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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 7164 | . . . . . . 7 ⊢ (𝑚 = 𝑀 → ((𝑝‘𝑥)𝑚𝑥) = ((𝑝‘𝑥)𝑀𝑥)) | |
2 | 1 | mpteq2dv 5164 | . . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥)) = (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑀𝑥))) |
3 | 2 | oveq2d 7174 | . . . . 5 ⊢ (𝑚 = 𝑀 → (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))) = (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑀𝑥)))) |
4 | 3 | oveq2d 7174 | . . . 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 5164 | . . 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 7174 | . 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 21197 | . 2 ⊢ 𝐷 = (𝑚 ∈ 𝐵 ↦ (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))))))) |
16 | ovex 7191 | . 2 ⊢ (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑀𝑥)))))) ∈ V | |
17 | 6, 15, 16 | fvmpt 6770 | 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 1537 ∈ wcel 2114 ↦ cmpt 5148 ∘ ccom 5561 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 .rcmulr 16568 Σg cgsu 16716 SymGrpcsymg 18497 pmSgncpsgn 18619 mulGrpcmgp 19241 ℤRHomczrh 20649 Mat cmat 21018 maDet cmdat 21195 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-slot 16489 df-base 16491 df-mat 21019 df-mdet 21196 |
This theorem is referenced by: mdetleib2 21199 m1detdiag 21208 mdetdiag 21210 mdetralt 21219 mdettpos 21222 chpmatval2 21443 mdetpmtr1 31090 |
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