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Mirrors > Home > MPE Home > Th. List > cpmatel | Structured version Visualization version GIF version |
Description: Property of a constant polynomial matrix. (Contributed by AV, 15-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 |
---|---|
cpmatel | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝑀 ∈ 𝑆 ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g‘𝑅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cpmat.s | . . . . . 6 ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) | |
2 | cpmat.p | . . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅) | |
3 | cpmat.c | . . . . . 6 ⊢ 𝐶 = (𝑁 Mat 𝑃) | |
4 | cpmat.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐶) | |
5 | 1, 2, 3, 4 | cpmat 22738 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑆 = {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g‘𝑅)}) |
6 | 5 | 3adant3 1132 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → 𝑆 = {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g‘𝑅)}) |
7 | 6 | eleq2d 2830 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝑀 ∈ 𝑆 ↔ 𝑀 ∈ {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g‘𝑅)})) |
8 | oveq 7456 | . . . . . . . . 9 ⊢ (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗)) | |
9 | 8 | fveq2d 6926 | . . . . . . . 8 ⊢ (𝑚 = 𝑀 → (coe1‘(𝑖𝑚𝑗)) = (coe1‘(𝑖𝑀𝑗))) |
10 | 9 | fveq1d 6924 | . . . . . . 7 ⊢ (𝑚 = 𝑀 → ((coe1‘(𝑖𝑚𝑗))‘𝑘) = ((coe1‘(𝑖𝑀𝑗))‘𝑘)) |
11 | 10 | eqeq1d 2742 | . . . . . 6 ⊢ (𝑚 = 𝑀 → (((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g‘𝑅) ↔ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g‘𝑅))) |
12 | 11 | ralbidv 3184 | . . . . 5 ⊢ (𝑚 = 𝑀 → (∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g‘𝑅) ↔ ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g‘𝑅))) |
13 | 12 | 2ralbidv 3227 | . . . 4 ⊢ (𝑚 = 𝑀 → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g‘𝑅) ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g‘𝑅))) |
14 | 13 | elrab 3708 | . . 3 ⊢ (𝑀 ∈ {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g‘𝑅)} ↔ (𝑀 ∈ 𝐵 ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g‘𝑅))) |
15 | 7, 14 | bitrdi 287 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝑀 ∈ 𝑆 ↔ (𝑀 ∈ 𝐵 ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g‘𝑅)))) |
16 | 15 | 3anibar 1329 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝑀 ∈ 𝑆 ↔ ∀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 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ∀wral 3067 {crab 3443 ‘cfv 6575 (class class class)co 7450 Fincfn 9005 ℕcn 12295 Basecbs 17260 0gc0g 17501 Poly1cpl1 22201 coe1cco1 22202 Mat cmat 22434 ConstPolyMat ccpmat 22732 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-iota 6527 df-fun 6577 df-fv 6583 df-ov 7453 df-oprab 7454 df-mpo 7455 df-cpmat 22735 |
This theorem is referenced by: cpmatelimp 22741 cpmatel2 22742 1elcpmat 22744 cpmatmcl 22748 m2cpm 22770 |
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