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Theorem marrepeval 21172
 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 = (0g𝑅)
Assertion
Ref Expression
marrepeval (((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁) ∧ (𝐼𝑁𝐽𝑁)) → (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.
StepHypRef 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 = (0g𝑅)
51, 2, 3, 4marrepval 21171 . . 3 (((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁)) → (𝐾(𝑀𝑄𝑆)𝐿) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗))))
653adant3 1129 . 2 (((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁) ∧ (𝐼𝑁𝐽𝑁)) → (𝐾(𝑀𝑄𝑆)𝐿) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗))))
7 simp3l 1198 . . 3 (((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁) ∧ (𝐼𝑁𝐽𝑁)) → 𝐼𝑁)
8 simpl3r 1226 . . 3 ((((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁) ∧ (𝐼𝑁𝐽𝑁)) ∧ 𝑖 = 𝐼) → 𝐽𝑁)
94fvexi 6663 . . . . . . . 8 0 ∈ V
10 ifexg 4475 . . . . . . . 8 ((𝑆 ∈ (Base‘𝑅) ∧ 0 ∈ V) → if(𝑗 = 𝐿, 𝑆, 0 ) ∈ V)
119, 10mpan2 690 . . . . . . 7 (𝑆 ∈ (Base‘𝑅) → if(𝑗 = 𝐿, 𝑆, 0 ) ∈ V)
12 ovexd 7174 . . . . . . 7 (𝑆 ∈ (Base‘𝑅) → (𝑖𝑀𝑗) ∈ V)
1311, 12ifcld 4473 . . . . . 6 (𝑆 ∈ (Base‘𝑅) → if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗)) ∈ V)
1413adantl 485 . . . . 5 ((𝑀𝐵𝑆 ∈ (Base‘𝑅)) → if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗)) ∈ V)
15143ad2ant1 1130 . . . 4 (((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁) ∧ (𝐼𝑁𝐽𝑁)) → if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗)) ∈ V)
1615adantr 484 . . 3 ((((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁) ∧ (𝐼𝑁𝐽𝑁)) ∧ (𝑖 = 𝐼𝑗 = 𝐽)) → if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗)) ∈ V)
17 eqeq1 2805 . . . . . 6 (𝑖 = 𝐼 → (𝑖 = 𝐾𝐼 = 𝐾))
1817adantr 484 . . . . 5 ((𝑖 = 𝐼𝑗 = 𝐽) → (𝑖 = 𝐾𝐼 = 𝐾))
19 eqeq1 2805 . . . . . . 7 (𝑗 = 𝐽 → (𝑗 = 𝐿𝐽 = 𝐿))
2019ifbid 4450 . . . . . 6 (𝑗 = 𝐽 → if(𝑗 = 𝐿, 𝑆, 0 ) = if(𝐽 = 𝐿, 𝑆, 0 ))
2120adantl 485 . . . . 5 ((𝑖 = 𝐼𝑗 = 𝐽) → if(𝑗 = 𝐿, 𝑆, 0 ) = if(𝐽 = 𝐿, 𝑆, 0 ))
22 oveq12 7148 . . . . 5 ((𝑖 = 𝐼𝑗 = 𝐽) → (𝑖𝑀𝑗) = (𝐼𝑀𝐽))
2318, 21, 22ifbieq12d 4455 . . . 4 ((𝑖 = 𝐼𝑗 = 𝐽) → if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗)) = if(𝐼 = 𝐾, if(𝐽 = 𝐿, 𝑆, 0 ), (𝐼𝑀𝐽)))
2423adantl 485 . . 3 ((((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁) ∧ (𝐼𝑁𝐽𝑁)) ∧ (𝑖 = 𝐼𝑗 = 𝐽)) → if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗)) = if(𝐼 = 𝐾, if(𝐽 = 𝐿, 𝑆, 0 ), (𝐼𝑀𝐽)))
257, 8, 16, 24ovmpodv2 7291 . 2 (((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁) ∧ (𝐼𝑁𝐽𝑁)) → ((𝐾(𝑀𝑄𝑆)𝐿) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗))) → (𝐼(𝐾(𝑀𝑄𝑆)𝐿)𝐽) = if(𝐼 = 𝐾, if(𝐽 = 𝐿, 𝑆, 0 ), (𝐼𝑀𝐽))))
266, 25mpd 15 1 (((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼(𝐾(𝑀𝑄𝑆)𝐿)𝐽) = if(𝐼 = 𝐾, if(𝐽 = 𝐿, 𝑆, 0 ), (𝐼𝑀𝐽)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112  Vcvv 3444  ifcif 4428  ‘cfv 6328  (class class class)co 7139   ∈ cmpo 7141  Basecbs 16479  0gc0g 16709   Mat cmat 21016   matRRep cmarrep 21165 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-slot 16483  df-base 16485  df-mat 21017  df-marrep 21167 This theorem is referenced by:  submatminr1  31167
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