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Mirrors > Home > MPE Home > Th. List > matval | Structured version Visualization version GIF version |
Description: Value of the matrix algebra. (Contributed by Stefan O'Rear, 4-Sep-2015.) |
Ref | Expression |
---|---|
matval.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
matval.g | ⊢ 𝐺 = (𝑅 freeLMod (𝑁 × 𝑁)) |
matval.t | ⊢ · = (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) |
Ref | Expression |
---|---|
matval | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐴 = (𝐺 sSet 〈(.r‘ndx), · 〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | matval.a | . 2 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
2 | elex 3352 | . . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V) | |
3 | id 22 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅) | |
4 | id 22 | . . . . . . . 8 ⊢ (𝑛 = 𝑁 → 𝑛 = 𝑁) | |
5 | 4 | sqxpeqd 5298 | . . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑛 × 𝑛) = (𝑁 × 𝑁)) |
6 | 3, 5 | oveqan12rd 6833 | . . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑟 freeLMod (𝑛 × 𝑛)) = (𝑅 freeLMod (𝑁 × 𝑁))) |
7 | matval.g | . . . . . 6 ⊢ 𝐺 = (𝑅 freeLMod (𝑁 × 𝑁)) | |
8 | 6, 7 | syl6eqr 2812 | . . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑟 freeLMod (𝑛 × 𝑛)) = 𝐺) |
9 | 4, 4, 4 | oteq123d 4568 | . . . . . . . 8 ⊢ (𝑛 = 𝑁 → 〈𝑛, 𝑛, 𝑛〉 = 〈𝑁, 𝑁, 𝑁〉) |
10 | 3, 9 | oveqan12rd 6833 | . . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑟 maMul 〈𝑛, 𝑛, 𝑛〉) = (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉)) |
11 | matval.t | . . . . . . 7 ⊢ · = (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) | |
12 | 10, 11 | syl6eqr 2812 | . . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑟 maMul 〈𝑛, 𝑛, 𝑛〉) = · ) |
13 | 12 | opeq2d 4560 | . . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → 〈(.r‘ndx), (𝑟 maMul 〈𝑛, 𝑛, 𝑛〉)〉 = 〈(.r‘ndx), · 〉) |
14 | 8, 13 | oveq12d 6831 | . . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ((𝑟 freeLMod (𝑛 × 𝑛)) sSet 〈(.r‘ndx), (𝑟 maMul 〈𝑛, 𝑛, 𝑛〉)〉) = (𝐺 sSet 〈(.r‘ndx), · 〉)) |
15 | df-mat 20416 | . . . 4 ⊢ Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet 〈(.r‘ndx), (𝑟 maMul 〈𝑛, 𝑛, 𝑛〉)〉)) | |
16 | ovex 6841 | . . . 4 ⊢ (𝐺 sSet 〈(.r‘ndx), · 〉) ∈ V | |
17 | 14, 15, 16 | ovmpt2a 6956 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝑁 Mat 𝑅) = (𝐺 sSet 〈(.r‘ndx), · 〉)) |
18 | 2, 17 | sylan2 492 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑁 Mat 𝑅) = (𝐺 sSet 〈(.r‘ndx), · 〉)) |
19 | 1, 18 | syl5eq 2806 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐴 = (𝐺 sSet 〈(.r‘ndx), · 〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 Vcvv 3340 〈cop 4327 〈cotp 4329 × cxp 5264 ‘cfv 6049 (class class class)co 6813 Fincfn 8121 ndxcnx 16056 sSet csts 16057 .rcmulr 16144 freeLMod cfrlm 20292 maMul cmmul 20391 Mat cmat 20415 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pr 5055 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-ot 4330 df-uni 4589 df-br 4805 df-opab 4865 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-iota 6012 df-fun 6051 df-fv 6057 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-mat 20416 |
This theorem is referenced by: matbas 20421 matplusg 20422 matsca 20423 matvsca 20424 matmulr 20446 |
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