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Theorem marrepeval 22064
Description: An entry of a matrix with a replaced row. (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
marrepeval (((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼(𝐾(𝑀𝑄𝑆)𝐿)𝐽) = if(𝐼 = 𝐾, if(𝐽 = 𝐿, 𝑆, 0 ), (𝐼𝑀𝐽)))

Proof of Theorem marrepeval
Dummy variables 𝑖 𝑗 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, 4marrepval 22063 . . 3 (((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁)) → (𝐾(𝑀𝑄𝑆)𝐿) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗))))
653adant3 1132 . 2 (((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁) ∧ (𝐼𝑁𝐽𝑁)) → (𝐾(𝑀𝑄𝑆)𝐿) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗))))
7 simp3l 1201 . . 3 (((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁) ∧ (𝐼𝑁𝐽𝑁)) → 𝐼𝑁)
8 simpl3r 1229 . . 3 ((((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁) ∧ (𝐼𝑁𝐽𝑁)) ∧ 𝑖 = 𝐼) → 𝐽𝑁)
94fvexi 6905 . . . . . . . 8 0 ∈ V
10 ifexg 4577 . . . . . . . 8 ((𝑆 ∈ (Base‘𝑅) ∧ 0 ∈ V) → if(𝑗 = 𝐿, 𝑆, 0 ) ∈ V)
119, 10mpan2 689 . . . . . . 7 (𝑆 ∈ (Base‘𝑅) → if(𝑗 = 𝐿, 𝑆, 0 ) ∈ V)
12 ovexd 7443 . . . . . . 7 (𝑆 ∈ (Base‘𝑅) → (𝑖𝑀𝑗) ∈ V)
1311, 12ifcld 4574 . . . . . 6 (𝑆 ∈ (Base‘𝑅) → if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗)) ∈ V)
1413adantl 482 . . . . 5 ((𝑀𝐵𝑆 ∈ (Base‘𝑅)) → if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗)) ∈ V)
15143ad2ant1 1133 . . . 4 (((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁) ∧ (𝐼𝑁𝐽𝑁)) → if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗)) ∈ V)
1615adantr 481 . . 3 ((((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁) ∧ (𝐼𝑁𝐽𝑁)) ∧ (𝑖 = 𝐼𝑗 = 𝐽)) → if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗)) ∈ V)
17 eqeq1 2736 . . . . . 6 (𝑖 = 𝐼 → (𝑖 = 𝐾𝐼 = 𝐾))
1817adantr 481 . . . . 5 ((𝑖 = 𝐼𝑗 = 𝐽) → (𝑖 = 𝐾𝐼 = 𝐾))
19 eqeq1 2736 . . . . . . 7 (𝑗 = 𝐽 → (𝑗 = 𝐿𝐽 = 𝐿))
2019ifbid 4551 . . . . . 6 (𝑗 = 𝐽 → if(𝑗 = 𝐿, 𝑆, 0 ) = if(𝐽 = 𝐿, 𝑆, 0 ))
2120adantl 482 . . . . 5 ((𝑖 = 𝐼𝑗 = 𝐽) → if(𝑗 = 𝐿, 𝑆, 0 ) = if(𝐽 = 𝐿, 𝑆, 0 ))
22 oveq12 7417 . . . . 5 ((𝑖 = 𝐼𝑗 = 𝐽) → (𝑖𝑀𝑗) = (𝐼𝑀𝐽))
2318, 21, 22ifbieq12d 4556 . . . 4 ((𝑖 = 𝐼𝑗 = 𝐽) → if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗)) = if(𝐼 = 𝐾, if(𝐽 = 𝐿, 𝑆, 0 ), (𝐼𝑀𝐽)))
2423adantl 482 . . 3 ((((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁) ∧ (𝐼𝑁𝐽𝑁)) ∧ (𝑖 = 𝐼𝑗 = 𝐽)) → if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗)) = if(𝐼 = 𝐾, if(𝐽 = 𝐿, 𝑆, 0 ), (𝐼𝑀𝐽)))
257, 8, 16, 24ovmpodv2 7565 . 2 (((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁) ∧ (𝐼𝑁𝐽𝑁)) → ((𝐾(𝑀𝑄𝑆)𝐿) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗))) → (𝐼(𝐾(𝑀𝑄𝑆)𝐿)𝐽) = if(𝐼 = 𝐾, if(𝐽 = 𝐿, 𝑆, 0 ), (𝐼𝑀𝐽))))
266, 25mpd 15 1 (((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼(𝐾(𝑀𝑄𝑆)𝐿)𝐽) = if(𝐼 = 𝐾, if(𝐽 = 𝐿, 𝑆, 0 ), (𝐼𝑀𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3474  ifcif 4528  cfv 6543  (class class class)co 7408  cmpo 7410  Basecbs 17143  0gc0g 17384   Mat cmat 21906   matRRep cmarrep 22057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-1cn 11167  ax-addcl 11169
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  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 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-nn 12212  df-slot 17114  df-ndx 17126  df-base 17144  df-mat 21907  df-marrep 22059
This theorem is referenced by:  submatminr1  32785
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