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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 3499 | . . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V) | |
3 | id 22 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅) | |
4 | id 22 | . . . . . . . 8 ⊢ (𝑛 = 𝑁 → 𝑛 = 𝑁) | |
5 | 4 | sqxpeqd 5721 | . . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑛 × 𝑛) = (𝑁 × 𝑁)) |
6 | 3, 5 | oveqan12rd 7451 | . . . . . 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 | eqtr4di 2793 | . . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑟 freeLMod (𝑛 × 𝑛)) = 𝐺) |
9 | 4, 4, 4 | oteq123d 4893 | . . . . . . . 8 ⊢ (𝑛 = 𝑁 → 〈𝑛, 𝑛, 𝑛〉 = 〈𝑁, 𝑁, 𝑁〉) |
10 | 3, 9 | oveqan12rd 7451 | . . . . . . 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 | eqtr4di 2793 | . . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑟 maMul 〈𝑛, 𝑛, 𝑛〉) = · ) |
13 | 12 | opeq2d 4885 | . . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → 〈(.r‘ndx), (𝑟 maMul 〈𝑛, 𝑛, 𝑛〉)〉 = 〈(.r‘ndx), · 〉) |
14 | 8, 13 | oveq12d 7449 | . . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ((𝑟 freeLMod (𝑛 × 𝑛)) sSet 〈(.r‘ndx), (𝑟 maMul 〈𝑛, 𝑛, 𝑛〉)〉) = (𝐺 sSet 〈(.r‘ndx), · 〉)) |
15 | df-mat 22428 | . . . 4 ⊢ Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet 〈(.r‘ndx), (𝑟 maMul 〈𝑛, 𝑛, 𝑛〉)〉)) | |
16 | ovex 7464 | . . . 4 ⊢ (𝐺 sSet 〈(.r‘ndx), · 〉) ∈ V | |
17 | 14, 15, 16 | ovmpoa 7588 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝑁 Mat 𝑅) = (𝐺 sSet 〈(.r‘ndx), · 〉)) |
18 | 2, 17 | sylan2 593 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑁 Mat 𝑅) = (𝐺 sSet 〈(.r‘ndx), · 〉)) |
19 | 1, 18 | eqtrid 2787 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐴 = (𝐺 sSet 〈(.r‘ndx), · 〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 〈cop 4637 〈cotp 4639 × cxp 5687 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 sSet csts 17197 ndxcnx 17227 .rcmulr 17299 freeLMod cfrlm 21784 maMul cmmul 22410 Mat cmat 22427 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-ot 4640 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-mat 22428 |
This theorem is referenced by: matbas 22433 matplusg 22434 matsca 22435 matscaOLD 22436 matvsca 22437 matvscaOLD 22438 matmulr 22460 |
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