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Theorem minmar1fval 20433
Description: First substitution for the definition of a matrix for a minor. (Contributed by AV, 31-Dec-2018.)
Hypotheses
Ref Expression
minmar1fval.a 𝐴 = (𝑁 Mat 𝑅)
minmar1fval.b 𝐵 = (Base‘𝐴)
minmar1fval.q 𝑄 = (𝑁 minMatR1 𝑅)
minmar1fval.o 1 = (1r𝑅)
minmar1fval.z 0 = (0g𝑅)
Assertion
Ref Expression
minmar1fval 𝑄 = (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗)))))
Distinct variable groups:   𝐵,𝑚   𝑖,𝑁,𝑗,𝑘,𝑙,𝑚   𝑅,𝑖,𝑗,𝑘,𝑙,𝑚
Allowed substitution hints:   𝐴(𝑖,𝑗,𝑘,𝑚,𝑙)   𝐵(𝑖,𝑗,𝑘,𝑙)   𝑄(𝑖,𝑗,𝑘,𝑚,𝑙)   1 (𝑖,𝑗,𝑘,𝑚,𝑙)   0 (𝑖,𝑗,𝑘,𝑚,𝑙)

Proof of Theorem minmar1fval
Dummy variables 𝑛 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 minmar1fval.q . 2 𝑄 = (𝑁 minMatR1 𝑅)
2 oveq12 6644 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
3 minmar1fval.a . . . . . . . 8 𝐴 = (𝑁 Mat 𝑅)
42, 3syl6eqr 2672 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = 𝐴)
54fveq2d 6182 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = (Base‘𝐴))
6 minmar1fval.b . . . . . 6 𝐵 = (Base‘𝐴)
75, 6syl6eqr 2672 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵)
8 simpl 473 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑛 = 𝑁)
9 fveq2 6178 . . . . . . . . . . 11 (𝑟 = 𝑅 → (1r𝑟) = (1r𝑅))
10 minmar1fval.o . . . . . . . . . . 11 1 = (1r𝑅)
119, 10syl6eqr 2672 . . . . . . . . . 10 (𝑟 = 𝑅 → (1r𝑟) = 1 )
12 fveq2 6178 . . . . . . . . . . 11 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
13 minmar1fval.z . . . . . . . . . . 11 0 = (0g𝑅)
1412, 13syl6eqr 2672 . . . . . . . . . 10 (𝑟 = 𝑅 → (0g𝑟) = 0 )
1511, 14ifeq12d 4097 . . . . . . . . 9 (𝑟 = 𝑅 → if(𝑗 = 𝑙, (1r𝑟), (0g𝑟)) = if(𝑗 = 𝑙, 1 , 0 ))
1615ifeq1d 4095 . . . . . . . 8 (𝑟 = 𝑅 → if(𝑖 = 𝑘, if(𝑗 = 𝑙, (1r𝑟), (0g𝑟)), (𝑖𝑚𝑗)) = if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗)))
1716adantl 482 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → if(𝑖 = 𝑘, if(𝑗 = 𝑙, (1r𝑟), (0g𝑟)), (𝑖𝑚𝑗)) = if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗)))
188, 8, 17mpt2eq123dv 6702 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑖𝑛, 𝑗𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, (1r𝑟), (0g𝑟)), (𝑖𝑚𝑗))) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗))))
198, 8, 18mpt2eq123dv 6702 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑘𝑛, 𝑙𝑛 ↦ (𝑖𝑛, 𝑗𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, (1r𝑟), (0g𝑟)), (𝑖𝑚𝑗)))) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗)))))
207, 19mpteq12dv 4724 . . . 4 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘𝑛, 𝑙𝑛 ↦ (𝑖𝑛, 𝑗𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, (1r𝑟), (0g𝑟)), (𝑖𝑚𝑗))))) = (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗))))))
21 df-minmar1 20422 . . . 4 minMatR1 = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘𝑛, 𝑙𝑛 ↦ (𝑖𝑛, 𝑗𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, (1r𝑟), (0g𝑟)), (𝑖𝑚𝑗))))))
22 fvex 6188 . . . . . 6 (Base‘𝐴) ∈ V
236, 22eqeltri 2695 . . . . 5 𝐵 ∈ V
2423mptex 6471 . . . 4 (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗))))) ∈ V
2520, 21, 24ovmpt2a 6776 . . 3 ((𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 minMatR1 𝑅) = (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗))))))
2621mpt2ndm0 6860 . . . . 5 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 minMatR1 𝑅) = ∅)
27 mpt0 6008 . . . . 5 (𝑚 ∈ ∅ ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗))))) = ∅
2826, 27syl6eqr 2672 . . . 4 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 minMatR1 𝑅) = (𝑚 ∈ ∅ ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗))))))
293fveq2i 6181 . . . . . . 7 (Base‘𝐴) = (Base‘(𝑁 Mat 𝑅))
306, 29eqtri 2642 . . . . . 6 𝐵 = (Base‘(𝑁 Mat 𝑅))
31 matbas0pc 20196 . . . . . 6 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅)
3230, 31syl5eq 2666 . . . . 5 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ∅)
3332mpteq1d 4729 . . . 4 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗))))) = (𝑚 ∈ ∅ ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗))))))
3428, 33eqtr4d 2657 . . 3 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 minMatR1 𝑅) = (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗))))))
3525, 34pm2.61i 176 . 2 (𝑁 minMatR1 𝑅) = (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗)))))
361, 35eqtri 2642 1 𝑄 = (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384   = wceq 1481  wcel 1988  Vcvv 3195  c0 3907  ifcif 4077  cmpt 4720  cfv 5876  (class class class)co 6635  cmpt2 6637  Basecbs 15838  0gc0g 16081  1rcur 18482   Mat cmat 20194   minMatR1 cminmar1 20420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-slot 15842  df-base 15844  df-mat 20195  df-minmar1 20422
This theorem is referenced by:  minmar1val0  20434
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