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Theorem marrepval0 20289
Description: Second 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
marrepval0 ((𝑀𝐵𝑆 ∈ (Base‘𝑅)) → (𝑀𝑄𝑆) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)))))
Distinct variable groups:   𝑖,𝑁,𝑗,𝑘,𝑙   𝑅,𝑖,𝑗,𝑘,𝑙   𝑖,𝑀,𝑗,𝑘,𝑙   𝑆,𝑖,𝑗,𝑘,𝑙
Allowed substitution hints:   𝐴(𝑖,𝑗,𝑘,𝑙)   𝐵(𝑖,𝑗,𝑘,𝑙)   𝑄(𝑖,𝑗,𝑘,𝑙)   0 (𝑖,𝑗,𝑘,𝑙)

Proof of Theorem marrepval0
Dummy variables 𝑚 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 marrepfval.a . . . . . . 7 𝐴 = (𝑁 Mat 𝑅)
2 marrepfval.b . . . . . . 7 𝐵 = (Base‘𝐴)
31, 2matrcl 20140 . . . . . 6 (𝑀𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
43simpld 475 . . . . 5 (𝑀𝐵𝑁 ∈ Fin)
54, 4jca 554 . . . 4 (𝑀𝐵 → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
65adantr 481 . . 3 ((𝑀𝐵𝑆 ∈ (Base‘𝑅)) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
7 mpt2exga 7194 . . 3 ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)))) ∈ V)
86, 7syl 17 . 2 ((𝑀𝐵𝑆 ∈ (Base‘𝑅)) → (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)))) ∈ V)
9 ifeq1 4064 . . . . . . 7 (𝑠 = 𝑆 → if(𝑗 = 𝑙, 𝑠, 0 ) = if(𝑗 = 𝑙, 𝑆, 0 ))
109adantl 482 . . . . . 6 ((𝑚 = 𝑀𝑠 = 𝑆) → if(𝑗 = 𝑙, 𝑠, 0 ) = if(𝑗 = 𝑙, 𝑆, 0 ))
11 oveq 6613 . . . . . . 7 (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
1211adantr 481 . . . . . 6 ((𝑚 = 𝑀𝑠 = 𝑆) → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
1310, 12ifeq12d 4080 . . . . 5 ((𝑚 = 𝑀𝑠 = 𝑆) → if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗)) = if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)))
1413mpt2eq3dv 6677 . . . 4 ((𝑚 = 𝑀𝑠 = 𝑆) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗))) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗))))
1514mpt2eq3dv 6677 . . 3 ((𝑚 = 𝑀𝑠 = 𝑆) → (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗)))) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)))))
16 marrepfval.q . . . 4 𝑄 = (𝑁 matRRep 𝑅)
17 marrepfval.z . . . 4 0 = (0g𝑅)
181, 2, 16, 17marrepfval 20288 . . 3 𝑄 = (𝑚𝐵, 𝑠 ∈ (Base‘𝑅) ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗)))))
1915, 18ovmpt2ga 6746 . 2 ((𝑀𝐵𝑆 ∈ (Base‘𝑅) ∧ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)))) ∈ V) → (𝑀𝑄𝑆) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)))))
208, 19mpd3an3 1422 1 ((𝑀𝐵𝑆 ∈ (Base‘𝑅)) → (𝑀𝑄𝑆) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  ifcif 4060  cfv 5849  (class class class)co 6607  cmpt2 6609  Fincfn 7902  Basecbs 15784  0gc0g 16024   Mat cmat 20135   matRRep cmarrep 20284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-1st 7116  df-2nd 7117  df-slot 15788  df-base 15789  df-mat 20136  df-marrep 20286
This theorem is referenced by:  marrepval  20290  minmar1marrep  20378
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