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Mirrors > Home > MPE Home > Th. List > cnnvs | Structured version Visualization version GIF version |
Description: The scalar product operation of the normed complex vector space of complex numbers. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cnnvs.6 | ⊢ 𝑈 = 〈〈 + , · 〉, abs〉 |
Ref | Expression |
---|---|
cnnvs | ⊢ · = ( ·𝑠OLD ‘𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . 3 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈) | |
2 | 1 | smfval 27460 | . 2 ⊢ ( ·𝑠OLD ‘𝑈) = (2nd ‘(1st ‘𝑈)) |
3 | cnnvs.6 | . . . . 5 ⊢ 𝑈 = 〈〈 + , · 〉, abs〉 | |
4 | 3 | fveq2i 6194 | . . . 4 ⊢ (1st ‘𝑈) = (1st ‘〈〈 + , · 〉, abs〉) |
5 | opex 4932 | . . . . 5 ⊢ 〈 + , · 〉 ∈ V | |
6 | absf 14077 | . . . . . 6 ⊢ abs:ℂ⟶ℝ | |
7 | cnex 10017 | . . . . . 6 ⊢ ℂ ∈ V | |
8 | fex 6490 | . . . . . 6 ⊢ ((abs:ℂ⟶ℝ ∧ ℂ ∈ V) → abs ∈ V) | |
9 | 6, 7, 8 | mp2an 708 | . . . . 5 ⊢ abs ∈ V |
10 | 5, 9 | op1st 7176 | . . . 4 ⊢ (1st ‘〈〈 + , · 〉, abs〉) = 〈 + , · 〉 |
11 | 4, 10 | eqtri 2644 | . . 3 ⊢ (1st ‘𝑈) = 〈 + , · 〉 |
12 | 11 | fveq2i 6194 | . 2 ⊢ (2nd ‘(1st ‘𝑈)) = (2nd ‘〈 + , · 〉) |
13 | addex 11830 | . . 3 ⊢ + ∈ V | |
14 | mulex 11831 | . . 3 ⊢ · ∈ V | |
15 | 13, 14 | op2nd 7177 | . 2 ⊢ (2nd ‘〈 + , · 〉) = · |
16 | 2, 12, 15 | 3eqtrri 2649 | 1 ⊢ · = ( ·𝑠OLD ‘𝑈) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∈ wcel 1990 Vcvv 3200 〈cop 4183 ⟶wf 5884 ‘cfv 5888 1st c1st 7166 2nd c2nd 7167 ℂcc 9934 ℝcr 9935 + caddc 9939 · cmul 9941 abscabs 13974 ·𝑠OLD cns 27442 |
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-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 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-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 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-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-sm 27452 |
This theorem is referenced by: cnnvm 27537 ipblnfi 27711 |
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