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Theorem lmatval 32206
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 3461 . 2 (𝑀 ∈ 𝑉 β†’ 𝑀 ∈ V)
2 fveq2 6839 . . . . 5 (π‘š = 𝑀 β†’ (β™―β€˜π‘š) = (β™―β€˜π‘€))
32oveq2d 7367 . . . 4 (π‘š = 𝑀 β†’ (1...(β™―β€˜π‘š)) = (1...(β™―β€˜π‘€)))
4 fveq1 6838 . . . . . 6 (π‘š = 𝑀 β†’ (π‘šβ€˜0) = (π‘€β€˜0))
54fveq2d 6843 . . . . 5 (π‘š = 𝑀 β†’ (β™―β€˜(π‘šβ€˜0)) = (β™―β€˜(π‘€β€˜0)))
65oveq2d 7367 . . . 4 (π‘š = 𝑀 β†’ (1...(β™―β€˜(π‘šβ€˜0))) = (1...(β™―β€˜(π‘€β€˜0))))
7 fveq1 6838 . . . . 5 (π‘š = 𝑀 β†’ (π‘šβ€˜(𝑖 βˆ’ 1)) = (π‘€β€˜(𝑖 βˆ’ 1)))
87fveq1d 6841 . . . 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 7426 . . 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 32205 . . 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 7384 . . . 4 (1...(β™―β€˜π‘€)) ∈ V
12 ovex 7384 . . . 4 (1...(β™―β€˜(π‘€β€˜0))) ∈ V
1311, 12mpoex 8004 . . 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 6945 . 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 1541   ∈ wcel 2106  Vcvv 3443  β€˜cfv 6493  (class class class)co 7351   ∈ cmpo 7353  0cc0 11009  1c1 11010   βˆ’ cmin 11343  ...cfz 13378  β™―chash 14184  litMatclmat 32204
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-1st 7913  df-2nd 7914  df-lmat 32205
This theorem is referenced by:  lmatfval  32207  lmatcl  32209
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