Theorem lmatval 34438

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 3472	. 2 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V)
2		fveq2 6883	. . . . 5 ⊢ (𝑚 = 𝑀 → (♯‘𝑚) = (♯‘𝑀))
3	2	oveq2d 7434	. . . 4 ⊢ (𝑚 = 𝑀 → (1...(♯‘𝑚)) = (1...(♯‘𝑀)))
4		fveq1 6882	. . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑚‘0) = (𝑀‘0))
5	4	fveq2d 6887	. . . . 5 ⊢ (𝑚 = 𝑀 → (♯‘(𝑚‘0)) = (♯‘(𝑀‘0)))
6	5	oveq2d 7434	. . . 4 ⊢ (𝑚 = 𝑀 → (1...(♯‘(𝑚‘0))) = (1...(♯‘(𝑀‘0))))
7		fveq1 6882	. . . . 5 ⊢ (𝑚 = 𝑀 → (𝑚‘(𝑖 − 1)) = (𝑀‘(𝑖 − 1)))
8	7	fveq1d 6885	. . . 4 ⊢ (𝑚 = 𝑀 → ((𝑚‘(𝑖 − 1))‘(𝑗 − 1)) = ((𝑀‘(𝑖 − 1))‘(𝑗 − 1)))
9	3, 6, 8	mpoeq123dv 7493	. . 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 34437	. . 3 ⊢ litMat = (𝑚 ∈ V ↦ (𝑖 ∈ (1...(♯‘𝑚)), 𝑗 ∈ (1...(♯‘(𝑚‘0))) ↦ ((𝑚‘(𝑖 − 1))‘(𝑗 − 1))))
11		ovex 7451	. . . 4 ⊢ (1...(♯‘𝑀)) ∈ V
12		ovex 7451	. . . 4 ⊢ (1...(♯‘(𝑀‘0))) ∈ V
13	11, 12	mpoex 8090	. . 3 ⊢ (𝑖 ∈ (1...(♯‘𝑀)), 𝑗 ∈ (1...(♯‘(𝑀‘0))) ↦ ((𝑀‘(𝑖 − 1))‘(𝑗 − 1))) ∈ V
14	9, 10, 13	fvmpt 6991	. 2 ⊢ (𝑀 ∈ V → (litMat‘𝑀) = (𝑖 ∈ (1...(♯‘𝑀)), 𝑗 ∈ (1...(♯‘(𝑀‘0))) ↦ ((𝑀‘(𝑖 − 1))‘(𝑗 − 1))))
15	1, 14	syl 18	1 ⊢ (𝑀 ∈ 𝑉 → (litMat‘𝑀) = (𝑖 ∈ (1...(♯‘𝑀)), 𝑗 ∈ (1...(♯‘(𝑀‘0))) ↦ ((𝑀‘(𝑖 − 1))‘(𝑗 − 1))))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  Vcvv 3451  ‘cfv 6537  (class class class)co 7418   ∈ cmpo 7420  0cc0 11193  1c1 11194   − cmin 11534  ...cfz 13632  ♯chash 14467  litMatclmat 34436

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-1st 7999  df-2nd 8000  df-lmat 34437

This theorem is used by:  lmatfval  34439  lmatcl  34441