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Theorem mdetleib 20616
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.)
Hypotheses
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 · = (.r𝑅)
mdetfval.u 𝑈 = (mulGrp‘𝑅)
Assertion
Ref Expression
mdetleib (𝑀𝐵 → (𝐷𝑀) = (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑀𝑥)))))))
Distinct variable groups:   𝑥,𝑝,𝑀   𝑁,𝑝,𝑥   𝑅,𝑝,𝑥
Allowed substitution hints:   𝐴(𝑥,𝑝)   𝐵(𝑥,𝑝)   𝐷(𝑥,𝑝)   𝑃(𝑥,𝑝)   𝑆(𝑥,𝑝)   · (𝑥,𝑝)   𝑈(𝑥,𝑝)   𝑌(𝑥,𝑝)

Proof of Theorem mdetleib
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 oveq 6821 . . . . . . 7 (𝑚 = 𝑀 → ((𝑝𝑥)𝑚𝑥) = ((𝑝𝑥)𝑀𝑥))
21mpteq2dv 4898 . . . . . 6 (𝑚 = 𝑀 → (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥)) = (𝑥𝑁 ↦ ((𝑝𝑥)𝑀𝑥)))
32oveq2d 6831 . . . . 5 (𝑚 = 𝑀 → (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥))) = (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑀𝑥))))
43oveq2d 6831 . . . 4 (𝑚 = 𝑀 → (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥)))) = (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑀𝑥)))))
54mpteq2dv 4898 . . 3 (𝑚 = 𝑀 → (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥))))) = (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑀𝑥))))))
65oveq2d 6831 . 2 (𝑚 = 𝑀 → (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥)))))) = (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)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 · = (.r𝑅)
14 mdetfval.u . . 3 𝑈 = (mulGrp‘𝑅)
157, 8, 9, 10, 11, 12, 13, 14mdetfval 20615 . 2 𝐷 = (𝑚𝐵 ↦ (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥)))))))
16 ovex 6843 . 2 (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑀𝑥)))))) ∈ V
176, 15, 16fvmpt 6446 1 (𝑀𝐵 → (𝐷𝑀) = (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑀𝑥)))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2140  cmpt 4882  ccom 5271  cfv 6050  (class class class)co 6815  Basecbs 16080  .rcmulr 16165   Σg cgsu 16324  SymGrpcsymg 18018  pmSgncpsgn 18130  mulGrpcmgp 18710  ℤRHomczrh 20071   Mat cmat 20436   maDet cmdat 20613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-slot 16084  df-base 16086  df-mat 20437  df-mdet 20614
This theorem is referenced by:  mdetleib2  20617  m1detdiag  20626  mdetdiag  20628  mdetralt  20637  mdettpos  20640  chpmatval2  20861  mdetpmtr1  30220
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