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 3510 | . 2 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V) | |
2 | fveq2 6663 | . . . . 5 ⊢ (𝑚 = 𝑀 → (♯‘𝑚) = (♯‘𝑀)) | |
3 | 2 | oveq2d 7161 | . . . 4 ⊢ (𝑚 = 𝑀 → (1...(♯‘𝑚)) = (1...(♯‘𝑀))) |
4 | fveq1 6662 | . . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑚‘0) = (𝑀‘0)) | |
5 | 4 | fveq2d 6667 | . . . . 5 ⊢ (𝑚 = 𝑀 → (♯‘(𝑚‘0)) = (♯‘(𝑀‘0))) |
6 | 5 | oveq2d 7161 | . . . 4 ⊢ (𝑚 = 𝑀 → (1...(♯‘(𝑚‘0))) = (1...(♯‘(𝑀‘0)))) |
7 | fveq1 6662 | . . . . 5 ⊢ (𝑚 = 𝑀 → (𝑚‘(𝑖 − 1)) = (𝑀‘(𝑖 − 1))) | |
8 | 7 | fveq1d 6665 | . . . 4 ⊢ (𝑚 = 𝑀 → ((𝑚‘(𝑖 − 1))‘(𝑗 − 1)) = ((𝑀‘(𝑖 − 1))‘(𝑗 − 1))) |
9 | 3, 6, 8 | mpoeq123dv 7218 | . . 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 30976 | . . 3 ⊢ litMat = (𝑚 ∈ V ↦ (𝑖 ∈ (1...(♯‘𝑚)), 𝑗 ∈ (1...(♯‘(𝑚‘0))) ↦ ((𝑚‘(𝑖 − 1))‘(𝑗 − 1)))) | |
11 | ovex 7178 | . . . 4 ⊢ (1...(♯‘𝑀)) ∈ V | |
12 | ovex 7178 | . . . 4 ⊢ (1...(♯‘(𝑀‘0))) ∈ V | |
13 | 11, 12 | mpoex 7766 | . . 3 ⊢ (𝑖 ∈ (1...(♯‘𝑀)), 𝑗 ∈ (1...(♯‘(𝑀‘0))) ↦ ((𝑀‘(𝑖 − 1))‘(𝑗 − 1))) ∈ V |
14 | 9, 10, 13 | fvmpt 6761 | . 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 1528 ∈ wcel 2105 Vcvv 3492 ‘cfv 6348 (class class class)co 7145 ∈ cmpo 7147 0cc0 10525 1c1 10526 − cmin 10858 ...cfz 12880 ♯chash 13678 litMatclmat 30975 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-lmat 30976 |
This theorem is referenced by: lmatfval 30978 lmatcl 30980 |
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