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Theorem lmatval 30188
Description: Value of the literal matrix conversion function. (Contributed by Thierry Arnoux, 28-Aug-2020.)
Assertion
Ref Expression
lmatval (𝑀𝑉 → (litMat‘𝑀) = (𝑖 ∈ (1...(♯‘𝑀)), 𝑗 ∈ (1...(♯‘(𝑀‘0))) ↦ ((𝑀‘(𝑖 − 1))‘(𝑗 − 1))))
Distinct variable group:   𝑖,𝑀,𝑗
Allowed substitution hints:   𝑉(𝑖,𝑗)

Proof of Theorem lmatval
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 elex 3352 . 2 (𝑀𝑉𝑀 ∈ V)
2 fveq2 6352 . . . . 5 (𝑚 = 𝑀 → (♯‘𝑚) = (♯‘𝑀))
32oveq2d 6829 . . . 4 (𝑚 = 𝑀 → (1...(♯‘𝑚)) = (1...(♯‘𝑀)))
4 fveq1 6351 . . . . . 6 (𝑚 = 𝑀 → (𝑚‘0) = (𝑀‘0))
54fveq2d 6356 . . . . 5 (𝑚 = 𝑀 → (♯‘(𝑚‘0)) = (♯‘(𝑀‘0)))
65oveq2d 6829 . . . 4 (𝑚 = 𝑀 → (1...(♯‘(𝑚‘0))) = (1...(♯‘(𝑀‘0))))
7 fveq1 6351 . . . . 5 (𝑚 = 𝑀 → (𝑚‘(𝑖 − 1)) = (𝑀‘(𝑖 − 1)))
87fveq1d 6354 . . . 4 (𝑚 = 𝑀 → ((𝑚‘(𝑖 − 1))‘(𝑗 − 1)) = ((𝑀‘(𝑖 − 1))‘(𝑗 − 1)))
93, 6, 8mpt2eq123dv 6882 . . 3 (𝑚 = 𝑀 → (𝑖 ∈ (1...(♯‘𝑚)), 𝑗 ∈ (1...(♯‘(𝑚‘0))) ↦ ((𝑚‘(𝑖 − 1))‘(𝑗 − 1))) = (𝑖 ∈ (1...(♯‘𝑀)), 𝑗 ∈ (1...(♯‘(𝑀‘0))) ↦ ((𝑀‘(𝑖 − 1))‘(𝑗 − 1))))
10 df-lmat 30187 . . 3 litMat = (𝑚 ∈ V ↦ (𝑖 ∈ (1...(♯‘𝑚)), 𝑗 ∈ (1...(♯‘(𝑚‘0))) ↦ ((𝑚‘(𝑖 − 1))‘(𝑗 − 1))))
11 ovex 6841 . . . 4 (1...(♯‘𝑀)) ∈ V
12 ovex 6841 . . . 4 (1...(♯‘(𝑀‘0))) ∈ V
1311, 12mpt2ex 7415 . . 3 (𝑖 ∈ (1...(♯‘𝑀)), 𝑗 ∈ (1...(♯‘(𝑀‘0))) ↦ ((𝑀‘(𝑖 − 1))‘(𝑗 − 1))) ∈ V
149, 10, 13fvmpt 6444 . 2 (𝑀 ∈ V → (litMat‘𝑀) = (𝑖 ∈ (1...(♯‘𝑀)), 𝑗 ∈ (1...(♯‘(𝑀‘0))) ↦ ((𝑀‘(𝑖 − 1))‘(𝑗 − 1))))
151, 14syl 17 1 (𝑀𝑉 → (litMat‘𝑀) = (𝑖 ∈ (1...(♯‘𝑀)), 𝑗 ∈ (1...(♯‘(𝑀‘0))) ↦ ((𝑀‘(𝑖 − 1))‘(𝑗 − 1))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  Vcvv 3340  cfv 6049  (class class class)co 6813  cmpt2 6815  0cc0 10128  1c1 10129  cmin 10458  ...cfz 12519  chash 13311  litMatclmat 30186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-lmat 30187
This theorem is referenced by:  lmatfval  30189  lmatcl  30191
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