Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  marrepfval Structured version   Visualization version   GIF version

Theorem marrepfval 21179
 Description: First substitution for the definition of the matrix row replacement function. (Contributed by AV, 12-Feb-2019.) (Proof shortened by AV, 2-Mar-2024.)
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
marrepfval 𝑄 = (𝑚𝐵, 𝑠 ∈ (Base‘𝑅) ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗)))))
Distinct variable groups:   𝐵,𝑚,𝑠   𝑖,𝑁,𝑗,𝑘,𝑙,𝑚,𝑠   𝑅,𝑖,𝑗,𝑘,𝑙,𝑚,𝑠
Allowed substitution hints:   𝐴(𝑖,𝑗,𝑘,𝑚,𝑠,𝑙)   𝐵(𝑖,𝑗,𝑘,𝑙)   𝑄(𝑖,𝑗,𝑘,𝑚,𝑠,𝑙)   0 (𝑖,𝑗,𝑘,𝑚,𝑠,𝑙)

Proof of Theorem marrepfval
Dummy variables 𝑛 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 marrepfval.q . 2 𝑄 = (𝑁 matRRep 𝑅)
2 marrepfval.b . . . . . 6 𝐵 = (Base‘𝐴)
32fvexi 6664 . . . . 5 𝐵 ∈ V
4 fvexd 6665 . . . . 5 ((𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
5 mpoexga 7765 . . . . 5 ((𝐵 ∈ V ∧ (Base‘𝑅) ∈ V) → (𝑚𝐵, 𝑠 ∈ (Base‘𝑅) ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗))))) ∈ V)
63, 4, 5sylancr 590 . . . 4 ((𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑚𝐵, 𝑠 ∈ (Base‘𝑅) ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗))))) ∈ V)
7 oveq12 7149 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
87fveq2d 6654 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = (Base‘(𝑁 Mat 𝑅)))
9 marrepfval.a . . . . . . . . 9 𝐴 = (𝑁 Mat 𝑅)
109fveq2i 6653 . . . . . . . 8 (Base‘𝐴) = (Base‘(𝑁 Mat 𝑅))
112, 10eqtri 2821 . . . . . . 7 𝐵 = (Base‘(𝑁 Mat 𝑅))
128, 11eqtr4di 2851 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵)
13 fveq2 6650 . . . . . . 7 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
1413adantl 485 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘𝑟) = (Base‘𝑅))
15 simpl 486 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑛 = 𝑁)
16 fveq2 6650 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
17 marrepfval.z . . . . . . . . . . . 12 0 = (0g𝑅)
1816, 17eqtr4di 2851 . . . . . . . . . . 11 (𝑟 = 𝑅 → (0g𝑟) = 0 )
1918ifeq2d 4444 . . . . . . . . . 10 (𝑟 = 𝑅 → if(𝑗 = 𝑙, 𝑠, (0g𝑟)) = if(𝑗 = 𝑙, 𝑠, 0 ))
2019ifeq1d 4443 . . . . . . . . 9 (𝑟 = 𝑅 → if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, (0g𝑟)), (𝑖𝑚𝑗)) = if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗)))
2120adantl 485 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, (0g𝑟)), (𝑖𝑚𝑗)) = if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗)))
2215, 15, 21mpoeq123dv 7213 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑖𝑛, 𝑗𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, (0g𝑟)), (𝑖𝑚𝑗))) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗))))
2315, 15, 22mpoeq123dv 7213 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑘𝑛, 𝑙𝑛 ↦ (𝑖𝑛, 𝑗𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, (0g𝑟)), (𝑖𝑚𝑗)))) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗)))))
2412, 14, 23mpoeq123dv 7213 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)), 𝑠 ∈ (Base‘𝑟) ↦ (𝑘𝑛, 𝑙𝑛 ↦ (𝑖𝑛, 𝑗𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, (0g𝑟)), (𝑖𝑚𝑗))))) = (𝑚𝐵, 𝑠 ∈ (Base‘𝑅) ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗))))))
25 df-marrep 21177 . . . . 5 matRRep = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)), 𝑠 ∈ (Base‘𝑟) ↦ (𝑘𝑛, 𝑙𝑛 ↦ (𝑖𝑛, 𝑗𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, (0g𝑟)), (𝑖𝑚𝑗))))))
2624, 25ovmpoga 7289 . . . 4 ((𝑁 ∈ V ∧ 𝑅 ∈ V ∧ (𝑚𝐵, 𝑠 ∈ (Base‘𝑅) ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗))))) ∈ V) → (𝑁 matRRep 𝑅) = (𝑚𝐵, 𝑠 ∈ (Base‘𝑅) ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗))))))
276, 26mpd3an3 1459 . . 3 ((𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 matRRep 𝑅) = (𝑚𝐵, 𝑠 ∈ (Base‘𝑅) ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗))))))
2825mpondm0 7372 . . . 4 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 matRRep 𝑅) = ∅)
29 matbas0pc 21028 . . . . . . 7 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅)
3011, 29syl5eq 2845 . . . . . 6 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ∅)
3130orcd 870 . . . . 5 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝐵 = ∅ ∨ (Base‘𝑅) = ∅))
32 0mpo0 7221 . . . . 5 ((𝐵 = ∅ ∨ (Base‘𝑅) = ∅) → (𝑚𝐵, 𝑠 ∈ (Base‘𝑅) ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗))))) = ∅)
3331, 32syl 17 . . . 4 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑚𝐵, 𝑠 ∈ (Base‘𝑅) ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗))))) = ∅)
3428, 33eqtr4d 2836 . . 3 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 matRRep 𝑅) = (𝑚𝐵, 𝑠 ∈ (Base‘𝑅) ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗))))))
3527, 34pm2.61i 185 . 2 (𝑁 matRRep 𝑅) = (𝑚𝐵, 𝑠 ∈ (Base‘𝑅) ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗)))))
361, 35eqtri 2821 1 𝑄 = (𝑚𝐵, 𝑠 ∈ (Base‘𝑅) ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗)))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111  Vcvv 3441  ∅c0 4243  ifcif 4425  ‘cfv 6327  (class class class)co 7140   ∈ cmpo 7142  Basecbs 16482  0gc0g 16712   Mat cmat 21026   matRRep cmarrep 21175 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7448 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7678  df-2nd 7679  df-slot 16486  df-base 16488  df-mat 21027  df-marrep 21177 This theorem is referenced by:  marrepval0  21180
 Copyright terms: Public domain W3C validator