Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > cnnvba | Structured version Visualization version GIF version |
Description: The base set of the normed complex vector space of complex numbers. (Contributed by NM, 7-Nov-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cnnvba.6 | β’ π = β¨β¨ + , Β· β©, absβ© |
Ref | Expression |
---|---|
cnnvba | β’ β = (BaseSetβπ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnnvba.6 | . . . 4 β’ π = β¨β¨ + , Β· β©, absβ© | |
2 | 1 | cnnvg 29418 | . . 3 β’ + = ( +π£ βπ) |
3 | 2 | rneqi 5888 | . 2 β’ ran + = ran ( +π£ βπ) |
4 | cnaddabloOLD 29321 | . . . 4 β’ + β AbelOp | |
5 | ablogrpo 29287 | . . . 4 β’ ( + β AbelOp β + β GrpOp) | |
6 | 4, 5 | ax-mp 5 | . . 3 β’ + β GrpOp |
7 | ax-addf 11063 | . . . 4 β’ + :(β Γ β)βΆβ | |
8 | 7 | fdmi 6675 | . . 3 β’ dom + = (β Γ β) |
9 | 6, 8 | grporn 29261 | . 2 β’ β = ran + |
10 | eqid 2737 | . . 3 β’ (BaseSetβπ) = (BaseSetβπ) | |
11 | eqid 2737 | . . 3 β’ ( +π£ βπ) = ( +π£ βπ) | |
12 | 10, 11 | bafval 29344 | . 2 β’ (BaseSetβπ) = ran ( +π£ βπ) |
13 | 3, 9, 12 | 3eqtr4i 2775 | 1 β’ β = (BaseSetβπ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 β wcel 2106 β¨cop 4590 Γ cxp 5628 ran crn 5631 βcfv 6491 βcc 10982 + caddc 10987 Β· cmul 10989 abscabs 15052 GrpOpcgr 29229 AbelOpcablo 29284 +π£ cpv 29325 BaseSetcba 29326 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7662 ax-cnex 11040 ax-resscn 11041 ax-1cn 11042 ax-icn 11043 ax-addcl 11044 ax-addrcl 11045 ax-mulcl 11046 ax-mulrcl 11047 ax-mulcom 11048 ax-addass 11049 ax-mulass 11050 ax-distr 11051 ax-i2m1 11052 ax-1ne0 11053 ax-1rid 11054 ax-rnegex 11055 ax-rrecex 11056 ax-cnre 11057 ax-pre-lttri 11058 ax-pre-lttrn 11059 ax-pre-ltadd 11060 ax-pre-mulgt0 11061 ax-pre-sup 11062 ax-addf 11063 ax-mulf 11064 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5528 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5585 df-we 5587 df-xp 5636 df-rel 5637 df-cnv 5638 df-co 5639 df-dm 5640 df-rn 5641 df-res 5642 df-ima 5643 df-pred 6249 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6443 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-riota 7305 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7793 df-1st 7911 df-2nd 7912 df-frecs 8179 df-wrecs 8210 df-recs 8284 df-rdg 8323 df-er 8581 df-en 8817 df-dom 8818 df-sdom 8819 df-sup 9311 df-pnf 11124 df-mnf 11125 df-xr 11126 df-ltxr 11127 df-le 11128 df-sub 11320 df-neg 11321 df-div 11746 df-nn 12087 df-2 12149 df-3 12150 df-n0 12347 df-z 12433 df-uz 12696 df-rp 12844 df-seq 13835 df-exp 13896 df-cj 14917 df-re 14918 df-im 14919 df-sqrt 15053 df-abs 15054 df-grpo 29233 df-ablo 29285 df-va 29335 df-ba 29336 |
This theorem is referenced by: cnnvm 29422 ipblnfi 29595 cnbn 29609 htthlem 29657 |
Copyright terms: Public domain | W3C validator |