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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 3493 | . 2 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V) | |
2 | fveq2 6889 | . . . . 5 ⊢ (𝑚 = 𝑀 → (♯‘𝑚) = (♯‘𝑀)) | |
3 | 2 | oveq2d 7422 | . . . 4 ⊢ (𝑚 = 𝑀 → (1...(♯‘𝑚)) = (1...(♯‘𝑀))) |
4 | fveq1 6888 | . . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑚‘0) = (𝑀‘0)) | |
5 | 4 | fveq2d 6893 | . . . . 5 ⊢ (𝑚 = 𝑀 → (♯‘(𝑚‘0)) = (♯‘(𝑀‘0))) |
6 | 5 | oveq2d 7422 | . . . 4 ⊢ (𝑚 = 𝑀 → (1...(♯‘(𝑚‘0))) = (1...(♯‘(𝑀‘0)))) |
7 | fveq1 6888 | . . . . 5 ⊢ (𝑚 = 𝑀 → (𝑚‘(𝑖 − 1)) = (𝑀‘(𝑖 − 1))) | |
8 | 7 | fveq1d 6891 | . . . 4 ⊢ (𝑚 = 𝑀 → ((𝑚‘(𝑖 − 1))‘(𝑗 − 1)) = ((𝑀‘(𝑖 − 1))‘(𝑗 − 1))) |
9 | 3, 6, 8 | mpoeq123dv 7481 | . . 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 32781 | . . 3 ⊢ litMat = (𝑚 ∈ V ↦ (𝑖 ∈ (1...(♯‘𝑚)), 𝑗 ∈ (1...(♯‘(𝑚‘0))) ↦ ((𝑚‘(𝑖 − 1))‘(𝑗 − 1)))) | |
11 | ovex 7439 | . . . 4 ⊢ (1...(♯‘𝑀)) ∈ V | |
12 | ovex 7439 | . . . 4 ⊢ (1...(♯‘(𝑀‘0))) ∈ V | |
13 | 11, 12 | mpoex 8063 | . . 3 ⊢ (𝑖 ∈ (1...(♯‘𝑀)), 𝑗 ∈ (1...(♯‘(𝑀‘0))) ↦ ((𝑀‘(𝑖 − 1))‘(𝑗 − 1))) ∈ V |
14 | 9, 10, 13 | fvmpt 6996 | . 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 1542 ∈ wcel 2107 Vcvv 3475 ‘cfv 6541 (class class class)co 7406 ∈ cmpo 7408 0cc0 11107 1c1 11108 − cmin 11441 ...cfz 13481 ♯chash 14287 litMatclmat 32780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7409 df-oprab 7410 df-mpo 7411 df-1st 7972 df-2nd 7973 df-lmat 32781 |
This theorem is referenced by: lmatfval 32783 lmatcl 32785 |
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