![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > smfval | Structured version Visualization version GIF version |
Description: Value of the function for the scalar multiplication operation on a normed complex vector space. (Contributed by NM, 24-Apr-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
smfval.4 | โข ๐ = ( ยท๐ OLD โ๐) |
Ref | Expression |
---|---|
smfval | โข ๐ = (2nd โ(1st โ๐)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smfval.4 | . 2 โข ๐ = ( ยท๐ OLD โ๐) | |
2 | df-sm 30117 | . . . . 5 โข ยท๐ OLD = (2nd โ 1st ) | |
3 | 2 | fveq1i 6891 | . . . 4 โข ( ยท๐ OLD โ๐) = ((2nd โ 1st )โ๐) |
4 | fo1st 7997 | . . . . . 6 โข 1st :VโontoโV | |
5 | fof 6804 | . . . . . 6 โข (1st :VโontoโV โ 1st :VโถV) | |
6 | 4, 5 | ax-mp 5 | . . . . 5 โข 1st :VโถV |
7 | fvco3 6989 | . . . . 5 โข ((1st :VโถV โง ๐ โ V) โ ((2nd โ 1st )โ๐) = (2nd โ(1st โ๐))) | |
8 | 6, 7 | mpan 686 | . . . 4 โข (๐ โ V โ ((2nd โ 1st )โ๐) = (2nd โ(1st โ๐))) |
9 | 3, 8 | eqtrid 2782 | . . 3 โข (๐ โ V โ ( ยท๐ OLD โ๐) = (2nd โ(1st โ๐))) |
10 | fvprc 6882 | . . . 4 โข (ยฌ ๐ โ V โ ( ยท๐ OLD โ๐) = โ ) | |
11 | fvprc 6882 | . . . . . 6 โข (ยฌ ๐ โ V โ (1st โ๐) = โ ) | |
12 | 11 | fveq2d 6894 | . . . . 5 โข (ยฌ ๐ โ V โ (2nd โ(1st โ๐)) = (2nd โโ )) |
13 | 2nd0 7984 | . . . . 5 โข (2nd โโ ) = โ | |
14 | 12, 13 | eqtr2di 2787 | . . . 4 โข (ยฌ ๐ โ V โ โ = (2nd โ(1st โ๐))) |
15 | 10, 14 | eqtrd 2770 | . . 3 โข (ยฌ ๐ โ V โ ( ยท๐ OLD โ๐) = (2nd โ(1st โ๐))) |
16 | 9, 15 | pm2.61i 182 | . 2 โข ( ยท๐ OLD โ๐) = (2nd โ(1st โ๐)) |
17 | 1, 16 | eqtri 2758 | 1 โข ๐ = (2nd โ(1st โ๐)) |
Colors of variables: wff setvar class |
Syntax hints: ยฌ wn 3 = wceq 1539 โ wcel 2104 Vcvv 3472 โ c0 4321 โ ccom 5679 โถwf 6538 โontoโwfo 6540 โcfv 6542 1st c1st 7975 2nd c2nd 7976 ยท๐ OLD cns 30107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fo 6548 df-fv 6550 df-1st 7977 df-2nd 7978 df-sm 30117 |
This theorem is referenced by: nvvop 30129 nvsf 30139 nvscl 30146 nvsid 30147 nvsass 30148 nvdi 30150 nvdir 30151 nv2 30152 nv0 30157 nvsz 30158 nvinv 30159 nvtri 30190 cnnvs 30200 phop 30338 ipdirilem 30349 h2hsm 30495 hhsssm 30778 |
Copyright terms: Public domain | W3C validator |