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Theorem mdetfval 22592
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 7440 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
3 mdetfval.a . . . . . . . 8 𝐴 = (𝑁 Mat 𝑅)
42, 3eqtr4di 2795 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = 𝐴)
54fveq2d 6910 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = (Base‘𝐴))
6 mdetfval.b . . . . . 6 𝐵 = (Base‘𝐴)
75, 6eqtr4di 2795 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵)
8 simpr 484 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑟 = 𝑅)
9 simpl 482 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑛 = 𝑁)
109fveq2d 6910 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (SymGrp‘𝑛) = (SymGrp‘𝑁))
1110fveq2d 6910 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(SymGrp‘𝑛)) = (Base‘(SymGrp‘𝑁)))
12 mdetfval.p . . . . . . . 8 𝑃 = (Base‘(SymGrp‘𝑁))
1311, 12eqtr4di 2795 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(SymGrp‘𝑛)) = 𝑃)
14 fveq2 6906 . . . . . . . . . 10 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
1514adantl 481 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (.r𝑟) = (.r𝑅))
16 mdetfval.t . . . . . . . . 9 · = (.r𝑅)
1715, 16eqtr4di 2795 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (.r𝑟) = · )
188fveq2d 6910 . . . . . . . . . . 11 ((𝑛 = 𝑁𝑟 = 𝑅) → (ℤRHom‘𝑟) = (ℤRHom‘𝑅))
19 mdetfval.y . . . . . . . . . . 11 𝑌 = (ℤRHom‘𝑅)
2018, 19eqtr4di 2795 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → (ℤRHom‘𝑟) = 𝑌)
21 fveq2 6906 . . . . . . . . . . . 12 (𝑛 = 𝑁 → (pmSgn‘𝑛) = (pmSgn‘𝑁))
2221adantr 480 . . . . . . . . . . 11 ((𝑛 = 𝑁𝑟 = 𝑅) → (pmSgn‘𝑛) = (pmSgn‘𝑁))
23 mdetfval.s . . . . . . . . . . 11 𝑆 = (pmSgn‘𝑁)
2422, 23eqtr4di 2795 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → (pmSgn‘𝑛) = 𝑆)
2520, 24coeq12d 5875 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → ((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛)) = (𝑌𝑆))
2625fveq1d 6908 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝) = ((𝑌𝑆)‘𝑝))
27 fveq2 6906 . . . . . . . . . . 11 (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
2827adantl 481 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → (mulGrp‘𝑟) = (mulGrp‘𝑅))
29 mdetfval.u . . . . . . . . . 10 𝑈 = (mulGrp‘𝑅)
3028, 29eqtr4di 2795 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (mulGrp‘𝑟) = 𝑈)
319mpteq1d 5237 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑥𝑛 ↦ ((𝑝𝑥)𝑚𝑥)) = (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥)))
3230, 31oveq12d 7449 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → ((mulGrp‘𝑟) Σg (𝑥𝑛 ↦ ((𝑝𝑥)𝑚𝑥))) = (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥))))
3317, 26, 32oveq123d 7452 . . . . . . 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 5233 . . . . . 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 7449 . . . . 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 5233 . . . 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 22591 . . . 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))))))))
386fvexi 6920 . . . . 5 𝐵 ∈ V
3938mptex 7243 . . . 4 (𝑚𝐵 ↦ (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥))))))) ∈ V
4036, 37, 39ovmpoa 7588 . . 3 ((𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 maDet 𝑅) = (𝑚𝐵 ↦ (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥))))))))
4137reldmmpo 7567 . . . . . 6 Rel dom maDet
4241ovprc 7469 . . . . 5 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 maDet 𝑅) = ∅)
43 mpt0 6710 . . . . 5 (𝑚 ∈ ∅ ↦ (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥))))))) = ∅
4442, 43eqtr4di 2795 . . . 4 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 maDet 𝑅) = (𝑚 ∈ ∅ ↦ (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥))))))))
45 df-mat 22412 . . . . . . . . . 10 Mat = (𝑦 ∈ Fin, 𝑧 ∈ V ↦ ((𝑧 freeLMod (𝑦 × 𝑦)) sSet ⟨(.r‘ndx), (𝑧 maMul ⟨𝑦, 𝑦, 𝑦⟩)⟩))
4645reldmmpo 7567 . . . . . . . . 9 Rel dom Mat
4746ovprc 7469 . . . . . . . 8 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 Mat 𝑅) = ∅)
483, 47eqtrid 2789 . . . . . . 7 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝐴 = ∅)
4948fveq2d 6910 . . . . . 6 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘𝐴) = (Base‘∅))
50 base0 17252 . . . . . 6 ∅ = (Base‘∅)
5149, 6, 503eqtr4g 2802 . . . . 5 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ∅)
5251mpteq1d 5237 . . . 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))))))))
5344, 52eqtr4d 2780 . . 3 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 maDet 𝑅) = (𝑚𝐵 ↦ (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥))))))))
5440, 53pm2.61i 182 . 2 (𝑁 maDet 𝑅) = (𝑚𝐵 ↦ (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥)))))))
551, 54eqtri 2765 1 𝐷 = (𝑚𝐵 ↦ (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥)))))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  c0 4333  cop 4632  cotp 4634  cmpt 5225   × cxp 5683  ccom 5689  cfv 6561  (class class class)co 7431  Fincfn 8985   sSet csts 17200  ndxcnx 17230  Basecbs 17247  .rcmulr 17298   Σg cgsu 17485  SymGrpcsymg 19386  pmSgncpsgn 19507  mulGrpcmgp 20137  ℤRHomczrh 21510   freeLMod cfrlm 21766   maMul cmmul 22394   Mat cmat 22411   maDet cmdat 22590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-1cn 11213  ax-addcl 11215
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-nn 12267  df-slot 17219  df-ndx 17231  df-base 17248  df-mat 22412  df-mdet 22591
This theorem is referenced by:  mdetleib  22593  nfimdetndef  22595  mdetfval1  22596  mdet0pr  22598  mdetf  22601
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