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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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