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Mirrors > Home > MPE Home > Th. List > vafval | Structured version Visualization version Unicode 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 |
Ref | Expression |
---|---|
vafval |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vafval.2 | . 2 | |
2 | df-va 27450 | . . . . 5 | |
3 | 2 | fveq1i 6192 | . . . 4 |
4 | fo1st 7188 | . . . . . 6 | |
5 | fof 6115 | . . . . . 6 | |
6 | 4, 5 | ax-mp 5 | . . . . 5 |
7 | fvco3 6275 | . . . . 5 | |
8 | 6, 7 | mpan 706 | . . . 4 |
9 | 3, 8 | syl5eq 2668 | . . 3 |
10 | fvprc 6185 | . . . 4 | |
11 | fvprc 6185 | . . . . . 6 | |
12 | 11 | fveq2d 6195 | . . . . 5 |
13 | 1st0 7174 | . . . . 5 | |
14 | 12, 13 | syl6req 2673 | . . . 4 |
15 | 10, 14 | eqtrd 2656 | . . 3 |
16 | 9, 15 | pm2.61i 176 | . 2 |
17 | 1, 16 | eqtri 2644 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wceq 1483 wcel 1990 cvv 3200 c0 3915 ccom 5118 wf 5884 wfo 5886 cfv 5888 c1st 7166 cpv 27440 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fo 5894 df-fv 5896 df-1st 7168 df-va 27450 |
This theorem is referenced by: nvvop 27464 nvablo 27471 nvsf 27474 nvscl 27481 nvsid 27482 nvsass 27483 nvdi 27485 nvdir 27486 nv2 27487 nv0 27492 nvsz 27493 nvinv 27494 cnnvg 27533 phop 27673 phpar 27679 ip0i 27680 ipdirilem 27684 h2hva 27831 hhssva 28114 hhshsslem1 28124 |
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