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Theorem marepveval 20193
 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 20192 . . 3 ((𝑀𝐵𝐶𝑉𝐾𝑁) → ((𝑀𝑄𝐶)‘𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝐾, (𝐶𝑖), (𝑖𝑀𝑗))))
65adantr 480 . 2 (((𝑀𝐵𝐶𝑉𝐾𝑁) ∧ (𝐼𝑁𝐽𝑁)) → ((𝑀𝑄𝐶)‘𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝐾, (𝐶𝑖), (𝑖𝑀𝑗))))
7 simprl 790 . . 3 (((𝑀𝐵𝐶𝑉𝐾𝑁) ∧ (𝐼𝑁𝐽𝑁)) → 𝐼𝑁)
8 simplrr 797 . . 3 ((((𝑀𝐵𝐶𝑉𝐾𝑁) ∧ (𝐼𝑁𝐽𝑁)) ∧ 𝑖 = 𝐼) → 𝐽𝑁)
9 fvex 6113 . . . . . 6 (𝐶𝑖) ∈ V
109a1i 11 . . . . 5 (((𝑀𝐵𝐶𝑉𝐾𝑁) ∧ (𝐼𝑁𝐽𝑁)) → (𝐶𝑖) ∈ V)
11 ovex 6577 . . . . . 6 (𝑖𝑀𝑗) ∈ V
1211a1i 11 . . . . 5 (((𝑀𝐵𝐶𝑉𝐾𝑁) ∧ (𝐼𝑁𝐽𝑁)) → (𝑖𝑀𝑗) ∈ V)
1310, 12ifcld 4081 . . . 4 (((𝑀𝐵𝐶𝑉𝐾𝑁) ∧ (𝐼𝑁𝐽𝑁)) → if(𝑗 = 𝐾, (𝐶𝑖), (𝑖𝑀𝑗)) ∈ V)
1413adantr 480 . . 3 ((((𝑀𝐵𝐶𝑉𝐾𝑁) ∧ (𝐼𝑁𝐽𝑁)) ∧ (𝑖 = 𝐼𝑗 = 𝐽)) → if(𝑗 = 𝐾, (𝐶𝑖), (𝑖𝑀𝑗)) ∈ V)
15 eqeq1 2614 . . . . . 6 (𝑗 = 𝐽 → (𝑗 = 𝐾𝐽 = 𝐾))
1615adantl 481 . . . . 5 ((𝑖 = 𝐼𝑗 = 𝐽) → (𝑗 = 𝐾𝐽 = 𝐾))
17 fveq2 6103 . . . . . 6 (𝑖 = 𝐼 → (𝐶𝑖) = (𝐶𝐼))
1817adantr 480 . . . . 5 ((𝑖 = 𝐼𝑗 = 𝐽) → (𝐶𝑖) = (𝐶𝐼))
19 oveq12 6558 . . . . 5 ((𝑖 = 𝐼𝑗 = 𝐽) → (𝑖𝑀𝑗) = (𝐼𝑀𝐽))
2016, 18, 19ifbieq12d 4063 . . . 4 ((𝑖 = 𝐼𝑗 = 𝐽) → if(𝑗 = 𝐾, (𝐶𝑖), (𝑖𝑀𝑗)) = if(𝐽 = 𝐾, (𝐶𝐼), (𝐼𝑀𝐽)))
2120adantl 481 . . 3 ((((𝑀𝐵𝐶𝑉𝐾𝑁) ∧ (𝐼𝑁𝐽𝑁)) ∧ (𝑖 = 𝐼𝑗 = 𝐽)) → if(𝑗 = 𝐾, (𝐶𝑖), (𝑖𝑀𝑗)) = if(𝐽 = 𝐾, (𝐶𝐼), (𝐼𝑀𝐽)))
227, 8, 14, 21ovmpt2dv2 6692 . 2 (((𝑀𝐵𝐶𝑉𝐾𝑁) ∧ (𝐼𝑁𝐽𝑁)) → (((𝑀𝑄𝐶)‘𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝐾, (𝐶𝑖), (𝑖𝑀𝑗))) → (𝐼((𝑀𝑄𝐶)‘𝐾)𝐽) = if(𝐽 = 𝐾, (𝐶𝐼), (𝐼𝑀𝐽))))
236, 22mpd 15 1 (((𝑀𝐵𝐶𝑉𝐾𝑁) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼((𝑀𝑄𝐶)‘𝐾)𝐽) = if(𝐽 = 𝐾, (𝐶𝐼), (𝐼𝑀𝐽)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ifcif 4036  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551   ↑𝑚 cmap 7744  Basecbs 15695   Mat cmat 20032   matRepV cmatrepV 20182 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-slot 15699  df-base 15700  df-mat 20033  df-marepv 20184 This theorem is referenced by:  ma1repveval  20196  1marepvsma1  20208
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