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Theorem mdetfval 20610
Description: First substitution for the determinant definition. (Contributed by Stefan O'Rear, 9-Sep-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
mdetfval 𝐷 = (𝑚𝐵 ↦ (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥)))))))
Distinct variable groups:   𝐵,𝑚   𝑚,𝑝,𝑥,𝑁   𝑃,𝑚   𝑅,𝑚,𝑝,𝑥   𝑆,𝑚   · ,𝑚   𝑈,𝑚   𝑚,𝑌
Allowed substitution hints:   𝐴(𝑥,𝑚,𝑝)   𝐵(𝑥,𝑝)   𝐷(𝑥,𝑚,𝑝)   𝑃(𝑥,𝑝)   𝑆(𝑥,𝑝)   · (𝑥,𝑝)   𝑈(𝑥,𝑝)   𝑌(𝑥,𝑝)

Proof of Theorem mdetfval
Dummy variables 𝑦 𝑧 𝑛 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mdetfval.d . 2 𝐷 = (𝑁 maDet 𝑅)
2 oveq12 6802 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
3 mdetfval.a . . . . . . . 8 𝐴 = (𝑁 Mat 𝑅)
42, 3syl6eqr 2823 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = 𝐴)
54fveq2d 6336 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = (Base‘𝐴))
6 mdetfval.b . . . . . 6 𝐵 = (Base‘𝐴)
75, 6syl6eqr 2823 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵)
8 simpr 471 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑟 = 𝑅)
9 simpl 468 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑛 = 𝑁)
109fveq2d 6336 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (SymGrp‘𝑛) = (SymGrp‘𝑁))
1110fveq2d 6336 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(SymGrp‘𝑛)) = (Base‘(SymGrp‘𝑁)))
12 mdetfval.p . . . . . . . 8 𝑃 = (Base‘(SymGrp‘𝑁))
1311, 12syl6eqr 2823 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(SymGrp‘𝑛)) = 𝑃)
14 fveq2 6332 . . . . . . . . . 10 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
1514adantl 467 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (.r𝑟) = (.r𝑅))
16 mdetfval.t . . . . . . . . 9 · = (.r𝑅)
1715, 16syl6eqr 2823 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (.r𝑟) = · )
188fveq2d 6336 . . . . . . . . . . 11 ((𝑛 = 𝑁𝑟 = 𝑅) → (ℤRHom‘𝑟) = (ℤRHom‘𝑅))
19 mdetfval.y . . . . . . . . . . 11 𝑌 = (ℤRHom‘𝑅)
2018, 19syl6eqr 2823 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → (ℤRHom‘𝑟) = 𝑌)
21 fveq2 6332 . . . . . . . . . . . 12 (𝑛 = 𝑁 → (pmSgn‘𝑛) = (pmSgn‘𝑁))
2221adantr 466 . . . . . . . . . . 11 ((𝑛 = 𝑁𝑟 = 𝑅) → (pmSgn‘𝑛) = (pmSgn‘𝑁))
23 mdetfval.s . . . . . . . . . . 11 𝑆 = (pmSgn‘𝑁)
2422, 23syl6eqr 2823 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → (pmSgn‘𝑛) = 𝑆)
2520, 24coeq12d 5425 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → ((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛)) = (𝑌𝑆))
2625fveq1d 6334 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝) = ((𝑌𝑆)‘𝑝))
27 fveq2 6332 . . . . . . . . . . 11 (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
2827adantl 467 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → (mulGrp‘𝑟) = (mulGrp‘𝑅))
29 mdetfval.u . . . . . . . . . 10 𝑈 = (mulGrp‘𝑅)
3028, 29syl6eqr 2823 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (mulGrp‘𝑟) = 𝑈)
319mpteq1d 4872 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑥𝑛 ↦ ((𝑝𝑥)𝑚𝑥)) = (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥)))
3230, 31oveq12d 6811 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → ((mulGrp‘𝑟) Σg (𝑥𝑛 ↦ ((𝑝𝑥)𝑚𝑥))) = (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥))))
3317, 26, 32oveq123d 6814 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r𝑟)((mulGrp‘𝑟) Σg (𝑥𝑛 ↦ ((𝑝𝑥)𝑚𝑥)))) = (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥)))))
3413, 33mpteq12dv 4867 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑝 ∈ (Base‘(SymGrp‘𝑛)) ↦ ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r𝑟)((mulGrp‘𝑟) Σg (𝑥𝑛 ↦ ((𝑝𝑥)𝑚𝑥))))) = (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥))))))
358, 34oveq12d 6811 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑟 Σg (𝑝 ∈ (Base‘(SymGrp‘𝑛)) ↦ ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r𝑟)((mulGrp‘𝑟) Σg (𝑥𝑛 ↦ ((𝑝𝑥)𝑚𝑥)))))) = (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥)))))))
367, 35mpteq12dv 4867 . . . 4 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑟 Σg (𝑝 ∈ (Base‘(SymGrp‘𝑛)) ↦ ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r𝑟)((mulGrp‘𝑟) Σg (𝑥𝑛 ↦ ((𝑝𝑥)𝑚𝑥))))))) = (𝑚𝐵 ↦ (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥))))))))
37 df-mdet 20609 . . . 4 maDet = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑟 Σg (𝑝 ∈ (Base‘(SymGrp‘𝑛)) ↦ ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r𝑟)((mulGrp‘𝑟) Σg (𝑥𝑛 ↦ ((𝑝𝑥)𝑚𝑥))))))))
38 fvex 6342 . . . . . 6 (Base‘𝐴) ∈ V
396, 38eqeltri 2846 . . . . 5 𝐵 ∈ V
4039mptex 6630 . . . 4 (𝑚𝐵 ↦ (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥))))))) ∈ V
4136, 37, 40ovmpt2a 6938 . . 3 ((𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 maDet 𝑅) = (𝑚𝐵 ↦ (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥))))))))
4237reldmmpt2 6918 . . . . . 6 Rel dom maDet
4342ovprc 6828 . . . . 5 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 maDet 𝑅) = ∅)
44 mpt0 6161 . . . . 5 (𝑚 ∈ ∅ ↦ (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥))))))) = ∅
4543, 44syl6eqr 2823 . . . 4 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 maDet 𝑅) = (𝑚 ∈ ∅ ↦ (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥))))))))
46 df-mat 20431 . . . . . . . . . 10 Mat = (𝑦 ∈ Fin, 𝑧 ∈ V ↦ ((𝑧 freeLMod (𝑦 × 𝑦)) sSet ⟨(.r‘ndx), (𝑧 maMul ⟨𝑦, 𝑦, 𝑦⟩)⟩))
4746reldmmpt2 6918 . . . . . . . . 9 Rel dom Mat
4847ovprc 6828 . . . . . . . 8 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 Mat 𝑅) = ∅)
493, 48syl5eq 2817 . . . . . . 7 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝐴 = ∅)
5049fveq2d 6336 . . . . . 6 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘𝐴) = (Base‘∅))
51 base0 16119 . . . . . 6 ∅ = (Base‘∅)
5250, 6, 513eqtr4g 2830 . . . . 5 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ∅)
5352mpteq1d 4872 . . . 4 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑚𝐵 ↦ (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥))))))) = (𝑚 ∈ ∅ ↦ (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥))))))))
5445, 53eqtr4d 2808 . . 3 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 maDet 𝑅) = (𝑚𝐵 ↦ (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥))))))))
5541, 54pm2.61i 176 . 2 (𝑁 maDet 𝑅) = (𝑚𝐵 ↦ (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥)))))))
561, 55eqtri 2793 1 𝐷 = (𝑚𝐵 ↦ (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥)))))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  c0 4063  cop 4322  cotp 4324  cmpt 4863   × cxp 5247  ccom 5253  cfv 6031  (class class class)co 6793  Fincfn 8109  ndxcnx 16061   sSet csts 16062  Basecbs 16064  .rcmulr 16150   Σg cgsu 16309  SymGrpcsymg 18004  pmSgncpsgn 18116  mulGrpcmgp 18697  ℤRHomczrh 20063   freeLMod cfrlm 20307   maMul cmmul 20406   Mat cmat 20430   maDet cmdat 20608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-slot 16068  df-base 16070  df-mat 20431  df-mdet 20609
This theorem is referenced by:  mdetleib  20611  nfimdetndef  20613  mdetfval1  20614  mdet0pr  20616  mdetf  20619
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