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Theorem lmatval 29207
 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 3185 . 2 (𝑀𝑉𝑀 ∈ V)
2 fveq2 6103 . . . . 5 (𝑚 = 𝑀 → (#‘𝑚) = (#‘𝑀))
32oveq2d 6565 . . . 4 (𝑚 = 𝑀 → (1...(#‘𝑚)) = (1...(#‘𝑀)))
4 fveq1 6102 . . . . . 6 (𝑚 = 𝑀 → (𝑚‘0) = (𝑀‘0))
54fveq2d 6107 . . . . 5 (𝑚 = 𝑀 → (#‘(𝑚‘0)) = (#‘(𝑀‘0)))
65oveq2d 6565 . . . 4 (𝑚 = 𝑀 → (1...(#‘(𝑚‘0))) = (1...(#‘(𝑀‘0))))
7 fveq1 6102 . . . . 5 (𝑚 = 𝑀 → (𝑚‘(𝑖 − 1)) = (𝑀‘(𝑖 − 1)))
87fveq1d 6105 . . . 4 (𝑚 = 𝑀 → ((𝑚‘(𝑖 − 1))‘(𝑗 − 1)) = ((𝑀‘(𝑖 − 1))‘(𝑗 − 1)))
93, 6, 8mpt2eq123dv 6615 . . 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 29206 . . 3 litMat = (𝑚 ∈ V ↦ (𝑖 ∈ (1...(#‘𝑚)), 𝑗 ∈ (1...(#‘(𝑚‘0))) ↦ ((𝑚‘(𝑖 − 1))‘(𝑗 − 1))))
11 ovex 6577 . . . 4 (1...(#‘𝑀)) ∈ V
12 ovex 6577 . . . 4 (1...(#‘(𝑀‘0))) ∈ V
1311, 12mpt2ex 7136 . . 3 (𝑖 ∈ (1...(#‘𝑀)), 𝑗 ∈ (1...(#‘(𝑀‘0))) ↦ ((𝑀‘(𝑖 − 1))‘(𝑗 − 1))) ∈ V
149, 10, 13fvmpt 6191 . 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 1475   ∈ wcel 1977  Vcvv 3173  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  0cc0 9815  1c1 9816   − cmin 10145  ...cfz 12197  #chash 12979  litMatclmat 29205 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-lmat 29206 This theorem is referenced by:  lmatfval  29208  lmatcl  29210
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