Theorem lmatval 33821

Description: Value of the literal matrix conversion function. (Contributed by Thierry Arnoux, 28-Aug-2020.)

Assertion

Ref	Expression
lmatval	⊢ (𝑀 ∈ 𝑉 → (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.

Step	Hyp	Ref	Expression
1		elex 3457	. 2 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V)
2		fveq2 6822	. . . . 5 ⊢ (𝑚 = 𝑀 → (♯‘𝑚) = (♯‘𝑀))
3	2	oveq2d 7362	. . . 4 ⊢ (𝑚 = 𝑀 → (1...(♯‘𝑚)) = (1...(♯‘𝑀)))
4		fveq1 6821	. . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑚‘0) = (𝑀‘0))
5	4	fveq2d 6826	. . . . 5 ⊢ (𝑚 = 𝑀 → (♯‘(𝑚‘0)) = (♯‘(𝑀‘0)))
6	5	oveq2d 7362	. . . 4 ⊢ (𝑚 = 𝑀 → (1...(♯‘(𝑚‘0))) = (1...(♯‘(𝑀‘0))))
7		fveq1 6821	. . . . 5 ⊢ (𝑚 = 𝑀 → (𝑚‘(𝑖 − 1)) = (𝑀‘(𝑖 − 1)))
8	7	fveq1d 6824	. . . 4 ⊢ (𝑚 = 𝑀 → ((𝑚‘(𝑖 − 1))‘(𝑗 − 1)) = ((𝑀‘(𝑖 − 1))‘(𝑗 − 1)))
9	3, 6, 8	mpoeq123dv 7421	. . 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 33820	. . 3 ⊢ litMat = (𝑚 ∈ V ↦ (𝑖 ∈ (1...(♯‘𝑚)), 𝑗 ∈ (1...(♯‘(𝑚‘0))) ↦ ((𝑚‘(𝑖 − 1))‘(𝑗 − 1))))
11		ovex 7379	. . . 4 ⊢ (1...(♯‘𝑀)) ∈ V
12		ovex 7379	. . . 4 ⊢ (1...(♯‘(𝑀‘0))) ∈ V
13	11, 12	mpoex 8011	. . 3 ⊢ (𝑖 ∈ (1...(♯‘𝑀)), 𝑗 ∈ (1...(♯‘(𝑀‘0))) ↦ ((𝑀‘(𝑖 − 1))‘(𝑗 − 1))) ∈ V
14	9, 10, 13	fvmpt 6929	. 2 ⊢ (𝑀 ∈ V → (litMat‘𝑀) = (𝑖 ∈ (1...(♯‘𝑀)), 𝑗 ∈ (1...(♯‘(𝑀‘0))) ↦ ((𝑀‘(𝑖 − 1))‘(𝑗 − 1))))
15	1, 14	syl 17	1 ⊢ (𝑀 ∈ 𝑉 → (litMat‘𝑀) = (𝑖 ∈ (1...(♯‘𝑀)), 𝑗 ∈ (1...(♯‘(𝑀‘0))) ↦ ((𝑀‘(𝑖 − 1))‘(𝑗 − 1))))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  Vcvv 3436  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348  0cc0 11003  1c1 11004   − cmin 11341  ...cfz 13404  ♯chash 14234  litMatclmat 33819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-lmat 33820

This theorem is referenced by:  lmatfval  33822  lmatcl  33824