Theorem maducoeval 21536

Description: An entry of 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
maducoeval	⊢ ((𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙)))))

Distinct variable groups:   𝑘,𝑁,𝑙   𝑅,𝑘,𝑙   𝑘,𝑀,𝑙   𝑘,𝐼,𝑙   𝑘,𝐻,𝑙

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

Proof of Theorem maducoeval

Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		madufval.a	. . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		madufval.d	. . . 4 ⊢ 𝐷 = (𝑁 maDet 𝑅)
3		madufval.j	. . . 4 ⊢ 𝐽 = (𝑁 maAdju 𝑅)
4		madufval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐴)
5		madufval.o	. . . 4 ⊢ 1 = (1r‘𝑅)
6		madufval.z	. . . 4 ⊢ 0 = (0g‘𝑅)
7	1, 2, 3, 4, 5, 6	maduval 21535	. . 3 ⊢ (𝑀 ∈ 𝐵 → (𝐽‘𝑀) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙))))))
8	7	3ad2ant1 1135	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → (𝐽‘𝑀) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙))))))
9		simp1r 1200	. . . . . . 7 ⊢ (((𝑖 = 𝐼 ∧ 𝑗 = 𝐻) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → 𝑗 = 𝐻)
10	9	eqeq2d 2748	. . . . . 6 ⊢ (((𝑖 = 𝐼 ∧ 𝑗 = 𝐻) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → (𝑘 = 𝑗 ↔ 𝑘 = 𝐻))
11		simp1l 1199	. . . . . . . 8 ⊢ (((𝑖 = 𝐼 ∧ 𝑗 = 𝐻) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → 𝑖 = 𝐼)
12	11	eqeq2d 2748	. . . . . . 7 ⊢ (((𝑖 = 𝐼 ∧ 𝑗 = 𝐻) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → (𝑙 = 𝑖 ↔ 𝑙 = 𝐼))
13	12	ifbid 4462	. . . . . 6 ⊢ (((𝑖 = 𝐼 ∧ 𝑗 = 𝐻) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → if(𝑙 = 𝑖, 1 , 0 ) = if(𝑙 = 𝐼, 1 , 0 ))
14	10, 13	ifbieq1d 4463	. . . . 5 ⊢ (((𝑖 = 𝐼 ∧ 𝑗 = 𝐻) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙)) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙)))
15	14	mpoeq3dva 7288	. . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑗 = 𝐻) → (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙))) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙))))
16	15	fveq2d 6721	. . 3 ⊢ ((𝑖 = 𝐼 ∧ 𝑗 = 𝐻) → (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙)))) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙)))))
17	16	adantl 485	. 2 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑖 = 𝐼 ∧ 𝑗 = 𝐻)) → (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙)))) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙)))))
18		simp2 1139	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → 𝐼 ∈ 𝑁)
19		simp3 1140	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → 𝐻 ∈ 𝑁)
20		fvexd 6732	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙)))) ∈ V)
21	8, 17, 18, 19, 20	ovmpod 7361	1 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙)))))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  Vcvv 3408  ifcif 4439  ‘cfv 6380  (class class class)co 7213   ∈ cmpo 7215  Basecbs 16760  0_gc0g 16944  1_rcur 19516   Mat cmat 21304   maDet cmdat 21481   maAdju cmadu 21529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-1cn 10787  ax-addcl 10789

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-nn 11831  df-slot 16735  df-ndx 16745  df-base 16761  df-mat 21305  df-madu 21531

This theorem is referenced by:  maducoeval2  21537  madugsum  21540  maducoevalmin1  21549