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Mirrors > Home > MPE Home > Th. List > mdetleib | Structured version Visualization version GIF version |
Description: Full substitution of our determinant definition (also known as Leibniz' Formula, expanding by columns). Proposition 4.6 in [Lang] p. 514. (Contributed by Stefan O'Rear, 3-Oct-2015.) (Revised by SO, 9-Jul-2018.) |
Ref | Expression |
---|---|
mdetfval.d | ⊢ 𝐷 = (𝑁 maDet 𝑅) |
mdetfval.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
mdetfval.b | ⊢ 𝐵 = (Base‘𝐴) |
mdetfval.p | ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) |
mdetfval.y | ⊢ 𝑌 = (ℤRHom‘𝑅) |
mdetfval.s | ⊢ 𝑆 = (pmSgn‘𝑁) |
mdetfval.t | ⊢ · = (.r‘𝑅) |
mdetfval.u | ⊢ 𝑈 = (mulGrp‘𝑅) |
Ref | Expression |
---|---|
mdetleib | ⊢ (𝑀 ∈ 𝐵 → (𝐷‘𝑀) = (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑀𝑥))))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq 7141 | . . . . . . 7 ⊢ (𝑚 = 𝑀 → ((𝑝‘𝑥)𝑚𝑥) = ((𝑝‘𝑥)𝑀𝑥)) | |
2 | 1 | mpteq2dv 5126 | . . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥)) = (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑀𝑥))) |
3 | 2 | oveq2d 7151 | . . . . 5 ⊢ (𝑚 = 𝑀 → (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))) = (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑀𝑥)))) |
4 | 3 | oveq2d 7151 | . . . 4 ⊢ (𝑚 = 𝑀 → (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥)))) = (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑀𝑥))))) |
5 | 4 | mpteq2dv 5126 | . . 3 ⊢ (𝑚 = 𝑀 → (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))))) = (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑀𝑥)))))) |
6 | 5 | oveq2d 7151 | . 2 ⊢ (𝑚 = 𝑀 → (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥)))))) = (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑀𝑥))))))) |
7 | mdetfval.d | . . 3 ⊢ 𝐷 = (𝑁 maDet 𝑅) | |
8 | mdetfval.a | . . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
9 | mdetfval.b | . . 3 ⊢ 𝐵 = (Base‘𝐴) | |
10 | mdetfval.p | . . 3 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) | |
11 | mdetfval.y | . . 3 ⊢ 𝑌 = (ℤRHom‘𝑅) | |
12 | mdetfval.s | . . 3 ⊢ 𝑆 = (pmSgn‘𝑁) | |
13 | mdetfval.t | . . 3 ⊢ · = (.r‘𝑅) | |
14 | mdetfval.u | . . 3 ⊢ 𝑈 = (mulGrp‘𝑅) | |
15 | 7, 8, 9, 10, 11, 12, 13, 14 | mdetfval 21191 | . 2 ⊢ 𝐷 = (𝑚 ∈ 𝐵 ↦ (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))))))) |
16 | ovex 7168 | . 2 ⊢ (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑀𝑥)))))) ∈ V | |
17 | 6, 15, 16 | fvmpt 6745 | 1 ⊢ (𝑀 ∈ 𝐵 → (𝐷‘𝑀) = (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑀𝑥))))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ↦ cmpt 5110 ∘ ccom 5523 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 .rcmulr 16558 Σg cgsu 16706 SymGrpcsymg 18487 pmSgncpsgn 18609 mulGrpcmgp 19232 ℤRHomczrh 20193 Mat cmat 21012 maDet cmdat 21189 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-slot 16479 df-base 16481 df-mat 21013 df-mdet 21190 |
This theorem is referenced by: mdetleib2 21193 m1detdiag 21202 mdetdiag 21204 mdetralt 21213 mdettpos 21216 chpmatval2 21438 mdetpmtr1 31176 |
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