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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 | β’ (π β π β (litMatβπ) = (π β (1...(β―βπ)), π β (1...(β―β(πβ0))) β¦ ((πβ(π β 1))β(π β 1)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3461 | . 2 β’ (π β π β π β V) | |
2 | fveq2 6839 | . . . . 5 β’ (π = π β (β―βπ) = (β―βπ)) | |
3 | 2 | oveq2d 7367 | . . . 4 β’ (π = π β (1...(β―βπ)) = (1...(β―βπ))) |
4 | fveq1 6838 | . . . . . 6 β’ (π = π β (πβ0) = (πβ0)) | |
5 | 4 | fveq2d 6843 | . . . . 5 β’ (π = π β (β―β(πβ0)) = (β―β(πβ0))) |
6 | 5 | oveq2d 7367 | . . . 4 β’ (π = π β (1...(β―β(πβ0))) = (1...(β―β(πβ0)))) |
7 | fveq1 6838 | . . . . 5 β’ (π = π β (πβ(π β 1)) = (πβ(π β 1))) | |
8 | 7 | fveq1d 6841 | . . . 4 β’ (π = π β ((πβ(π β 1))β(π β 1)) = ((πβ(π β 1))β(π β 1))) |
9 | 3, 6, 8 | mpoeq123dv 7426 | . . 3 β’ (π = π β (π β (1...(β―βπ)), π β (1...(β―β(πβ0))) β¦ ((πβ(π β 1))β(π β 1))) = (π β (1...(β―βπ)), π β (1...(β―β(πβ0))) β¦ ((πβ(π β 1))β(π β 1)))) |
10 | df-lmat 32205 | . . 3 β’ litMat = (π β V β¦ (π β (1...(β―βπ)), π β (1...(β―β(πβ0))) β¦ ((πβ(π β 1))β(π β 1)))) | |
11 | ovex 7384 | . . . 4 β’ (1...(β―βπ)) β V | |
12 | ovex 7384 | . . . 4 β’ (1...(β―β(πβ0))) β V | |
13 | 11, 12 | mpoex 8004 | . . 3 β’ (π β (1...(β―βπ)), π β (1...(β―β(πβ0))) β¦ ((πβ(π β 1))β(π β 1))) β V |
14 | 9, 10, 13 | fvmpt 6945 | . 2 β’ (π β V β (litMatβπ) = (π β (1...(β―βπ)), π β (1...(β―β(πβ0))) β¦ ((πβ(π β 1))β(π β 1)))) |
15 | 1, 14 | syl 17 | 1 β’ (π β π β (litMatβπ) = (π β (1...(β―βπ)), π β (1...(β―β(πβ0))) β¦ ((πβ(π β 1))β(π β 1)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1541 β wcel 2106 Vcvv 3443 βcfv 6493 (class class class)co 7351 β cmpo 7353 0cc0 11009 1c1 11010 β cmin 11343 ...cfz 13378 β―chash 14184 litMatclmat 32204 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7354 df-oprab 7355 df-mpo 7356 df-1st 7913 df-2nd 7914 df-lmat 32205 |
This theorem is referenced by: lmatfval 32207 lmatcl 32209 |
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