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Theorem marepveval 20576
Description: An entry of a matrix with a replaced column. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 26-Feb-2019.)
Hypotheses
Ref Expression
marepvfval.a 𝐴 = (𝑁 Mat 𝑅)
marepvfval.b 𝐵 = (Base‘𝐴)
marepvfval.q 𝑄 = (𝑁 matRepV 𝑅)
marepvfval.v 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁)
Assertion
Ref Expression
marepveval (((𝑀𝐵𝐶𝑉𝐾𝑁) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼((𝑀𝑄𝐶)‘𝐾)𝐽) = if(𝐽 = 𝐾, (𝐶𝐼), (𝐼𝑀𝐽)))

Proof of Theorem marepveval
Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 marepvfval.a . . . 4 𝐴 = (𝑁 Mat 𝑅)
2 marepvfval.b . . . 4 𝐵 = (Base‘𝐴)
3 marepvfval.q . . . 4 𝑄 = (𝑁 matRepV 𝑅)
4 marepvfval.v . . . 4 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁)
51, 2, 3, 4marepvval 20575 . . 3 ((𝑀𝐵𝐶𝑉𝐾𝑁) → ((𝑀𝑄𝐶)‘𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝐾, (𝐶𝑖), (𝑖𝑀𝑗))))
65adantr 472 . 2 (((𝑀𝐵𝐶𝑉𝐾𝑁) ∧ (𝐼𝑁𝐽𝑁)) → ((𝑀𝑄𝐶)‘𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝐾, (𝐶𝑖), (𝑖𝑀𝑗))))
7 simprl 811 . . 3 (((𝑀𝐵𝐶𝑉𝐾𝑁) ∧ (𝐼𝑁𝐽𝑁)) → 𝐼𝑁)
8 simplrr 820 . . 3 ((((𝑀𝐵𝐶𝑉𝐾𝑁) ∧ (𝐼𝑁𝐽𝑁)) ∧ 𝑖 = 𝐼) → 𝐽𝑁)
9 fvexd 6364 . . . . 5 (((𝑀𝐵𝐶𝑉𝐾𝑁) ∧ (𝐼𝑁𝐽𝑁)) → (𝐶𝑖) ∈ V)
10 ovexd 6843 . . . . 5 (((𝑀𝐵𝐶𝑉𝐾𝑁) ∧ (𝐼𝑁𝐽𝑁)) → (𝑖𝑀𝑗) ∈ V)
119, 10ifcld 4275 . . . 4 (((𝑀𝐵𝐶𝑉𝐾𝑁) ∧ (𝐼𝑁𝐽𝑁)) → if(𝑗 = 𝐾, (𝐶𝑖), (𝑖𝑀𝑗)) ∈ V)
1211adantr 472 . . 3 ((((𝑀𝐵𝐶𝑉𝐾𝑁) ∧ (𝐼𝑁𝐽𝑁)) ∧ (𝑖 = 𝐼𝑗 = 𝐽)) → if(𝑗 = 𝐾, (𝐶𝑖), (𝑖𝑀𝑗)) ∈ V)
13 eqeq1 2764 . . . . . 6 (𝑗 = 𝐽 → (𝑗 = 𝐾𝐽 = 𝐾))
1413adantl 473 . . . . 5 ((𝑖 = 𝐼𝑗 = 𝐽) → (𝑗 = 𝐾𝐽 = 𝐾))
15 fveq2 6352 . . . . . 6 (𝑖 = 𝐼 → (𝐶𝑖) = (𝐶𝐼))
1615adantr 472 . . . . 5 ((𝑖 = 𝐼𝑗 = 𝐽) → (𝐶𝑖) = (𝐶𝐼))
17 oveq12 6822 . . . . 5 ((𝑖 = 𝐼𝑗 = 𝐽) → (𝑖𝑀𝑗) = (𝐼𝑀𝐽))
1814, 16, 17ifbieq12d 4257 . . . 4 ((𝑖 = 𝐼𝑗 = 𝐽) → if(𝑗 = 𝐾, (𝐶𝑖), (𝑖𝑀𝑗)) = if(𝐽 = 𝐾, (𝐶𝐼), (𝐼𝑀𝐽)))
1918adantl 473 . . 3 ((((𝑀𝐵𝐶𝑉𝐾𝑁) ∧ (𝐼𝑁𝐽𝑁)) ∧ (𝑖 = 𝐼𝑗 = 𝐽)) → if(𝑗 = 𝐾, (𝐶𝑖), (𝑖𝑀𝑗)) = if(𝐽 = 𝐾, (𝐶𝐼), (𝐼𝑀𝐽)))
207, 8, 12, 19ovmpt2dv2 6959 . 2 (((𝑀𝐵𝐶𝑉𝐾𝑁) ∧ (𝐼𝑁𝐽𝑁)) → (((𝑀𝑄𝐶)‘𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝐾, (𝐶𝑖), (𝑖𝑀𝑗))) → (𝐼((𝑀𝑄𝐶)‘𝐾)𝐽) = if(𝐽 = 𝐾, (𝐶𝐼), (𝐼𝑀𝐽))))
216, 20mpd 15 1 (((𝑀𝐵𝐶𝑉𝐾𝑁) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼((𝑀𝑄𝐶)‘𝐾)𝐽) = if(𝐽 = 𝐾, (𝐶𝐼), (𝐼𝑀𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  Vcvv 3340  ifcif 4230  cfv 6049  (class class class)co 6813  cmpt2 6815  𝑚 cmap 8023  Basecbs 16059   Mat cmat 20415   matRepV cmatrepV 20565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-slot 16063  df-base 16065  df-mat 20416  df-marepv 20567
This theorem is referenced by:  ma1repveval  20579  1marepvsma1  20591
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