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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 3501 | . . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V) | |
| 3 | id 22 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅) | |
| 4 | id 22 | . . . . . . . 8 ⊢ (𝑛 = 𝑁 → 𝑛 = 𝑁) | |
| 5 | 4 | sqxpeqd 5717 | . . . . . . 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 2795 | . . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑟 freeLMod (𝑛 × 𝑛)) = 𝐺) |
| 9 | 4, 4, 4 | oteq123d 4888 | . . . . . . . 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 2795 | . . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑟 maMul 〈𝑛, 𝑛, 𝑛〉) = · ) |
| 13 | 12 | opeq2d 4880 | . . . . 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 22412 | . . . 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 2789 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐴 = (𝐺 sSet 〈(.r‘ndx), · 〉)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 〈cotp 4634 × cxp 5683 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 sSet csts 17200 ndxcnx 17230 .rcmulr 17298 freeLMod cfrlm 21766 maMul cmmul 22394 Mat cmat 22411 |
| 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-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-ot 4635 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-mat 22412 |
| This theorem is referenced by: matbas 22417 matplusg 22418 matsca 22419 matscaOLD 22420 matvsca 22421 matvscaOLD 22422 matmulr 22444 |
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