Theorem minmar1eval 22689

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 22688	. . . 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 1132	. . 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 1144	. 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 1214	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 𝐼 ∈ 𝑁)
10		simpl3r 1242	. . 3 ⊢ (((𝑀 ∈ 𝐵 ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) ∧ 𝑖 = 𝐼) → 𝐽 ∈ 𝑁)
11	4	fvexi 6877	. . . . . 6 ⊢ 1 ∈ V
12	5	fvexi 6877	. . . . . 6 ⊢ 0 ∈ V
13	11, 12	ifex 4530	. . . . 5 ⊢ if(𝑗 = 𝐿, 1 , 0 ) ∈ V
14		ovex 7425	. . . . 5 ⊢ (𝑖𝑀𝑗) ∈ V
15	13, 14	ifex 4530	. . . 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 2765	. . . . . 6 ⊢ (𝑖 = 𝐼 → (𝑖 = 𝐾 ↔ 𝐼 = 𝐾))
18	17	adantr 484	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑗 = 𝐽) → (𝑖 = 𝐾 ↔ 𝐼 = 𝐾))
19		eqeq1 2765	. . . . . . 7 ⊢ (𝑗 = 𝐽 → (𝑗 = 𝐿 ↔ 𝐽 = 𝐿))
20	19	adantl 485	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑗 = 𝐽) → (𝑗 = 𝐿 ↔ 𝐽 = 𝐿))
21	20	ifbid 4503	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑗 = 𝐽) → if(𝑗 = 𝐿, 1 , 0 ) = if(𝐽 = 𝐿, 1 , 0 ))
22		oveq12 7401	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑗 = 𝐽) → (𝑖𝑀𝑗) = (𝐼𝑀𝐽))
23	18, 21, 22	ifbieq12d 4508	. . . 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 485	. . 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 7550	. 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 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  Vcvv 3453  ifcif 4479  ‘cfv 6517  (class class class)co 7392   ∈ cmpo 7394  Basecbs 17228  0_gc0g 17451  1_rcur 20210   Mat cmat 22447   minMatR1 cminmar1 22673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-cnex 11126  ax-1cn 11128  ax-addcl 11130

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-nn 12208  df-slot 17201  df-ndx 17213  df-base 17229  df-mat 22448  df-minmar1 22675

This theorem is referenced by:  madjusmdetlem1  34085