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Theorem marepveval 20568
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 20567 . . 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 6356 . . . . 5 (((𝑀𝐵𝐶𝑉𝐾𝑁) ∧ (𝐼𝑁𝐽𝑁)) → (𝐶𝑖) ∈ V)
10 ovexd 6835 . . . . 5 (((𝑀𝐵𝐶𝑉𝐾𝑁) ∧ (𝐼𝑁𝐽𝑁)) → (𝑖𝑀𝑗) ∈ V)
119, 10ifcld 4267 . . . 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 2756 . . . . . 6 (𝑗 = 𝐽 → (𝑗 = 𝐾𝐽 = 𝐾))
1413adantl 473 . . . . 5 ((𝑖 = 𝐼𝑗 = 𝐽) → (𝑗 = 𝐾𝐽 = 𝐾))
15 fveq2 6344 . . . . . 6 (𝑖 = 𝐼 → (𝐶𝑖) = (𝐶𝐼))
1615adantr 472 . . . . 5 ((𝑖 = 𝐼𝑗 = 𝐽) → (𝐶𝑖) = (𝐶𝐼))
17 oveq12 6814 . . . . 5 ((𝑖 = 𝐼𝑗 = 𝐽) → (𝑖𝑀𝑗) = (𝐼𝑀𝐽))
1814, 16, 17ifbieq12d 4249 . . . 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 6951 . 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 1624  wcel 2131  Vcvv 3332  ifcif 4222  cfv 6041  (class class class)co 6805  cmpt2 6807  𝑚 cmap 8015  Basecbs 16051   Mat cmat 20407   matRepV cmatrepV 20557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-1st 7325  df-2nd 7326  df-slot 16055  df-base 16057  df-mat 20408  df-marepv 20559
This theorem is referenced by:  ma1repveval  20571  1marepvsma1  20583
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