Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lmatval | Structured version Visualization version GIF version |
Description: Value of the literal matrix conversion function. (Contributed by Thierry Arnoux, 28-Aug-2020.) |
Ref | Expression |
---|---|
lmatval | ⊢ (𝑀 ∈ 𝑉 → (litMat‘𝑀) = (𝑖 ∈ (1...(#‘𝑀)), 𝑗 ∈ (1...(#‘(𝑀‘0))) ↦ ((𝑀‘(𝑖 − 1))‘(𝑗 − 1)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3185 | . 2 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V) | |
2 | fveq2 6103 | . . . . 5 ⊢ (𝑚 = 𝑀 → (#‘𝑚) = (#‘𝑀)) | |
3 | 2 | oveq2d 6565 | . . . 4 ⊢ (𝑚 = 𝑀 → (1...(#‘𝑚)) = (1...(#‘𝑀))) |
4 | fveq1 6102 | . . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑚‘0) = (𝑀‘0)) | |
5 | 4 | fveq2d 6107 | . . . . 5 ⊢ (𝑚 = 𝑀 → (#‘(𝑚‘0)) = (#‘(𝑀‘0))) |
6 | 5 | oveq2d 6565 | . . . 4 ⊢ (𝑚 = 𝑀 → (1...(#‘(𝑚‘0))) = (1...(#‘(𝑀‘0)))) |
7 | fveq1 6102 | . . . . 5 ⊢ (𝑚 = 𝑀 → (𝑚‘(𝑖 − 1)) = (𝑀‘(𝑖 − 1))) | |
8 | 7 | fveq1d 6105 | . . . 4 ⊢ (𝑚 = 𝑀 → ((𝑚‘(𝑖 − 1))‘(𝑗 − 1)) = ((𝑀‘(𝑖 − 1))‘(𝑗 − 1))) |
9 | 3, 6, 8 | mpt2eq123dv 6615 | . . 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 29206 | . . 3 ⊢ litMat = (𝑚 ∈ V ↦ (𝑖 ∈ (1...(#‘𝑚)), 𝑗 ∈ (1...(#‘(𝑚‘0))) ↦ ((𝑚‘(𝑖 − 1))‘(𝑗 − 1)))) | |
11 | ovex 6577 | . . . 4 ⊢ (1...(#‘𝑀)) ∈ V | |
12 | ovex 6577 | . . . 4 ⊢ (1...(#‘(𝑀‘0))) ∈ V | |
13 | 11, 12 | mpt2ex 7136 | . . 3 ⊢ (𝑖 ∈ (1...(#‘𝑀)), 𝑗 ∈ (1...(#‘(𝑀‘0))) ↦ ((𝑀‘(𝑖 − 1))‘(𝑗 − 1))) ∈ V |
14 | 9, 10, 13 | fvmpt 6191 | . 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)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 0cc0 9815 1c1 9816 − cmin 10145 ...cfz 12197 #chash 12979 litMatclmat 29205 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-lmat 29206 |
This theorem is referenced by: lmatfval 29208 lmatcl 29210 |
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