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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 2823 | . . 3 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈) | |
2 | 1 | smfval 28384 | . 2 ⊢ ( ·𝑠OLD ‘𝑈) = (2nd ‘(1st ‘𝑈)) |
3 | cnnvs.6 | . . . . 5 ⊢ 𝑈 = 〈〈 + , · 〉, abs〉 | |
4 | 3 | fveq2i 6675 | . . . 4 ⊢ (1st ‘𝑈) = (1st ‘〈〈 + , · 〉, abs〉) |
5 | opex 5358 | . . . . 5 ⊢ 〈 + , · 〉 ∈ V | |
6 | absf 14699 | . . . . . 6 ⊢ abs:ℂ⟶ℝ | |
7 | cnex 10620 | . . . . . 6 ⊢ ℂ ∈ V | |
8 | fex 6991 | . . . . . 6 ⊢ ((abs:ℂ⟶ℝ ∧ ℂ ∈ V) → abs ∈ V) | |
9 | 6, 7, 8 | mp2an 690 | . . . . 5 ⊢ abs ∈ V |
10 | 5, 9 | op1st 7699 | . . . 4 ⊢ (1st ‘〈〈 + , · 〉, abs〉) = 〈 + , · 〉 |
11 | 4, 10 | eqtri 2846 | . . 3 ⊢ (1st ‘𝑈) = 〈 + , · 〉 |
12 | 11 | fveq2i 6675 | . 2 ⊢ (2nd ‘(1st ‘𝑈)) = (2nd ‘〈 + , · 〉) |
13 | addex 12390 | . . 3 ⊢ + ∈ V | |
14 | mulex 12391 | . . 3 ⊢ · ∈ V | |
15 | 13, 14 | op2nd 7700 | . 2 ⊢ (2nd ‘〈 + , · 〉) = · |
16 | 2, 12, 15 | 3eqtrri 2851 | 1 ⊢ · = ( ·𝑠OLD ‘𝑈) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 Vcvv 3496 〈cop 4575 ⟶wf 6353 ‘cfv 6357 1st c1st 7689 2nd c2nd 7690 ℂcc 10537 ℝcr 10538 + caddc 10542 · cmul 10544 abscabs 14595 ·𝑠OLD cns 28366 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-sm 28376 |
This theorem is referenced by: cnnvm 28461 ipblnfi 28634 |
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