Theorem marrepeval 22536

Description: An entry of a matrix with a replaced row. (Contributed by AV, 12-Feb-2019.)

Hypotheses

Ref	Expression
marrepfval.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
marrepfval.b	⊢ 𝐵 = (Base‘𝐴)
marrepfval.q	⊢ 𝑄 = (𝑁 matRRep 𝑅)
marrepfval.z	⊢ 0 = (0g‘𝑅)

Assertion

Ref	Expression
marrepeval	⊢ (((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝐾(𝑀𝑄𝑆)𝐿)𝐽) = if(𝐼 = 𝐾, if(𝐽 = 𝐿, 𝑆, 0 ), (𝐼𝑀𝐽)))

Proof of Theorem marrepeval

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

Step	Hyp	Ref	Expression
1		marrepfval.a	. . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		marrepfval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐴)
3		marrepfval.q	. . . 4 ⊢ 𝑄 = (𝑁 matRRep 𝑅)
4		marrepfval.z	. . . 4 ⊢ 0 = (0g‘𝑅)
5	1, 2, 3, 4	marrepval 22535	. . 3 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) → (𝐾(𝑀𝑄𝑆)𝐿) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗))))
6	5	3adant3 1132	. 2 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐾(𝑀𝑄𝑆)𝐿) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗))))
7		simp3l 1201	. . 3 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 𝐼 ∈ 𝑁)
8		simpl3r 1229	. . 3 ⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) ∧ 𝑖 = 𝐼) → 𝐽 ∈ 𝑁)
9	4	fvexi 6901	. . . . . . . 8 ⊢ 0 ∈ V
10		ifexg 4557	. . . . . . . 8 ⊢ ((𝑆 ∈ (Base‘𝑅) ∧ 0 ∈ V) → if(𝑗 = 𝐿, 𝑆, 0 ) ∈ V)
11	9, 10	mpan2 691	. . . . . . 7 ⊢ (𝑆 ∈ (Base‘𝑅) → if(𝑗 = 𝐿, 𝑆, 0 ) ∈ V)
12		ovexd 7449	. . . . . . 7 ⊢ (𝑆 ∈ (Base‘𝑅) → (𝑖𝑀𝑗) ∈ V)
13	11, 12	ifcld 4554	. . . . . 6 ⊢ (𝑆 ∈ (Base‘𝑅) → if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗)) ∈ V)
14	13	adantl 481	. . . . 5 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) → if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗)) ∈ V)
15	14	3ad2ant1 1133	. . . 4 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗)) ∈ V)
16	15	adantr 480	. . 3 ⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) ∧ (𝑖 = 𝐼 ∧ 𝑗 = 𝐽)) → if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗)) ∈ V)
17		eqeq1 2738	. . . . . 6 ⊢ (𝑖 = 𝐼 → (𝑖 = 𝐾 ↔ 𝐼 = 𝐾))
18	17	adantr 480	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑗 = 𝐽) → (𝑖 = 𝐾 ↔ 𝐼 = 𝐾))
19		eqeq1 2738	. . . . . . 7 ⊢ (𝑗 = 𝐽 → (𝑗 = 𝐿 ↔ 𝐽 = 𝐿))
20	19	ifbid 4531	. . . . . 6 ⊢ (𝑗 = 𝐽 → if(𝑗 = 𝐿, 𝑆, 0 ) = if(𝐽 = 𝐿, 𝑆, 0 ))
21	20	adantl 481	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑗 = 𝐽) → if(𝑗 = 𝐿, 𝑆, 0 ) = if(𝐽 = 𝐿, 𝑆, 0 ))
22		oveq12 7423	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑗 = 𝐽) → (𝑖𝑀𝑗) = (𝐼𝑀𝐽))
23	18, 21, 22	ifbieq12d 4536	. . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑗 = 𝐽) → if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗)) = if(𝐼 = 𝐾, if(𝐽 = 𝐿, 𝑆, 0 ), (𝐼𝑀𝐽)))
24	23	adantl 481	. . 3 ⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) ∧ (𝑖 = 𝐼 ∧ 𝑗 = 𝐽)) → if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗)) = if(𝐼 = 𝐾, if(𝐽 = 𝐿, 𝑆, 0 ), (𝐼𝑀𝐽)))
25	7, 8, 16, 24	ovmpodv2 7574	. 2 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → ((𝐾(𝑀𝑄𝑆)𝐿) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗))) → (𝐼(𝐾(𝑀𝑄𝑆)𝐿)𝐽) = if(𝐼 = 𝐾, if(𝐽 = 𝐿, 𝑆, 0 ), (𝐼𝑀𝐽))))
26	6, 25	mpd 15	1 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝐾(𝑀𝑄𝑆)𝐿)𝐽) = if(𝐼 = 𝐾, if(𝐽 = 𝐿, 𝑆, 0 ), (𝐼𝑀𝐽)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  Vcvv 3464  ifcif 4507  ‘cfv 6542  (class class class)co 7414   ∈ cmpo 7416  Basecbs 17230  0_gc0g 17460   Mat cmat 22378   matRRep cmarrep 22529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5261  ax-sep 5278  ax-nul 5288  ax-pow 5347  ax-pr 5414  ax-un 7738  ax-cnex 11194  ax-1cn 11196  ax-addcl 11198

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3773  df-csb 3882  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-pss 3953  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-iun 4975  df-br 5126  df-opab 5188  df-mpt 5208  df-tr 5242  df-id 5560  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6303  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7871  df-1st 7997  df-2nd 7998  df-frecs 8289  df-wrecs 8320  df-recs 8394  df-rdg 8433  df-nn 12250  df-slot 17202  df-ndx 17214  df-base 17231  df-mat 22379  df-marrep 22531

This theorem is referenced by:  submatminr1  33750