Theorem lmatval 34492

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 3471	. 2 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V)
2		fveq2 6886	. . . . 5 ⊢ (𝑚 = 𝑀 → (♯‘𝑚) = (♯‘𝑀))
3	2	oveq2d 7437	. . . 4 ⊢ (𝑚 = 𝑀 → (1...(♯‘𝑚)) = (1...(♯‘𝑀)))
4		fveq1 6885	. . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑚‘0) = (𝑀‘0))
5	4	fveq2d 6890	. . . . 5 ⊢ (𝑚 = 𝑀 → (♯‘(𝑚‘0)) = (♯‘(𝑀‘0)))
6	5	oveq2d 7437	. . . 4 ⊢ (𝑚 = 𝑀 → (1...(♯‘(𝑚‘0))) = (1...(♯‘(𝑀‘0))))
7		fveq1 6885	. . . . 5 ⊢ (𝑚 = 𝑀 → (𝑚‘(𝑖 − 1)) = (𝑀‘(𝑖 − 1)))
8	7	fveq1d 6888	. . . 4 ⊢ (𝑚 = 𝑀 → ((𝑚‘(𝑖 − 1))‘(𝑗 − 1)) = ((𝑀‘(𝑖 − 1))‘(𝑗 − 1)))
9	3, 6, 8	mpoeq123dv 7496	. . 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 34491	. . 3 ⊢ litMat = (𝑚 ∈ V ↦ (𝑖 ∈ (1...(♯‘𝑚)), 𝑗 ∈ (1...(♯‘(𝑚‘0))) ↦ ((𝑚‘(𝑖 − 1))‘(𝑗 − 1))))
11		ovex 7454	. . . 4 ⊢ (1...(♯‘𝑀)) ∈ V
12		ovex 7454	. . . 4 ⊢ (1...(♯‘(𝑀‘0))) ∈ V
13	11, 12	mpoex 8093	. . 3 ⊢ (𝑖 ∈ (1...(♯‘𝑀)), 𝑗 ∈ (1...(♯‘(𝑀‘0))) ↦ ((𝑀‘(𝑖 − 1))‘(𝑗 − 1))) ∈ V
14	9, 10, 13	fvmpt 6994	. 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 1569   ∈ wcel 2144  Vcvv 3450  ‘cfv 6539  (class class class)co 7421   ∈ cmpo 7423  0cc0 11204  1c1 11205   − cmin 11545  ...cfz 13643  ♯chash 14478  litMatclmat 34490

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-1st 8002  df-2nd 8003  df-lmat 34491

This theorem is used by:  lmatfval  34493  lmatcl  34495