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Mirrors > Home > MPE Home > Th. List > cnnvg | Structured version Visualization version GIF version |
Description: The vector addition (group) operation of the normed complex vector space of complex numbers. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cnnvg.6 | ⊢ 𝑈 = 〈〈 + , · 〉, abs〉 |
Ref | Expression |
---|---|
cnnvg | ⊢ + = ( +𝑣 ‘𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . . 3 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
2 | 1 | vafval 29492 | . 2 ⊢ ( +𝑣 ‘𝑈) = (1st ‘(1st ‘𝑈)) |
3 | cnnvg.6 | . . . . 5 ⊢ 𝑈 = 〈〈 + , · 〉, abs〉 | |
4 | 3 | fveq2i 6845 | . . . 4 ⊢ (1st ‘𝑈) = (1st ‘〈〈 + , · 〉, abs〉) |
5 | opex 5421 | . . . . 5 ⊢ 〈 + , · 〉 ∈ V | |
6 | absf 15221 | . . . . . 6 ⊢ abs:ℂ⟶ℝ | |
7 | cnex 11131 | . . . . . 6 ⊢ ℂ ∈ V | |
8 | fex 7175 | . . . . . 6 ⊢ ((abs:ℂ⟶ℝ ∧ ℂ ∈ V) → abs ∈ V) | |
9 | 6, 7, 8 | mp2an 690 | . . . . 5 ⊢ abs ∈ V |
10 | 5, 9 | op1st 7928 | . . . 4 ⊢ (1st ‘〈〈 + , · 〉, abs〉) = 〈 + , · 〉 |
11 | 4, 10 | eqtri 2764 | . . 3 ⊢ (1st ‘𝑈) = 〈 + , · 〉 |
12 | 11 | fveq2i 6845 | . 2 ⊢ (1st ‘(1st ‘𝑈)) = (1st ‘〈 + , · 〉) |
13 | addex 12912 | . . 3 ⊢ + ∈ V | |
14 | mulex 12913 | . . 3 ⊢ · ∈ V | |
15 | 13, 14 | op1st 7928 | . 2 ⊢ (1st ‘〈 + , · 〉) = + |
16 | 2, 12, 15 | 3eqtrri 2769 | 1 ⊢ + = ( +𝑣 ‘𝑈) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 Vcvv 3445 〈cop 4592 ⟶wf 6492 ‘cfv 6496 1st c1st 7918 ℂcc 11048 ℝcr 11049 + caddc 11053 · cmul 11055 abscabs 15118 +𝑣 cpv 29474 |
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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 ax-pre-sup 11128 ax-addf 11129 ax-mulf 11130 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7802 df-1st 7920 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-er 8647 df-en 8883 df-dom 8884 df-sdom 8885 df-sup 9377 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-div 11812 df-nn 12153 df-2 12215 df-3 12216 df-n0 12413 df-z 12499 df-uz 12763 df-rp 12915 df-seq 13906 df-exp 13967 df-cj 14983 df-re 14984 df-im 14985 df-sqrt 15119 df-abs 15120 df-va 29484 |
This theorem is referenced by: cnnvba 29568 cnnvm 29571 ipblnfi 29744 |
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