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| Mirrors > Home > MPE Home > Th. List > cpmatelimp | Structured version Visualization version GIF version | ||
| Description: Implication of a set being a constant polynomial matrix. (Contributed by AV, 18-Nov-2019.) | 
| Ref | Expression | 
|---|---|
| cpmat.s | ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) | 
| cpmat.p | ⊢ 𝑃 = (Poly1‘𝑅) | 
| cpmat.c | ⊢ 𝐶 = (𝑁 Mat 𝑃) | 
| cpmat.b | ⊢ 𝐵 = (Base‘𝐶) | 
| Ref | Expression | 
|---|---|
| cpmatelimp | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑀 ∈ 𝑆 → (𝑀 ∈ 𝐵 ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g‘𝑅)))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | cpmat.s | . . . . 5 ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) | |
| 2 | cpmat.p | . . . . 5 ⊢ 𝑃 = (Poly1‘𝑅) | |
| 3 | cpmat.c | . . . . 5 ⊢ 𝐶 = (𝑁 Mat 𝑃) | |
| 4 | cpmat.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐶) | |
| 5 | 1, 2, 3, 4 | cpmatpmat 22716 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → 𝑀 ∈ 𝐵) | 
| 6 | 5 | 3expa 1119 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑀 ∈ 𝑆) → 𝑀 ∈ 𝐵) | 
| 7 | 1, 2, 3, 4 | cpmatel 22717 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑀 ∈ 𝑆 ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g‘𝑅))) | 
| 8 | 7 | 3expa 1119 | . . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑀 ∈ 𝐵) → (𝑀 ∈ 𝑆 ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g‘𝑅))) | 
| 9 | 8 | biimpd 229 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑀 ∈ 𝐵) → (𝑀 ∈ 𝑆 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g‘𝑅))) | 
| 10 | 9 | impancom 451 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑀 ∈ 𝑆) → (𝑀 ∈ 𝐵 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g‘𝑅))) | 
| 11 | 6, 10 | jcai 516 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑀 ∈ 𝑆) → (𝑀 ∈ 𝐵 ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g‘𝑅))) | 
| 12 | 11 | ex 412 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑀 ∈ 𝑆 → (𝑀 ∈ 𝐵 ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g‘𝑅)))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 ℕcn 12266 Basecbs 17247 0gc0g 17484 Ringcrg 20230 Poly1cpl1 22178 coe1cco1 22179 Mat cmat 22411 ConstPolyMat ccpmat 22709 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-cpmat 22712 | 
| This theorem is referenced by: cpmatmcllem 22724 m2cpminvid2lem 22760 | 
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