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Theorem mdetleib 21172
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 7136 . . . . . . 7 (𝑚 = 𝑀 → ((𝑝𝑥)𝑚𝑥) = ((𝑝𝑥)𝑀𝑥))
21mpteq2dv 5135 . . . . . 6 (𝑚 = 𝑀 → (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥)) = (𝑥𝑁 ↦ ((𝑝𝑥)𝑀𝑥)))
32oveq2d 7146 . . . . 5 (𝑚 = 𝑀 → (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥))) = (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑀𝑥))))
43oveq2d 7146 . . . 4 (𝑚 = 𝑀 → (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥)))) = (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑀𝑥)))))
54mpteq2dv 5135 . . 3 (𝑚 = 𝑀 → (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥))))) = (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑀𝑥))))))
65oveq2d 7146 . 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 21171 . 2 𝐷 = (𝑚𝐵 ↦ (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥)))))))
16 ovex 7163 . 2 (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑀𝑥)))))) ∈ V
176, 15, 16fvmpt 6741 1 (𝑀𝐵 → (𝐷𝑀) = (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑀𝑥)))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  cmpt 5119  ccom 5532  cfv 6328  (class class class)co 7130  Basecbs 16462  .rcmulr 16545   Σg cgsu 16693  SymGrpcsymg 18474  pmSgncpsgn 18596  mulGrpcmgp 19218  ℤRHomczrh 20623   Mat cmat 20992   maDet cmdat 21169
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303
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 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-slot 16466  df-base 16468  df-mat 20993  df-mdet 21170
This theorem is referenced by:  mdetleib2  21173  m1detdiag  21182  mdetdiag  21184  mdetralt  21193  mdettpos  21196  chpmatval2  21417  mdetpmtr1  31099
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