![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 7437 | . . . . . . 7 ⊢ (𝑚 = 𝑀 → ((𝑝‘𝑥)𝑚𝑥) = ((𝑝‘𝑥)𝑀𝑥)) | |
2 | 1 | mpteq2dv 5250 | . . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥)) = (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑀𝑥))) |
3 | 2 | oveq2d 7447 | . . . . 5 ⊢ (𝑚 = 𝑀 → (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))) = (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑀𝑥)))) |
4 | 3 | oveq2d 7447 | . . . 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 5250 | . . 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 7447 | . 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 22608 | . 2 ⊢ 𝐷 = (𝑚 ∈ 𝐵 ↦ (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))))))) |
16 | ovex 7464 | . 2 ⊢ (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑀𝑥)))))) ∈ V | |
17 | 6, 15, 16 | fvmpt 7016 | 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 2106 ↦ cmpt 5231 ∘ ccom 5693 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 .rcmulr 17299 Σg cgsu 17487 SymGrpcsymg 19401 pmSgncpsgn 19522 mulGrpcmgp 20152 ℤRHomczrh 21528 Mat cmat 22427 maDet cmdat 22606 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-1cn 11211 ax-addcl 11213 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-nn 12265 df-slot 17216 df-ndx 17228 df-base 17246 df-mat 22428 df-mdet 22607 |
This theorem is referenced by: mdetleib2 22610 m1detdiag 22619 mdetdiag 22621 mdetralt 22630 mdettpos 22633 chpmatval2 22855 mdetpmtr1 33784 |
Copyright terms: Public domain | W3C validator |