
16 | lpolset.h | . . . 4 ⊢ 𝐻 = (LSHyp‘𝑊) | |
17 | lpolset.p | . . . 4 ⊢ 𝑃 = (LPol‘𝑊) | |
18 | 12, 13, 14, 15, 16, 17 | islpolN 36772 | . . 3 ⊢ (𝑊 ∈ 𝑋 → ( ⊥ ∈ 𝑃 ↔ ( ⊥ :𝒫 𝑉⟶𝑆 ∧ (( ⊥ ‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))))) |
19 | 11, 18 | syl 17 | . 2 ⊢ (𝜑 → ( ⊥ ∈ 𝑃 ↔ ( ⊥ :𝒫 𝑉⟶𝑆 ∧ (( ⊥ ‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))))) |
20 | 1, 10, 19 | mpbir2and 957 | 1 ⊢ (𝜑 → ⊥ ∈ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 ∀wal 1481 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ⊆ wss 3574 𝒫 cpw 4158 {csn 4177 ⟶wf 5884 ‘cfv 5888 Basecbs 15857 0gc0g 16100 LSubSpclss 18932 LSAtomsclsa 34261 LSHypclsh 34262 LPolclpoN 36769 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-map 7859 df-lpolN 36770 |
This theorem is referenced by: dochpolN 36779 |
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