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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 29406 | . . 3 β’ + = ( +π£ βπ) |
3 | 2 | rneqi 5889 | . 2 β’ ran + = ran ( +π£ βπ) |
4 | cnaddabloOLD 29309 | . . . 4 β’ + β AbelOp | |
5 | ablogrpo 29275 | . . . 4 β’ ( + β AbelOp β + β GrpOp) | |
6 | 4, 5 | ax-mp 5 | . . 3 β’ + β GrpOp |
7 | ax-addf 11064 | . . . 4 β’ + :(β Γ β)βΆβ | |
8 | 7 | fdmi 6676 | . . 3 β’ dom + = (β Γ β) |
9 | 6, 8 | grporn 29249 | . 2 β’ β = ran + |
10 | eqid 2738 | . . 3 β’ (BaseSetβπ) = (BaseSetβπ) | |
11 | eqid 2738 | . . 3 β’ ( +π£ βπ) = ( +π£ βπ) | |
12 | 10, 11 | bafval 29332 | . 2 β’ (BaseSetβπ) = ran ( +π£ βπ) |
13 | 3, 9, 12 | 3eqtr4i 2776 | 1 β’ β = (BaseSetβπ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 β wcel 2107 β¨cop 4591 Γ cxp 5629 ran crn 5632 βcfv 6492 βcc 10983 + caddc 10988 Β· cmul 10990 abscabs 15053 GrpOpcgr 29217 AbelOpcablo 29272 +π£ cpv 29313 BaseSetcba 29314 |
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 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 ax-pre-sup 11063 ax-addf 11064 ax-mulf 11065 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-1st 7912 df-2nd 7913 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-er 8582 df-en 8818 df-dom 8819 df-sdom 8820 df-sup 9312 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-div 11747 df-nn 12088 df-2 12150 df-3 12151 df-n0 12348 df-z 12434 df-uz 12697 df-rp 12845 df-seq 13836 df-exp 13897 df-cj 14918 df-re 14919 df-im 14920 df-sqrt 15054 df-abs 15055 df-grpo 29221 df-ablo 29273 df-va 29323 df-ba 29324 |
This theorem is referenced by: cnnvm 29410 ipblnfi 29583 cnbn 29597 htthlem 29645 |
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