![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 3493 | . 2 β’ (π β π β π β V) | |
2 | fveq2 6892 | . . . . 5 β’ (π = π β (β―βπ) = (β―βπ)) | |
3 | 2 | oveq2d 7425 | . . . 4 β’ (π = π β (1...(β―βπ)) = (1...(β―βπ))) |
4 | fveq1 6891 | . . . . . 6 β’ (π = π β (πβ0) = (πβ0)) | |
5 | 4 | fveq2d 6896 | . . . . 5 β’ (π = π β (β―β(πβ0)) = (β―β(πβ0))) |
6 | 5 | oveq2d 7425 | . . . 4 β’ (π = π β (1...(β―β(πβ0))) = (1...(β―β(πβ0)))) |
7 | fveq1 6891 | . . . . 5 β’ (π = π β (πβ(π β 1)) = (πβ(π β 1))) | |
8 | 7 | fveq1d 6894 | . . . 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 7484 | . . 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 32792 | . . 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 7442 | . . . 4 β’ (1...(β―βπ)) β V | |
12 | ovex 7442 | . . . 4 β’ (1...(β―β(πβ0))) β V | |
13 | 11, 12 | mpoex 8066 | . . 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 6999 | . 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 1542 β wcel 2107 Vcvv 3475 βcfv 6544 (class class class)co 7409 β cmpo 7411 0cc0 11110 1c1 11111 β cmin 11444 ...cfz 13484 β―chash 14290 litMatclmat 32791 |
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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
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 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-lmat 32792 |
This theorem is referenced by: lmatfval 32794 lmatcl 32796 |
Copyright terms: Public domain | W3C validator |