Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lmatval Structured version   Visualization version   GIF version

Theorem lmatval 32793
Description: Value of the literal matrix conversion function. (Contributed by Thierry Arnoux, 28-Aug-2020.)
Assertion
Ref Expression
lmatval (𝑀 ∈ 𝑉 β†’ (litMatβ€˜π‘€) = (𝑖 ∈ (1...(β™―β€˜π‘€)), 𝑗 ∈ (1...(β™―β€˜(π‘€β€˜0))) ↦ ((π‘€β€˜(𝑖 βˆ’ 1))β€˜(𝑗 βˆ’ 1))))
Distinct variable group:   𝑖,𝑀,𝑗
Allowed substitution hints:   𝑉(𝑖,𝑗)

Proof of Theorem lmatval
Dummy variable π‘š is distinct from all other variables.
StepHypRef Expression
1 elex 3493 . 2 (𝑀 ∈ 𝑉 β†’ 𝑀 ∈ V)
2 fveq2 6892 . . . . 5 (π‘š = 𝑀 β†’ (β™―β€˜π‘š) = (β™―β€˜π‘€))
32oveq2d 7425 . . . 4 (π‘š = 𝑀 β†’ (1...(β™―β€˜π‘š)) = (1...(β™―β€˜π‘€)))
4 fveq1 6891 . . . . . 6 (π‘š = 𝑀 β†’ (π‘šβ€˜0) = (π‘€β€˜0))
54fveq2d 6896 . . . . 5 (π‘š = 𝑀 β†’ (β™―β€˜(π‘šβ€˜0)) = (β™―β€˜(π‘€β€˜0)))
65oveq2d 7425 . . . 4 (π‘š = 𝑀 β†’ (1...(β™―β€˜(π‘šβ€˜0))) = (1...(β™―β€˜(π‘€β€˜0))))
7 fveq1 6891 . . . . 5 (π‘š = 𝑀 β†’ (π‘šβ€˜(𝑖 βˆ’ 1)) = (π‘€β€˜(𝑖 βˆ’ 1)))
87fveq1d 6894 . . . 4 (π‘š = 𝑀 β†’ ((π‘šβ€˜(𝑖 βˆ’ 1))β€˜(𝑗 βˆ’ 1)) = ((π‘€β€˜(𝑖 βˆ’ 1))β€˜(𝑗 βˆ’ 1)))
93, 6, 8mpoeq123dv 7484 . . 3 (π‘š = 𝑀 β†’ (𝑖 ∈ (1...(β™―β€˜π‘š)), 𝑗 ∈ (1...(β™―β€˜(π‘šβ€˜0))) ↦ ((π‘šβ€˜(𝑖 βˆ’ 1))β€˜(𝑗 βˆ’ 1))) = (𝑖 ∈ (1...(β™―β€˜π‘€)), 𝑗 ∈ (1...(β™―β€˜(π‘€β€˜0))) ↦ ((π‘€β€˜(𝑖 βˆ’ 1))β€˜(𝑗 βˆ’ 1))))
10 df-lmat 32792 . . 3 litMat = (π‘š ∈ V ↦ (𝑖 ∈ (1...(β™―β€˜π‘š)), 𝑗 ∈ (1...(β™―β€˜(π‘šβ€˜0))) ↦ ((π‘šβ€˜(𝑖 βˆ’ 1))β€˜(𝑗 βˆ’ 1))))
11 ovex 7442 . . . 4 (1...(β™―β€˜π‘€)) ∈ V
12 ovex 7442 . . . 4 (1...(β™―β€˜(π‘€β€˜0))) ∈ V
1311, 12mpoex 8066 . . 3 (𝑖 ∈ (1...(β™―β€˜π‘€)), 𝑗 ∈ (1...(β™―β€˜(π‘€β€˜0))) ↦ ((π‘€β€˜(𝑖 βˆ’ 1))β€˜(𝑗 βˆ’ 1))) ∈ V
149, 10, 13fvmpt 6999 . 2 (𝑀 ∈ V β†’ (litMatβ€˜π‘€) = (𝑖 ∈ (1...(β™―β€˜π‘€)), 𝑗 ∈ (1...(β™―β€˜(π‘€β€˜0))) ↦ ((π‘€β€˜(𝑖 βˆ’ 1))β€˜(𝑗 βˆ’ 1))))
151, 14syl 17 1 (𝑀 ∈ 𝑉 β†’ (litMatβ€˜π‘€) = (𝑖 ∈ (1...(β™―β€˜π‘€)), 𝑗 ∈ (1...(β™―β€˜(π‘€β€˜0))) ↦ ((π‘€β€˜(𝑖 βˆ’ 1))β€˜(𝑗 βˆ’ 1))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  Vcvv 3475  β€˜cfv 6544  (class class class)co 7409   ∈ cmpo 7411  0cc0 11110  1c1 11111   βˆ’ cmin 11444  ...cfz 13484  β™―chash 14290  litMatclmat 32791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-lmat 32792
This theorem is referenced by:  lmatfval  32794  lmatcl  32796
  Copyright terms: Public domain W3C validator