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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 29850 | . . . . 5 โข ยท๐ OLD = (2nd โ 1st ) | |
3 | 2 | fveq1i 6893 | . . . 4 โข ( ยท๐ OLD โ๐) = ((2nd โ 1st )โ๐) |
4 | fo1st 7995 | . . . . . 6 โข 1st :VโontoโV | |
5 | fof 6806 | . . . . . 6 โข (1st :VโontoโV โ 1st :VโถV) | |
6 | 4, 5 | ax-mp 5 | . . . . 5 โข 1st :VโถV |
7 | fvco3 6991 | . . . . 5 โข ((1st :VโถV โง ๐ โ V) โ ((2nd โ 1st )โ๐) = (2nd โ(1st โ๐))) | |
8 | 6, 7 | mpan 689 | . . . 4 โข (๐ โ V โ ((2nd โ 1st )โ๐) = (2nd โ(1st โ๐))) |
9 | 3, 8 | eqtrid 2785 | . . 3 โข (๐ โ V โ ( ยท๐ OLD โ๐) = (2nd โ(1st โ๐))) |
10 | fvprc 6884 | . . . 4 โข (ยฌ ๐ โ V โ ( ยท๐ OLD โ๐) = โ ) | |
11 | fvprc 6884 | . . . . . 6 โข (ยฌ ๐ โ V โ (1st โ๐) = โ ) | |
12 | 11 | fveq2d 6896 | . . . . 5 โข (ยฌ ๐ โ V โ (2nd โ(1st โ๐)) = (2nd โโ )) |
13 | 2nd0 7982 | . . . . 5 โข (2nd โโ ) = โ | |
14 | 12, 13 | eqtr2di 2790 | . . . 4 โข (ยฌ ๐ โ V โ โ = (2nd โ(1st โ๐))) |
15 | 10, 14 | eqtrd 2773 | . . 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 2761 | 1 โข ๐ = (2nd โ(1st โ๐)) |
Colors of variables: wff setvar class |
Syntax hints: ยฌ wn 3 = wceq 1542 โ wcel 2107 Vcvv 3475 โ c0 4323 โ ccom 5681 โถwf 6540 โontoโwfo 6542 โcfv 6544 1st c1st 7973 2nd c2nd 7974 ยท๐ OLD cns 29840 |
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-sep 5300 ax-nul 5307 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-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 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-fo 6550 df-fv 6552 df-1st 7975 df-2nd 7976 df-sm 29850 |
This theorem is referenced by: nvvop 29862 nvsf 29872 nvscl 29879 nvsid 29880 nvsass 29881 nvdi 29883 nvdir 29884 nv2 29885 nv0 29890 nvsz 29891 nvinv 29892 nvtri 29923 cnnvs 29933 phop 30071 ipdirilem 30082 h2hsm 30228 hhsssm 30511 |
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