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Theorem mdetfval 22309
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 7421 . . . . . . . 8 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (𝑛 Mat π‘Ÿ) = (𝑁 Mat 𝑅))
3 mdetfval.a . . . . . . . 8 𝐴 = (𝑁 Mat 𝑅)
42, 3eqtr4di 2789 . . . . . . 7 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (𝑛 Mat π‘Ÿ) = 𝐴)
54fveq2d 6895 . . . . . 6 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (Baseβ€˜(𝑛 Mat π‘Ÿ)) = (Baseβ€˜π΄))
6 mdetfval.b . . . . . 6 𝐡 = (Baseβ€˜π΄)
75, 6eqtr4di 2789 . . . . 5 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (Baseβ€˜(𝑛 Mat π‘Ÿ)) = 𝐡)
8 simpr 484 . . . . . 6 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ π‘Ÿ = 𝑅)
9 simpl 482 . . . . . . . . . 10 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ 𝑛 = 𝑁)
109fveq2d 6895 . . . . . . . . 9 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (SymGrpβ€˜π‘›) = (SymGrpβ€˜π‘))
1110fveq2d 6895 . . . . . . . 8 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (Baseβ€˜(SymGrpβ€˜π‘›)) = (Baseβ€˜(SymGrpβ€˜π‘)))
12 mdetfval.p . . . . . . . 8 𝑃 = (Baseβ€˜(SymGrpβ€˜π‘))
1311, 12eqtr4di 2789 . . . . . . 7 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (Baseβ€˜(SymGrpβ€˜π‘›)) = 𝑃)
14 fveq2 6891 . . . . . . . . . 10 (π‘Ÿ = 𝑅 β†’ (.rβ€˜π‘Ÿ) = (.rβ€˜π‘…))
1514adantl 481 . . . . . . . . 9 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (.rβ€˜π‘Ÿ) = (.rβ€˜π‘…))
16 mdetfval.t . . . . . . . . 9 Β· = (.rβ€˜π‘…)
1715, 16eqtr4di 2789 . . . . . . . 8 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (.rβ€˜π‘Ÿ) = Β· )
188fveq2d 6895 . . . . . . . . . . 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 2789 . . . . . . . . . 10 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (β„€RHomβ€˜π‘Ÿ) = π‘Œ)
21 fveq2 6891 . . . . . . . . . . . 12 (𝑛 = 𝑁 β†’ (pmSgnβ€˜π‘›) = (pmSgnβ€˜π‘))
2221adantr 480 . . . . . . . . . . 11 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (pmSgnβ€˜π‘›) = (pmSgnβ€˜π‘))
23 mdetfval.s . . . . . . . . . . 11 𝑆 = (pmSgnβ€˜π‘)
2422, 23eqtr4di 2789 . . . . . . . . . 10 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (pmSgnβ€˜π‘›) = 𝑆)
2520, 24coeq12d 5864 . . . . . . . . 9 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ ((β„€RHomβ€˜π‘Ÿ) ∘ (pmSgnβ€˜π‘›)) = (π‘Œ ∘ 𝑆))
2625fveq1d 6893 . . . . . . . 8 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (((β„€RHomβ€˜π‘Ÿ) ∘ (pmSgnβ€˜π‘›))β€˜π‘) = ((π‘Œ ∘ 𝑆)β€˜π‘))
27 fveq2 6891 . . . . . . . . . . 11 (π‘Ÿ = 𝑅 β†’ (mulGrpβ€˜π‘Ÿ) = (mulGrpβ€˜π‘…))
2827adantl 481 . . . . . . . . . 10 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (mulGrpβ€˜π‘Ÿ) = (mulGrpβ€˜π‘…))
29 mdetfval.u . . . . . . . . . 10 π‘ˆ = (mulGrpβ€˜π‘…)
3028, 29eqtr4di 2789 . . . . . . . . 9 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (mulGrpβ€˜π‘Ÿ) = π‘ˆ)
319mpteq1d 5243 . . . . . . . . 9 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (π‘₯ ∈ 𝑛 ↦ ((π‘β€˜π‘₯)π‘šπ‘₯)) = (π‘₯ ∈ 𝑁 ↦ ((π‘β€˜π‘₯)π‘šπ‘₯)))
3230, 31oveq12d 7430 . . . . . . . 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 7433 . . . . . . 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 5239 . . . . . 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 7430 . . . . 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 5239 . . . 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 22308 . . . 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 6905 . . . . 5 𝐡 ∈ V
3938mptex 7227 . . . 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 7566 . . 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 7546 . . . . . 6 Rel dom maDet
4241ovprc 7450 . . . . 5 (Β¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ (𝑁 maDet 𝑅) = βˆ…)
43 mpt0 6692 . . . . 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 2789 . . . 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 22129 . . . . . . . . . 10 Mat = (𝑦 ∈ Fin, 𝑧 ∈ V ↦ ((𝑧 freeLMod (𝑦 Γ— 𝑦)) sSet ⟨(.rβ€˜ndx), (𝑧 maMul βŸ¨π‘¦, 𝑦, π‘¦βŸ©)⟩))
4645reldmmpo 7546 . . . . . . . . 9 Rel dom Mat
4746ovprc 7450 . . . . . . . 8 (Β¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ (𝑁 Mat 𝑅) = βˆ…)
483, 47eqtrid 2783 . . . . . . 7 (Β¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ 𝐴 = βˆ…)
4948fveq2d 6895 . . . . . 6 (Β¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ (Baseβ€˜π΄) = (Baseβ€˜βˆ…))
50 base0 17154 . . . . . 6 βˆ… = (Baseβ€˜βˆ…)
5149, 6, 503eqtr4g 2796 . . . . 5 (Β¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) β†’ 𝐡 = βˆ…)
5251mpteq1d 5243 . . . 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 2774 . . 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 2759 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 395   = wceq 1540   ∈ wcel 2105  Vcvv 3473  βˆ…c0 4322  βŸ¨cop 4634  βŸ¨cotp 4636   ↦ cmpt 5231   Γ— cxp 5674   ∘ ccom 5680  β€˜cfv 6543  (class class class)co 7412  Fincfn 8943   sSet csts 17101  ndxcnx 17131  Basecbs 17149  .rcmulr 17203   Ξ£g cgsu 17391  SymGrpcsymg 19276  pmSgncpsgn 19399  mulGrpcmgp 20029  β„€RHomczrh 21269   freeLMod cfrlm 21521   maMul cmmul 22106   Mat cmat 22128   maDet cmdat 22307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11170  ax-1cn 11172  ax-addcl 11174
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-nn 12218  df-slot 17120  df-ndx 17132  df-base 17150  df-mat 22129  df-mdet 22308
This theorem is referenced by:  mdetleib  22310  nfimdetndef  22312  mdetfval1  22313  mdet0pr  22315  mdetf  22318
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