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Mirrors > Home > MPE Home > Th. List > vafval | Structured version Visualization version GIF version |
Description: Value of the function for the vector addition (group) operation on a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
vafval.2 | ā¢ šŗ = ( +š£ āš) |
Ref | Expression |
---|---|
vafval | ā¢ šŗ = (1st ā(1st āš)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vafval.2 | . 2 ā¢ šŗ = ( +š£ āš) | |
2 | df-va 29848 | . . . . 5 ā¢ +š£ = (1st ā 1st ) | |
3 | 2 | fveq1i 6893 | . . . 4 ā¢ ( +š£ āš) = ((1st ā 1st )āš) |
4 | fo1st 7995 | . . . . . 6 ā¢ 1st :VāontoāV | |
5 | fof 6806 | . . . . . 6 ā¢ (1st :VāontoāV ā 1st :Vā¶V) | |
6 | 4, 5 | ax-mp 5 | . . . . 5 ā¢ 1st :Vā¶V |
7 | fvco3 6991 | . . . . 5 ā¢ ((1st :Vā¶V ā§ š ā V) ā ((1st ā 1st )āš) = (1st ā(1st āš))) | |
8 | 6, 7 | mpan 689 | . . . 4 ā¢ (š ā V ā ((1st ā 1st )āš) = (1st ā(1st āš))) |
9 | 3, 8 | eqtrid 2785 | . . 3 ā¢ (š ā V ā ( +š£ āš) = (1st ā(1st āš))) |
10 | fvprc 6884 | . . . 4 ā¢ (Ā¬ š ā V ā ( +š£ āš) = ā ) | |
11 | fvprc 6884 | . . . . . 6 ā¢ (Ā¬ š ā V ā (1st āš) = ā ) | |
12 | 11 | fveq2d 6896 | . . . . 5 ā¢ (Ā¬ š ā V ā (1st ā(1st āš)) = (1st āā )) |
13 | 1st0 7981 | . . . . 5 ā¢ (1st āā ) = ā | |
14 | 12, 13 | eqtr2di 2790 | . . . 4 ā¢ (Ā¬ š ā V ā ā = (1st ā(1st āš))) |
15 | 10, 14 | eqtrd 2773 | . . 3 ā¢ (Ā¬ š ā V ā ( +š£ āš) = (1st ā(1st āš))) |
16 | 9, 15 | pm2.61i 182 | . 2 ā¢ ( +š£ āš) = (1st ā(1st āš)) |
17 | 1, 16 | eqtri 2761 | 1 ā¢ šŗ = (1st ā(1st āš)) |
Colors of variables: wff setvar class |
Syntax hints: Ā¬ wn 3 = wceq 1542 ā wcel 2107 Vcvv 3475 ā c0 4323 ā ccom 5681 ā¶wf 6540 āontoāwfo 6542 ācfv 6544 1st c1st 7973 +š£ cpv 29838 |
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-va 29848 |
This theorem is referenced by: nvvop 29862 nvablo 29869 nvsf 29872 nvscl 29879 nvsid 29880 nvsass 29881 nvdi 29883 nvdir 29884 nv2 29885 nv0 29890 nvsz 29891 nvinv 29892 cnnvg 29931 phop 30071 ip0i 30078 ipdirilem 30082 h2hva 30227 hhssva 30510 hhshsslem1 30520 |
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