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Theorem mdetfval 21958
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 𝐡 = (Baseβ€˜π΄)
mdetfval.p 𝑃 = (Baseβ€˜(SymGrpβ€˜π‘))
mdetfval.y π‘Œ = (β„€RHomβ€˜π‘…)
mdetfval.s 𝑆 = (pmSgnβ€˜π‘)
mdetfval.t Β· = (.rβ€˜π‘…)
mdetfval.u π‘ˆ = (mulGrpβ€˜π‘…)
Assertion
Ref Expression
mdetfval 𝐷 = (π‘š ∈ 𝐡 ↦ (𝑅 Ξ£g (𝑝 ∈ 𝑃 ↦ (((π‘Œ ∘ 𝑆)β€˜π‘) Β· (π‘ˆ Ξ£g (π‘₯ ∈ 𝑁 ↦ ((π‘β€˜π‘₯)π‘šπ‘₯)))))))
Distinct variable groups:   𝐡,π‘š   π‘š,𝑝,π‘₯,𝑁   𝑃,π‘š   𝑅,π‘š,𝑝,π‘₯   𝑆,π‘š   Β· ,π‘š   π‘ˆ,π‘š   π‘š,π‘Œ
Allowed substitution hints:   𝐴(π‘₯,π‘š,𝑝)   𝐡(π‘₯,𝑝)   𝐷(π‘₯,π‘š,𝑝)   𝑃(π‘₯,𝑝)   𝑆(π‘₯,𝑝)   Β· (π‘₯,𝑝)   π‘ˆ(π‘₯,𝑝)   π‘Œ(π‘₯,𝑝)

Proof of Theorem mdetfval
Dummy variables 𝑦 𝑧 𝑛 π‘Ÿ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mdetfval.d . 2 𝐷 = (𝑁 maDet 𝑅)
2 oveq12 7370 . . . . . . . 8 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (𝑛 Mat π‘Ÿ) = (𝑁 Mat 𝑅))
3 mdetfval.a . . . . . . . 8 𝐴 = (𝑁 Mat 𝑅)
42, 3eqtr4di 2791 . . . . . . 7 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (𝑛 Mat π‘Ÿ) = 𝐴)
54fveq2d 6850 . . . . . 6 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (Baseβ€˜(𝑛 Mat π‘Ÿ)) = (Baseβ€˜π΄))
6 mdetfval.b . . . . . 6 𝐡 = (Baseβ€˜π΄)
75, 6eqtr4di 2791 . . . . 5 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (Baseβ€˜(𝑛 Mat π‘Ÿ)) = 𝐡)
8 simpr 486 . . . . . 6 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ π‘Ÿ = 𝑅)
9 simpl 484 . . . . . . . . . 10 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ 𝑛 = 𝑁)
109fveq2d 6850 . . . . . . . . 9 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (SymGrpβ€˜π‘›) = (SymGrpβ€˜π‘))
1110fveq2d 6850 . . . . . . . 8 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (Baseβ€˜(SymGrpβ€˜π‘›)) = (Baseβ€˜(SymGrpβ€˜π‘)))
12 mdetfval.p . . . . . . . 8 𝑃 = (Baseβ€˜(SymGrpβ€˜π‘))
1311, 12eqtr4di 2791 . . . . . . 7 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (Baseβ€˜(SymGrpβ€˜π‘›)) = 𝑃)
14 fveq2 6846 . . . . . . . . . 10 (π‘Ÿ = 𝑅 β†’ (.rβ€˜π‘Ÿ) = (.rβ€˜π‘…))
1514adantl 483 . . . . . . . . 9 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (.rβ€˜π‘Ÿ) = (.rβ€˜π‘…))
16 mdetfval.t . . . . . . . . 9 Β· = (.rβ€˜π‘…)
1715, 16eqtr4di 2791 . . . . . . . 8 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (.rβ€˜π‘Ÿ) = Β· )
188fveq2d 6850 . . . . . . . . . . 11 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (β„€RHomβ€˜π‘Ÿ) = (β„€RHomβ€˜π‘…))
19 mdetfval.y . . . . . . . . . . 11 π‘Œ = (β„€RHomβ€˜π‘…)
2018, 19eqtr4di 2791 . . . . . . . . . 10 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (β„€RHomβ€˜π‘Ÿ) = π‘Œ)
21 fveq2 6846 . . . . . . . . . . . 12 (𝑛 = 𝑁 β†’ (pmSgnβ€˜π‘›) = (pmSgnβ€˜π‘))
2221adantr 482 . . . . . . . . . . 11 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (pmSgnβ€˜π‘›) = (pmSgnβ€˜π‘))
23 mdetfval.s . . . . . . . . . . 11 𝑆 = (pmSgnβ€˜π‘)
2422, 23eqtr4di 2791 . . . . . . . . . 10 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (pmSgnβ€˜π‘›) = 𝑆)
2520, 24coeq12d 5824 . . . . . . . . 9 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ ((β„€RHomβ€˜π‘Ÿ) ∘ (pmSgnβ€˜π‘›)) = (π‘Œ ∘ 𝑆))
2625fveq1d 6848 . . . . . . . 8 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (((β„€RHomβ€˜π‘Ÿ) ∘ (pmSgnβ€˜π‘›))β€˜π‘) = ((π‘Œ ∘ 𝑆)β€˜π‘))
27 fveq2 6846 . . . . . . . . . . 11 (π‘Ÿ = 𝑅 β†’ (mulGrpβ€˜π‘Ÿ) = (mulGrpβ€˜π‘…))
2827adantl 483 . . . . . . . . . 10 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (mulGrpβ€˜π‘Ÿ) = (mulGrpβ€˜π‘…))
29 mdetfval.u . . . . . . . . . 10 π‘ˆ = (mulGrpβ€˜π‘…)
3028, 29eqtr4di 2791 . . . . . . . . 9 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (mulGrpβ€˜π‘Ÿ) = π‘ˆ)
319mpteq1d 5204 . . . . . . . . 9 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (π‘₯ ∈ 𝑛 ↦ ((π‘β€˜π‘₯)π‘šπ‘₯)) = (π‘₯ ∈ 𝑁 ↦ ((π‘β€˜π‘₯)π‘šπ‘₯)))
3230, 31oveq12d 7379 . . . . . . . 8 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ ((mulGrpβ€˜π‘Ÿ) Ξ£g (π‘₯ ∈ 𝑛 ↦ ((π‘β€˜π‘₯)π‘šπ‘₯))) = (π‘ˆ Ξ£g (π‘₯ ∈ 𝑁 ↦ ((π‘β€˜π‘₯)π‘šπ‘₯))))
3317, 26, 32oveq123d 7382 . . . . . . 7 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ ((((β„€RHomβ€˜π‘Ÿ) ∘ (pmSgnβ€˜π‘›))β€˜π‘)(.rβ€˜π‘Ÿ)((mulGrpβ€˜π‘Ÿ) Ξ£g (π‘₯ ∈ 𝑛 ↦ ((π‘β€˜π‘₯)π‘šπ‘₯)))) = (((π‘Œ ∘ 𝑆)β€˜π‘) Β· (π‘ˆ Ξ£g (π‘₯ ∈ 𝑁 ↦ ((π‘β€˜π‘₯)π‘šπ‘₯)))))
3413, 33mpteq12dv 5200 . . . . . 6 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (𝑝 ∈ (Baseβ€˜(SymGrpβ€˜π‘›)) ↦ ((((β„€RHomβ€˜π‘Ÿ) ∘ (pmSgnβ€˜π‘›))β€˜π‘)(.rβ€˜π‘Ÿ)((mulGrpβ€˜π‘Ÿ) Ξ£g (π‘₯ ∈ 𝑛 ↦ ((π‘β€˜π‘₯)π‘šπ‘₯))))) = (𝑝 ∈ 𝑃 ↦ (((π‘Œ ∘ 𝑆)β€˜π‘) Β· (π‘ˆ Ξ£g (π‘₯ ∈ 𝑁 ↦ ((π‘β€˜π‘₯)π‘šπ‘₯))))))
358, 34oveq12d 7379 . . . . 5 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (π‘Ÿ Ξ£g (𝑝 ∈ (Baseβ€˜(SymGrpβ€˜π‘›)) ↦ ((((β„€RHomβ€˜π‘Ÿ) ∘ (pmSgnβ€˜π‘›))β€˜π‘)(.rβ€˜π‘Ÿ)((mulGrpβ€˜π‘Ÿ) Ξ£g (π‘₯ ∈ 𝑛 ↦ ((π‘β€˜π‘₯)π‘šπ‘₯)))))) = (𝑅 Ξ£g (𝑝 ∈ 𝑃 ↦ (((π‘Œ ∘ 𝑆)β€˜π‘) Β· (π‘ˆ Ξ£g (π‘₯ ∈ 𝑁 ↦ ((π‘β€˜π‘₯)π‘šπ‘₯)))))))
367, 35mpteq12dv 5200 . . . 4 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (π‘š ∈ (Baseβ€˜(𝑛 Mat π‘Ÿ)) ↦ (π‘Ÿ Ξ£g (𝑝 ∈ (Baseβ€˜(SymGrpβ€˜π‘›)) ↦ ((((β„€RHomβ€˜π‘Ÿ) ∘ (pmSgnβ€˜π‘›))β€˜π‘)(.rβ€˜π‘Ÿ)((mulGrpβ€˜π‘Ÿ) Ξ£g (π‘₯ ∈ 𝑛 ↦ ((π‘β€˜π‘₯)π‘šπ‘₯))))))) = (π‘š ∈ 𝐡 ↦ (𝑅 Ξ£g (𝑝 ∈ 𝑃 ↦ (((π‘Œ ∘ 𝑆)β€˜π‘) Β· (π‘ˆ Ξ£g (π‘₯ ∈ 𝑁 ↦ ((π‘β€˜π‘₯)π‘šπ‘₯))))))))
37 df-mdet 21957 . . . 4 maDet = (𝑛 ∈ V, π‘Ÿ ∈ V ↦ (π‘š ∈ (Baseβ€˜(𝑛 Mat π‘Ÿ)) ↦ (π‘Ÿ Ξ£g (𝑝 ∈ (Baseβ€˜(SymGrpβ€˜π‘›)) ↦ ((((β„€RHomβ€˜π‘Ÿ) ∘ (pmSgnβ€˜π‘›))β€˜π‘)(.rβ€˜π‘Ÿ)((mulGrpβ€˜π‘Ÿ) Ξ£g (π‘₯ ∈ 𝑛 ↦ ((π‘β€˜π‘₯)π‘šπ‘₯))))))))
386fvexi 6860 . . . . 5 𝐡 ∈ V
3938mptex 7177 . . . 4 (π‘š ∈ 𝐡 ↦ (𝑅 Ξ£g (𝑝 ∈ 𝑃 ↦ (((π‘Œ ∘ 𝑆)β€˜π‘) Β· (π‘ˆ Ξ£g (π‘₯ ∈ 𝑁 ↦ ((π‘β€˜π‘₯)π‘šπ‘₯))))))) ∈ V
4036, 37, 39ovmpoa 7514 . . 3 ((𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ (𝑁 maDet 𝑅) = (π‘š ∈ 𝐡 ↦ (𝑅 Ξ£g (𝑝 ∈ 𝑃 ↦ (((π‘Œ ∘ 𝑆)β€˜π‘) Β· (π‘ˆ Ξ£g (π‘₯ ∈ 𝑁 ↦ ((π‘β€˜π‘₯)π‘šπ‘₯))))))))
4137reldmmpo 7494 . . . . . 6 Rel dom maDet
4241ovprc 7399 . . . . 5 (Β¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ (𝑁 maDet 𝑅) = βˆ…)
43 mpt0 6647 . . . . 5 (π‘š ∈ βˆ… ↦ (𝑅 Ξ£g (𝑝 ∈ 𝑃 ↦ (((π‘Œ ∘ 𝑆)β€˜π‘) Β· (π‘ˆ Ξ£g (π‘₯ ∈ 𝑁 ↦ ((π‘β€˜π‘₯)π‘šπ‘₯))))))) = βˆ…
4442, 43eqtr4di 2791 . . . 4 (Β¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ (𝑁 maDet 𝑅) = (π‘š ∈ βˆ… ↦ (𝑅 Ξ£g (𝑝 ∈ 𝑃 ↦ (((π‘Œ ∘ 𝑆)β€˜π‘) Β· (π‘ˆ Ξ£g (π‘₯ ∈ 𝑁 ↦ ((π‘β€˜π‘₯)π‘šπ‘₯))))))))
45 df-mat 21778 . . . . . . . . . 10 Mat = (𝑦 ∈ Fin, 𝑧 ∈ V ↦ ((𝑧 freeLMod (𝑦 Γ— 𝑦)) sSet ⟨(.rβ€˜ndx), (𝑧 maMul βŸ¨π‘¦, 𝑦, π‘¦βŸ©)⟩))
4645reldmmpo 7494 . . . . . . . . 9 Rel dom Mat
4746ovprc 7399 . . . . . . . 8 (Β¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ (𝑁 Mat 𝑅) = βˆ…)
483, 47eqtrid 2785 . . . . . . 7 (Β¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ 𝐴 = βˆ…)
4948fveq2d 6850 . . . . . 6 (Β¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ (Baseβ€˜π΄) = (Baseβ€˜βˆ…))
50 base0 17096 . . . . . 6 βˆ… = (Baseβ€˜βˆ…)
5149, 6, 503eqtr4g 2798 . . . . 5 (Β¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ 𝐡 = βˆ…)
5251mpteq1d 5204 . . . 4 (Β¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ (π‘š ∈ 𝐡 ↦ (𝑅 Ξ£g (𝑝 ∈ 𝑃 ↦ (((π‘Œ ∘ 𝑆)β€˜π‘) Β· (π‘ˆ Ξ£g (π‘₯ ∈ 𝑁 ↦ ((π‘β€˜π‘₯)π‘šπ‘₯))))))) = (π‘š ∈ βˆ… ↦ (𝑅 Ξ£g (𝑝 ∈ 𝑃 ↦ (((π‘Œ ∘ 𝑆)β€˜π‘) Β· (π‘ˆ Ξ£g (π‘₯ ∈ 𝑁 ↦ ((π‘β€˜π‘₯)π‘šπ‘₯))))))))
5344, 52eqtr4d 2776 . . 3 (Β¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ (𝑁 maDet 𝑅) = (π‘š ∈ 𝐡 ↦ (𝑅 Ξ£g (𝑝 ∈ 𝑃 ↦ (((π‘Œ ∘ 𝑆)β€˜π‘) Β· (π‘ˆ Ξ£g (π‘₯ ∈ 𝑁 ↦ ((π‘β€˜π‘₯)π‘šπ‘₯))))))))
5440, 53pm2.61i 182 . 2 (𝑁 maDet 𝑅) = (π‘š ∈ 𝐡 ↦ (𝑅 Ξ£g (𝑝 ∈ 𝑃 ↦ (((π‘Œ ∘ 𝑆)β€˜π‘) Β· (π‘ˆ Ξ£g (π‘₯ ∈ 𝑁 ↦ ((π‘β€˜π‘₯)π‘šπ‘₯)))))))
551, 54eqtri 2761 1 𝐷 = (π‘š ∈ 𝐡 ↦ (𝑅 Ξ£g (𝑝 ∈ 𝑃 ↦ (((π‘Œ ∘ 𝑆)β€˜π‘) Β· (π‘ˆ Ξ£g (π‘₯ ∈ 𝑁 ↦ ((π‘β€˜π‘₯)π‘šπ‘₯)))))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3447  βˆ…c0 4286  βŸ¨cop 4596  βŸ¨cotp 4598   ↦ cmpt 5192   Γ— cxp 5635   ∘ ccom 5641  β€˜cfv 6500  (class class class)co 7361  Fincfn 8889   sSet csts 17043  ndxcnx 17073  Basecbs 17091  .rcmulr 17142   Ξ£g cgsu 17330  SymGrpcsymg 19156  pmSgncpsgn 19279  mulGrpcmgp 19904  β„€RHomczrh 20923   freeLMod cfrlm 21175   maMul cmmul 21755   Mat cmat 21777   maDet cmdat 21956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-1cn 11117  ax-addcl 11119
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-nn 12162  df-slot 17062  df-ndx 17074  df-base 17092  df-mat 21778  df-mdet 21957
This theorem is referenced by:  mdetleib  21959  nfimdetndef  21961  mdetfval1  21962  mdet0pr  21964  mdetf  21967
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