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Theorem mdetleib 20150
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 6529 . . . . . . 7 (𝑚 = 𝑀 → ((𝑝𝑥)𝑚𝑥) = ((𝑝𝑥)𝑀𝑥))
21mpteq2dv 4663 . . . . . 6 (𝑚 = 𝑀 → (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥)) = (𝑥𝑁 ↦ ((𝑝𝑥)𝑀𝑥)))
32oveq2d 6539 . . . . 5 (𝑚 = 𝑀 → (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥))) = (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑀𝑥))))
43oveq2d 6539 . . . 4 (𝑚 = 𝑀 → (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥)))) = (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑀𝑥)))))
54mpteq2dv 4663 . . 3 (𝑚 = 𝑀 → (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥))))) = (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑀𝑥))))))
65oveq2d 6539 . 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 20149 . 2 𝐷 = (𝑚𝐵 ↦ (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥)))))))
16 ovex 6551 . 2 (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑀𝑥)))))) ∈ V
176, 15, 16fvmpt 6172 1 (𝑀𝐵 → (𝐷𝑀) = (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑀𝑥)))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1975  cmpt 4633  ccom 5028  cfv 5786  (class class class)co 6523  Basecbs 15637  .rcmulr 15711   Σg cgsu 15866  SymGrpcsymg 17562  pmSgncpsgn 17674  mulGrpcmgp 18254  ℤRHomczrh 19608   Mat cmat 19970   maDet cmdat 20147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-slot 15641  df-base 15642  df-mat 19971  df-mdet 20148
This theorem is referenced by:  mdetleib2  20151  m1detdiag  20160  mdetdiag  20162  mdetralt  20171  mdettpos  20174  chpmatval2  20395  mdetpmtr1  29019
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