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Mirrors > Home > MPE Home > Th. List > cnaddinv | Structured version Visualization version GIF version |
Description: Value of the group inverse of complex number addition. See also cnfldneg 20117. (Contributed by Steve Rodriguez, 3-Dec-2006.) (Revised by AV, 26-Aug-2021.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cnaddabl.g | ⊢ 𝐺 = {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉} |
Ref | Expression |
---|---|
cnaddinv | ⊢ (𝐴 ∈ ℂ → ((invg‘𝐺)‘𝐴) = -𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negid 10922 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴 + -𝐴) = 0) | |
2 | cnaddabl.g | . . . . 5 ⊢ 𝐺 = {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉} | |
3 | 2 | cnaddabl 18982 | . . . 4 ⊢ 𝐺 ∈ Abel |
4 | ablgrp 18903 | . . . 4 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ 𝐺 ∈ Grp |
6 | id 22 | . . 3 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
7 | negcl 10875 | . . 3 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
8 | cnex 10607 | . . . . 5 ⊢ ℂ ∈ V | |
9 | 2 | grpbase 16602 | . . . . 5 ⊢ (ℂ ∈ V → ℂ = (Base‘𝐺)) |
10 | 8, 9 | ax-mp 5 | . . . 4 ⊢ ℂ = (Base‘𝐺) |
11 | addex 12375 | . . . . 5 ⊢ + ∈ V | |
12 | 2 | grpplusg 16603 | . . . . 5 ⊢ ( + ∈ V → + = (+g‘𝐺)) |
13 | 11, 12 | ax-mp 5 | . . . 4 ⊢ + = (+g‘𝐺) |
14 | 2 | cnaddid 18983 | . . . . 5 ⊢ (0g‘𝐺) = 0 |
15 | 14 | eqcomi 2807 | . . . 4 ⊢ 0 = (0g‘𝐺) |
16 | eqid 2798 | . . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
17 | 10, 13, 15, 16 | grpinvid1 18146 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ ℂ ∧ -𝐴 ∈ ℂ) → (((invg‘𝐺)‘𝐴) = -𝐴 ↔ (𝐴 + -𝐴) = 0)) |
18 | 5, 6, 7, 17 | mp3an2i 1463 | . 2 ⊢ (𝐴 ∈ ℂ → (((invg‘𝐺)‘𝐴) = -𝐴 ↔ (𝐴 + -𝐴) = 0)) |
19 | 1, 18 | mpbird 260 | 1 ⊢ (𝐴 ∈ ℂ → ((invg‘𝐺)‘𝐴) = -𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 Vcvv 3441 {cpr 4527 〈cop 4531 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 0cc0 10526 + caddc 10529 -cneg 10860 ndxcnx 16472 Basecbs 16475 +gcplusg 16557 0gc0g 16705 Grpcgrp 18095 invgcminusg 18096 Abelcabl 18899 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-addf 10605 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-plusg 16570 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-cmn 18900 df-abl 18901 |
This theorem is referenced by: (None) |
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