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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 6911 | . . . . . . 7 ⊢ (𝑚 = 𝑀 → ((𝑝‘𝑥)𝑚𝑥) = ((𝑝‘𝑥)𝑀𝑥)) | |
2 | 1 | mpteq2dv 4968 | . . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥)) = (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑀𝑥))) |
3 | 2 | oveq2d 6921 | . . . . 5 ⊢ (𝑚 = 𝑀 → (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))) = (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑀𝑥)))) |
4 | 3 | oveq2d 6921 | . . . 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 4968 | . . 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 6921 | . 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 20760 | . 2 ⊢ 𝐷 = (𝑚 ∈ 𝐵 ↦ (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))))))) |
16 | ovex 6937 | . 2 ⊢ (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑀𝑥)))))) ∈ V | |
17 | 6, 15, 16 | fvmpt 6529 | 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 1658 ∈ wcel 2166 ↦ cmpt 4952 ∘ ccom 5346 ‘cfv 6123 (class class class)co 6905 Basecbs 16222 .rcmulr 16306 Σg cgsu 16454 SymGrpcsymg 18147 pmSgncpsgn 18259 mulGrpcmgp 18843 ℤRHomczrh 20208 Mat cmat 20580 maDet cmdat 20758 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-slot 16226 df-base 16228 df-mat 20581 df-mdet 20759 |
This theorem is referenced by: mdetleib2 20762 m1detdiag 20771 mdetdiag 20773 mdetralt 20782 mdettpos 20785 chpmatval2 21008 mdetpmtr1 30434 |
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