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Theorem marrepval 22285
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 22284 . . 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 768 . . 3 (((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁)) → 𝐾𝑁)
8 simplrr 775 . . 3 ((((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁)) ∧ 𝑘 = 𝐾) → 𝐿𝑁)
91, 2matrcl 22133 . . . . . . 7 (𝑀𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
109simpld 494 . . . . . 6 (𝑀𝐵𝑁 ∈ Fin)
1110, 10jca 511 . . . . 5 (𝑀𝐵 → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
1211ad3antrrr 727 . . . 4 ((((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁)) ∧ (𝑘 = 𝐾𝑙 = 𝐿)) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
13 mpoexga 8068 . . . 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 2743 . . . . . . 7 (𝑘 = 𝐾 → (𝑖 = 𝑘𝑖 = 𝐾))
1615adantr 480 . . . . . 6 ((𝑘 = 𝐾𝑙 = 𝐿) → (𝑖 = 𝑘𝑖 = 𝐾))
17 eqeq2 2743 . . . . . . . 8 (𝑙 = 𝐿 → (𝑗 = 𝑙𝑗 = 𝐿))
1817ifbid 4551 . . . . . . 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 4552 . . . . 5 ((𝑘 = 𝐾𝑙 = 𝐿) → if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)) = if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗)))
2120mpoeq3dv 7491 . . . 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, 22ovmpodv2 7569 . 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 205  wa 395   = wceq 1540  wcel 2105  Vcvv 3473  ifcif 4528  cfv 6543  (class class class)co 7412  cmpo 7414  Fincfn 8943  Basecbs 17149  0gc0g 17390   Mat cmat 22128   matRRep cmarrep 22279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11170  ax-1cn 11172  ax-addcl 11174
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-nn 12218  df-slot 17120  df-ndx 17132  df-base 17150  df-mat 22129  df-marrep 22281
This theorem is referenced by:  marrepeval  22286  marrepcl  22287  1marepvmarrepid  22298  smadiadetglem1  22394  smadiadetglem2  22395  madjusmdetlem1  33106
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