Theorem maduval 21787

Description: Second substitution for the adjunct (cofactor) matrix. (Contributed by SO, 11-Jul-2018.)

Hypotheses

Ref	Expression
madufval.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
madufval.d	⊢ 𝐷 = (𝑁 maDet 𝑅)
madufval.j	⊢ 𝐽 = (𝑁 maAdju 𝑅)
madufval.b	⊢ 𝐵 = (Base‘𝐴)
madufval.o	⊢ 1 = (1r‘𝑅)
madufval.z	⊢ 0 = (0g‘𝑅)

Assertion

Ref	Expression
maduval	⊢ (𝑀 ∈ 𝐵 → (𝐽‘𝑀) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙))))))

Distinct variable groups:   𝑖,𝑁,𝑗,𝑘,𝑙   𝑅,𝑖,𝑗,𝑘,𝑙   𝑖,𝑀,𝑗,𝑘,𝑙

Allowed substitution hints:   𝐴(𝑖,𝑗,𝑘,𝑙)   𝐵(𝑖,𝑗,𝑘,𝑙)   𝐷(𝑖,𝑗,𝑘,𝑙)   1 (𝑖,𝑗,𝑘,𝑙)   𝐽(𝑖,𝑗,𝑘,𝑙)   0 (𝑖,𝑗,𝑘,𝑙)

Proof of Theorem maduval

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		madufval.a	. . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		madufval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐴)
3	1, 2	matrcl 21559	. . . 4 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
4	3	simpld 495	. . 3 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin)
5		mpoexga 7918	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙))))) ∈ V)
6	4, 4, 5	syl2anc 584	. 2 ⊢ (𝑀 ∈ 𝐵 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙))))) ∈ V)
7		oveq 7281	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (𝑘𝑚𝑙) = (𝑘𝑀𝑙))
8	7	ifeq2d 4479	. . . . . . 7 ⊢ (𝑚 = 𝑀 → if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙)) = if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙)))
9	8	mpoeq3dv 7354	. . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙))) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙))))
10	9	3ad2ant1 1132	. . . . 5 ⊢ ((𝑚 = 𝑀 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙))) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙))))
11	10	fveq2d 6778	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙)))) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙)))))
12	11	mpoeq3dva 7352	. . 3 ⊢ (𝑚 = 𝑀 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙))))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙))))))
13		madufval.d	. . . 4 ⊢ 𝐷 = (𝑁 maDet 𝑅)
14		madufval.j	. . . 4 ⊢ 𝐽 = (𝑁 maAdju 𝑅)
15		madufval.o	. . . 4 ⊢ 1 = (1r‘𝑅)
16		madufval.z	. . . 4 ⊢ 0 = (0g‘𝑅)
17	1, 13, 14, 2, 15, 16	madufval 21786	. . 3 ⊢ 𝐽 = (𝑚 ∈ 𝐵 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙))))))
18	12, 17	fvmptg 6873	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙))))) ∈ V) → (𝐽‘𝑀) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙))))))
19	6, 18	mpdan 684	1 ⊢ (𝑀 ∈ 𝐵 → (𝐽‘𝑀) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙))))))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  Vcvv 3432  ifcif 4459  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  Fincfn 8733  Basecbs 16912  0_gc0g 17150  1_rcur 19737   Mat cmat 21554   maDet cmdat 21733   maAdju cmadu 21781

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-1cn 10929  ax-addcl 10931

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-nn 11974  df-slot 16883  df-ndx 16895  df-base 16913  df-mat 21555  df-madu 21783

This theorem is referenced by:  maducoeval  21788