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Theorem marrepval 20187
 Description: Third substitution for the definition of the matrix row replacement function. (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
marrepval (((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁)) → (𝐾(𝑀𝑄𝑆)𝐿) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗))))
Distinct variable groups:   𝑖,𝑁,𝑗   𝑅,𝑖,𝑗   𝑖,𝑀,𝑗   𝑆,𝑖,𝑗   𝑖,𝐾,𝑗   𝑖,𝐿,𝑗
Allowed substitution hints:   𝐴(𝑖,𝑗)   𝐵(𝑖,𝑗)   𝑄(𝑖,𝑗)   0 (𝑖,𝑗)

Proof of Theorem marrepval
Dummy variables 𝑘 𝑙 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, 4marrepval0 20186 . . 3 ((𝑀𝐵𝑆 ∈ (Base‘𝑅)) → (𝑀𝑄𝑆) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)))))
65adantr 480 . 2 (((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁)) → (𝑀𝑄𝑆) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)))))
7 simprl 790 . . 3 (((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁)) → 𝐾𝑁)
8 simplrr 797 . . 3 ((((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁)) ∧ 𝑘 = 𝐾) → 𝐿𝑁)
91, 2matrcl 20037 . . . . . . 7 (𝑀𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
109simpld 474 . . . . . 6 (𝑀𝐵𝑁 ∈ Fin)
1110, 10jca 553 . . . . 5 (𝑀𝐵 → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
1211ad3antrrr 762 . . . 4 ((((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁)) ∧ (𝑘 = 𝐾𝑙 = 𝐿)) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
13 mpt2exga 7135 . . . 4 ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗))) ∈ V)
1412, 13syl 17 . . 3 ((((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁)) ∧ (𝑘 = 𝐾𝑙 = 𝐿)) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗))) ∈ V)
15 eqeq2 2621 . . . . . . 7 (𝑘 = 𝐾 → (𝑖 = 𝑘𝑖 = 𝐾))
1615adantr 480 . . . . . 6 ((𝑘 = 𝐾𝑙 = 𝐿) → (𝑖 = 𝑘𝑖 = 𝐾))
17 eqeq2 2621 . . . . . . . 8 (𝑙 = 𝐿 → (𝑗 = 𝑙𝑗 = 𝐿))
1817ifbid 4058 . . . . . . 7 (𝑙 = 𝐿 → if(𝑗 = 𝑙, 𝑆, 0 ) = if(𝑗 = 𝐿, 𝑆, 0 ))
1918adantl 481 . . . . . 6 ((𝑘 = 𝐾𝑙 = 𝐿) → if(𝑗 = 𝑙, 𝑆, 0 ) = if(𝑗 = 𝐿, 𝑆, 0 ))
2016, 19ifbieq1d 4059 . . . . 5 ((𝑘 = 𝐾𝑙 = 𝐿) → if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)) = if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗)))
2120mpt2eq3dv 6619 . . . 4 ((𝑘 = 𝐾𝑙 = 𝐿) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗))) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗))))
2221adantl 481 . . 3 ((((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁)) ∧ (𝑘 = 𝐾𝑙 = 𝐿)) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗))) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗))))
237, 8, 14, 22ovmpt2dv2 6692 . 2 (((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁)) → ((𝑀𝑄𝑆) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)))) → (𝐾(𝑀𝑄𝑆)𝐿) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗)))))
246, 23mpd 15 1 (((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁)) → (𝐾(𝑀𝑄𝑆)𝐿) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ifcif 4036  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  Fincfn 7841  Basecbs 15695  0gc0g 15923   Mat cmat 20032   matRRep cmarrep 20181 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-marrep 20183 This theorem is referenced by:  marrepeval  20188  marrepcl  20189  1marepvmarrepid  20200  smadiadetglem1  20296  smadiadetglem2  20297  madjusmdetlem1  29221
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