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Theorem mdetfval 20311
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 6613 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
3 mdetfval.a . . . . . . . 8 𝐴 = (𝑁 Mat 𝑅)
42, 3syl6eqr 2673 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = 𝐴)
54fveq2d 6152 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = (Base‘𝐴))
6 mdetfval.b . . . . . 6 𝐵 = (Base‘𝐴)
75, 6syl6eqr 2673 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵)
8 simpr 477 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑟 = 𝑅)
9 simpl 473 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑛 = 𝑁)
109fveq2d 6152 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (SymGrp‘𝑛) = (SymGrp‘𝑁))
1110fveq2d 6152 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(SymGrp‘𝑛)) = (Base‘(SymGrp‘𝑁)))
12 mdetfval.p . . . . . . . 8 𝑃 = (Base‘(SymGrp‘𝑁))
1311, 12syl6eqr 2673 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(SymGrp‘𝑛)) = 𝑃)
14 fveq2 6148 . . . . . . . . . 10 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
1514adantl 482 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (.r𝑟) = (.r𝑅))
16 mdetfval.t . . . . . . . . 9 · = (.r𝑅)
1715, 16syl6eqr 2673 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (.r𝑟) = · )
188fveq2d 6152 . . . . . . . . . . 11 ((𝑛 = 𝑁𝑟 = 𝑅) → (ℤRHom‘𝑟) = (ℤRHom‘𝑅))
19 mdetfval.y . . . . . . . . . . 11 𝑌 = (ℤRHom‘𝑅)
2018, 19syl6eqr 2673 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → (ℤRHom‘𝑟) = 𝑌)
21 fveq2 6148 . . . . . . . . . . . 12 (𝑛 = 𝑁 → (pmSgn‘𝑛) = (pmSgn‘𝑁))
2221adantr 481 . . . . . . . . . . 11 ((𝑛 = 𝑁𝑟 = 𝑅) → (pmSgn‘𝑛) = (pmSgn‘𝑁))
23 mdetfval.s . . . . . . . . . . 11 𝑆 = (pmSgn‘𝑁)
2422, 23syl6eqr 2673 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → (pmSgn‘𝑛) = 𝑆)
2520, 24coeq12d 5246 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → ((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛)) = (𝑌𝑆))
2625fveq1d 6150 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝) = ((𝑌𝑆)‘𝑝))
27 fveq2 6148 . . . . . . . . . . 11 (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
2827adantl 482 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → (mulGrp‘𝑟) = (mulGrp‘𝑅))
29 mdetfval.u . . . . . . . . . 10 𝑈 = (mulGrp‘𝑅)
3028, 29syl6eqr 2673 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (mulGrp‘𝑟) = 𝑈)
319mpteq1d 4698 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑥𝑛 ↦ ((𝑝𝑥)𝑚𝑥)) = (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥)))
3230, 31oveq12d 6622 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → ((mulGrp‘𝑟) Σg (𝑥𝑛 ↦ ((𝑝𝑥)𝑚𝑥))) = (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥))))
3317, 26, 32oveq123d 6625 . . . . . . 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 4693 . . . . . 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 6622 . . . . 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 4693 . . . 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 20310 . . . 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 6158 . . . . . 6 (Base‘𝐴) ∈ V
396, 38eqeltri 2694 . . . . 5 𝐵 ∈ V
4039mptex 6440 . . . 4 (𝑚𝐵 ↦ (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥))))))) ∈ V
4136, 37, 40ovmpt2a 6744 . . 3 ((𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 maDet 𝑅) = (𝑚𝐵 ↦ (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥))))))))
4237reldmmpt2 6724 . . . . . 6 Rel dom maDet
4342ovprc 6636 . . . . 5 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 maDet 𝑅) = ∅)
44 mpt0 5978 . . . . 5 (𝑚 ∈ ∅ ↦ (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥))))))) = ∅
4543, 44syl6eqr 2673 . . . 4 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 maDet 𝑅) = (𝑚 ∈ ∅ ↦ (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥))))))))
46 df-mat 20133 . . . . . . . . . 10 Mat = (𝑦 ∈ Fin, 𝑧 ∈ V ↦ ((𝑧 freeLMod (𝑦 × 𝑦)) sSet ⟨(.r‘ndx), (𝑧 maMul ⟨𝑦, 𝑦, 𝑦⟩)⟩))
4746reldmmpt2 6724 . . . . . . . . 9 Rel dom Mat
4847ovprc 6636 . . . . . . . 8 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 Mat 𝑅) = ∅)
493, 48syl5eq 2667 . . . . . . 7 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝐴 = ∅)
5049fveq2d 6152 . . . . . 6 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘𝐴) = (Base‘∅))
51 base0 15833 . . . . . 6 ∅ = (Base‘∅)
5250, 6, 513eqtr4g 2680 . . . . 5 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ∅)
5352mpteq1d 4698 . . . 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 2658 . . 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 2643 1 𝐷 = (𝑚𝐵 ↦ (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥)))))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  c0 3891  cop 4154  cotp 4156  cmpt 4673   × cxp 5072  ccom 5078  cfv 5847  (class class class)co 6604  Fincfn 7899  ndxcnx 15778   sSet csts 15779  Basecbs 15781  .rcmulr 15863   Σg cgsu 16022  SymGrpcsymg 17718  pmSgncpsgn 17830  mulGrpcmgp 18410  ℤRHomczrh 19767   freeLMod cfrlm 20009   maMul cmmul 20108   Mat cmat 20132   maDet cmdat 20309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-slot 15785  df-base 15786  df-mat 20133  df-mdet 20310
This theorem is referenced by:  mdetleib  20312  nfimdetndef  20314  mdetfval1  20315  mdet0pr  20317  mdetf  20320
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