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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 3463 | . . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V) | |
3 | id 22 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅) | |
4 | id 22 | . . . . . . . 8 ⊢ (𝑛 = 𝑁 → 𝑛 = 𝑁) | |
5 | 4 | sqxpeqd 5665 | . . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑛 × 𝑛) = (𝑁 × 𝑁)) |
6 | 3, 5 | oveqan12rd 7377 | . . . . . 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 2794 | . . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑟 freeLMod (𝑛 × 𝑛)) = 𝐺) |
9 | 4, 4, 4 | oteq123d 4845 | . . . . . . . 8 ⊢ (𝑛 = 𝑁 → 〈𝑛, 𝑛, 𝑛〉 = 〈𝑁, 𝑁, 𝑁〉) |
10 | 3, 9 | oveqan12rd 7377 | . . . . . . 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 2794 | . . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑟 maMul 〈𝑛, 𝑛, 𝑛〉) = · ) |
13 | 12 | opeq2d 4837 | . . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → 〈(.r‘ndx), (𝑟 maMul 〈𝑛, 𝑛, 𝑛〉)〉 = 〈(.r‘ndx), · 〉) |
14 | 8, 13 | oveq12d 7375 | . . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ((𝑟 freeLMod (𝑛 × 𝑛)) sSet 〈(.r‘ndx), (𝑟 maMul 〈𝑛, 𝑛, 𝑛〉)〉) = (𝐺 sSet 〈(.r‘ndx), · 〉)) |
15 | df-mat 21755 | . . . 4 ⊢ Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet 〈(.r‘ndx), (𝑟 maMul 〈𝑛, 𝑛, 𝑛〉)〉)) | |
16 | ovex 7390 | . . . 4 ⊢ (𝐺 sSet 〈(.r‘ndx), · 〉) ∈ V | |
17 | 14, 15, 16 | ovmpoa 7510 | . . 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 2788 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐴 = (𝐺 sSet 〈(.r‘ndx), · 〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3445 〈cop 4592 〈cotp 4594 × cxp 5631 ‘cfv 6496 (class class class)co 7357 Fincfn 8883 sSet csts 17035 ndxcnx 17065 .rcmulr 17134 freeLMod cfrlm 21152 maMul cmmul 21732 Mat cmat 21754 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-sbc 3740 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-ot 4595 df-uni 4866 df-br 5106 df-opab 5168 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-iota 6448 df-fun 6498 df-fv 6504 df-ov 7360 df-oprab 7361 df-mpo 7362 df-mat 21755 |
This theorem is referenced by: matbas 21760 matplusg 21761 matsca 21762 matscaOLD 21763 matvsca 21764 matvscaOLD 21765 matmulr 21787 |
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