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Mirrors > Home > MPE Home > Th. List > bafval | Structured version Visualization version GIF version |
Description: Value of the function for the base set of a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bafval.1 | β’ π = (BaseSetβπ) |
bafval.2 | β’ πΊ = ( +π£ βπ) |
Ref | Expression |
---|---|
bafval | β’ π = ran πΊ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6846 | . . . . 5 β’ (π’ = π β ( +π£ βπ’) = ( +π£ βπ)) | |
2 | 1 | rneqd 5897 | . . . 4 β’ (π’ = π β ran ( +π£ βπ’) = ran ( +π£ βπ)) |
3 | df-ba 29587 | . . . 4 β’ BaseSet = (π’ β V β¦ ran ( +π£ βπ’)) | |
4 | fvex 6859 | . . . . 5 β’ ( +π£ βπ) β V | |
5 | 4 | rnex 7853 | . . . 4 β’ ran ( +π£ βπ) β V |
6 | 2, 3, 5 | fvmpt 6952 | . . 3 β’ (π β V β (BaseSetβπ) = ran ( +π£ βπ)) |
7 | rn0 5885 | . . . . 5 β’ ran β = β | |
8 | 7 | eqcomi 2742 | . . . 4 β’ β = ran β |
9 | fvprc 6838 | . . . 4 β’ (Β¬ π β V β (BaseSetβπ) = β ) | |
10 | fvprc 6838 | . . . . 5 β’ (Β¬ π β V β ( +π£ βπ) = β ) | |
11 | 10 | rneqd 5897 | . . . 4 β’ (Β¬ π β V β ran ( +π£ βπ) = ran β ) |
12 | 8, 9, 11 | 3eqtr4a 2799 | . . 3 β’ (Β¬ π β V β (BaseSetβπ) = ran ( +π£ βπ)) |
13 | 6, 12 | pm2.61i 182 | . 2 β’ (BaseSetβπ) = ran ( +π£ βπ) |
14 | bafval.1 | . 2 β’ π = (BaseSetβπ) | |
15 | bafval.2 | . . 3 β’ πΊ = ( +π£ βπ) | |
16 | 15 | rneqi 5896 | . 2 β’ ran πΊ = ran ( +π£ βπ) |
17 | 13, 14, 16 | 3eqtr4i 2771 | 1 β’ π = ran πΊ |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 = wceq 1542 β wcel 2107 Vcvv 3447 β c0 4286 ran crn 5638 βcfv 6500 +π£ cpv 29576 BaseSetcba 29577 |
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 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 |
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 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-iota 6452 df-fun 6502 df-fv 6508 df-ba 29587 |
This theorem is referenced by: nvi 29605 nvgf 29609 nvsf 29610 nvgcl 29611 nvcom 29612 nvass 29613 nvadd32 29614 nvrcan 29615 nvadd4 29616 nvscl 29617 nvsid 29618 nvsass 29619 nvdi 29621 nvdir 29622 nv2 29623 nvzcl 29625 nv0rid 29626 nv0lid 29627 nv0 29628 nvsz 29629 nvinv 29630 nvinvfval 29631 nvmval 29633 nvmfval 29635 nvnnncan1 29638 nvnegneg 29640 nvrinv 29642 nvlinv 29643 nvaddsub 29646 cnnvba 29670 sspba 29718 isph 29813 phpar 29815 ip0i 29816 ipdirilem 29820 hhba 30158 hhssabloilem 30252 hhshsslem1 30258 |
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