Theorem minmar1eval 20455

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 20454	. . . 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 1266	. . 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 1081	. 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 1089	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 𝐼 ∈ 𝑁)
10		simpl3r 1117	. . 3 ⊢ (((𝑀 ∈ 𝐵 ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) ∧ 𝑖 = 𝐼) → 𝐽 ∈ 𝑁)
11		fvex 6201	. . . . . . 7 ⊢ (1r‘𝑅) ∈ V
12	4, 11	eqeltri 2697	. . . . . 6 ⊢ 1 ∈ V
13		fvex 6201	. . . . . . 7 ⊢ (0g‘𝑅) ∈ V
14	5, 13	eqeltri 2697	. . . . . 6 ⊢ 0 ∈ V
15	12, 14	ifex 4156	. . . . 5 ⊢ if(𝑗 = 𝐿, 1 , 0 ) ∈ V
16		ovex 6678	. . . . 5 ⊢ (𝑖𝑀𝑗) ∈ V
17	15, 16	ifex 4156	. . . 4 ⊢ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)) ∈ V
18	17	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)
19		eqeq1 2626	. . . . . 6 ⊢ (𝑖 = 𝐼 → (𝑖 = 𝐾 ↔ 𝐼 = 𝐾))
20	19	adantr 481	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑗 = 𝐽) → (𝑖 = 𝐾 ↔ 𝐼 = 𝐾))
21		eqeq1 2626	. . . . . . 7 ⊢ (𝑗 = 𝐽 → (𝑗 = 𝐿 ↔ 𝐽 = 𝐿))
22	21	adantl 482	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑗 = 𝐽) → (𝑗 = 𝐿 ↔ 𝐽 = 𝐿))
23	22	ifbid 4108	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑗 = 𝐽) → if(𝑗 = 𝐿, 1 , 0 ) = if(𝐽 = 𝐿, 1 , 0 ))
24		oveq12 6659	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑗 = 𝐽) → (𝑖𝑀𝑗) = (𝐼𝑀𝐽))
25	20, 23, 24	ifbieq12d 4113	. . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑗 = 𝐽) → if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)) = if(𝐼 = 𝐾, if(𝐽 = 𝐿, 1 , 0 ), (𝐼𝑀𝐽)))
26	25	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)))
27	9, 10, 18, 26	ovmpt2dv2 6794	. 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))))
28	8, 27	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 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ifcif 4086  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  Basecbs 15857  0_gc0g 16100  1_rcur 18501   Mat cmat 20213   minMatR1 cminmar1 20439

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-slot 15861  df-base 15863  df-mat 20214  df-minmar1 20441

This theorem is referenced by:  madjusmdetlem1  29893