Theorem minmar1eval 22639

Description: An entry of a matrix for a minor. (Contributed by AV, 31-Dec-2018.)

Hypotheses

Ref	Expression
minmar1fval.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
minmar1fval.b	⊢ 𝐵 = (Base‘𝐴)
minmar1fval.q	⊢ 𝑄 = (𝑁 minMatR1 𝑅)
minmar1fval.o	⊢ 1 = (1r‘𝑅)
minmar1fval.z	⊢ 0 = (0g‘𝑅)

Assertion

Ref	Expression
minmar1eval	⊢ ((𝑀 ∈ 𝐵 ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝐾(𝑄‘𝑀)𝐿)𝐽) = if(𝐼 = 𝐾, if(𝐽 = 𝐿, 1 , 0 ), (𝐼𝑀𝐽)))

Proof of Theorem minmar1eval

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

Step	Hyp	Ref	Expression
1		minmar1fval.a	. . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		minmar1fval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐴)
3		minmar1fval.q	. . . . 5 ⊢ 𝑄 = (𝑁 minMatR1 𝑅)
4		minmar1fval.o	. . . . 5 ⊢ 1 = (1r‘𝑅)
5		minmar1fval.z	. . . . 5 ⊢ 0 = (0g‘𝑅)
6	1, 2, 3, 4, 5	minmar1val 22638	. . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝐾(𝑄‘𝑀)𝐿) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗))))
7	6	3expb 1126	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) → (𝐾(𝑄‘𝑀)𝐿) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗))))
8	7	3adant3 1138	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐾(𝑄‘𝑀)𝐿) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗))))
9		simp3l 1208	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 𝐼 ∈ 𝑁)
10		simpl3r 1236	. . 3 ⊢ (((𝑀 ∈ 𝐵 ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) ∧ 𝑖 = 𝐼) → 𝐽 ∈ 𝑁)
11	4	fvexi 6848	. . . . . 6 ⊢ 1 ∈ V
12	5	fvexi 6848	. . . . . 6 ⊢ 0 ∈ V
13	11, 12	ifex 4512	. . . . 5 ⊢ if(𝑗 = 𝐿, 1 , 0 ) ∈ V
14		ovex 7396	. . . . 5 ⊢ (𝑖𝑀𝑗) ∈ V
15	13, 14	ifex 4512	. . . 4 ⊢ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)) ∈ V
16	15	a1i 11	. . 3 ⊢ (((𝑀 ∈ 𝐵 ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) ∧ (𝑖 = 𝐼 ∧ 𝑗 = 𝐽)) → if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)) ∈ V)
17		eqeq1 2744	. . . . . 6 ⊢ (𝑖 = 𝐼 → (𝑖 = 𝐾 ↔ 𝐼 = 𝐾))
18	17	adantr 481	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑗 = 𝐽) → (𝑖 = 𝐾 ↔ 𝐼 = 𝐾))
19		eqeq1 2744	. . . . . . 7 ⊢ (𝑗 = 𝐽 → (𝑗 = 𝐿 ↔ 𝐽 = 𝐿))
20	19	adantl 482	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑗 = 𝐽) → (𝑗 = 𝐿 ↔ 𝐽 = 𝐿))
21	20	ifbid 4485	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑗 = 𝐽) → if(𝑗 = 𝐿, 1 , 0 ) = if(𝐽 = 𝐿, 1 , 0 ))
22		oveq12 7372	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑗 = 𝐽) → (𝑖𝑀𝑗) = (𝐼𝑀𝐽))
23	18, 21, 22	ifbieq12d 4490	. . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑗 = 𝐽) → if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)) = if(𝐼 = 𝐾, if(𝐽 = 𝐿, 1 , 0 ), (𝐼𝑀𝐽)))
24	23	adantl 482	. . 3 ⊢ (((𝑀 ∈ 𝐵 ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) ∧ (𝑖 = 𝐼 ∧ 𝑗 = 𝐽)) → if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)) = if(𝐼 = 𝐾, if(𝐽 = 𝐿, 1 , 0 ), (𝐼𝑀𝐽)))
25	9, 10, 16, 24	ovmpodv2 7521	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → ((𝐾(𝑄‘𝑀)𝐿) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗))) → (𝐼(𝐾(𝑄‘𝑀)𝐿)𝐽) = if(𝐼 = 𝐾, if(𝐽 = 𝐿, 1 , 0 ), (𝐼𝑀𝐽))))
26	8, 25	mpd 15	1 ⊢ ((𝑀 ∈ 𝐵 ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝐾(𝑄‘𝑀)𝐿)𝐽) = if(𝐼 = 𝐾, if(𝐽 = 𝐿, 1 , 0 ), (𝐼𝑀𝐽)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  Vcvv 3432  ifcif 4461  ‘cfv 6492  (class class class)co 7363   ∈ cmpo 7365  Basecbs 17177  0_gc0g 17400  1_rcur 20160   Mat cmat 22397   minMatR1 cminmar1 22623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-1cn 11094  ax-addcl 11096

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-nn 12173  df-slot 17150  df-ndx 17162  df-base 17178  df-mat 22398  df-minmar1 22625

This theorem is referenced by:  madjusmdetlem1  34018