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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 20570. (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 10932 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴 + -𝐴) = 0) | |
2 | cnaddabl.g | . . . . 5 ⊢ 𝐺 = {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉} | |
3 | 2 | cnaddabl 18988 | . . . 4 ⊢ 𝐺 ∈ Abel |
4 | ablgrp 18910 | . . . 4 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ 𝐺 ∈ Grp |
6 | id 22 | . . 3 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
7 | negcl 10885 | . . 3 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
8 | cnex 10617 | . . . . 5 ⊢ ℂ ∈ V | |
9 | 2 | grpbase 16609 | . . . . 5 ⊢ (ℂ ∈ V → ℂ = (Base‘𝐺)) |
10 | 8, 9 | ax-mp 5 | . . . 4 ⊢ ℂ = (Base‘𝐺) |
11 | addex 12386 | . . . . 5 ⊢ + ∈ V | |
12 | 2 | grpplusg 16610 | . . . . 5 ⊢ ( + ∈ V → + = (+g‘𝐺)) |
13 | 11, 12 | ax-mp 5 | . . . 4 ⊢ + = (+g‘𝐺) |
14 | 2 | cnaddid 18989 | . . . . 5 ⊢ (0g‘𝐺) = 0 |
15 | 14 | eqcomi 2830 | . . . 4 ⊢ 0 = (0g‘𝐺) |
16 | eqid 2821 | . . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
17 | 10, 13, 15, 16 | grpinvid1 18153 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ ℂ ∧ -𝐴 ∈ ℂ) → (((invg‘𝐺)‘𝐴) = -𝐴 ↔ (𝐴 + -𝐴) = 0)) |
18 | 5, 6, 7, 17 | mp3an2i 1462 | . 2 ⊢ (𝐴 ∈ ℂ → (((invg‘𝐺)‘𝐴) = -𝐴 ↔ (𝐴 + -𝐴) = 0)) |
19 | 1, 18 | mpbird 259 | 1 ⊢ (𝐴 ∈ ℂ → ((invg‘𝐺)‘𝐴) = -𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 Vcvv 3494 {cpr 4568 〈cop 4572 ‘cfv 6354 (class class class)co 7155 ℂcc 10534 0cc0 10536 + caddc 10539 -cneg 10870 ndxcnx 16479 Basecbs 16482 +gcplusg 16564 0gc0g 16712 Grpcgrp 18102 invgcminusg 18103 Abelcabl 18906 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-addf 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-plusg 16577 df-0g 16714 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-grp 18105 df-minusg 18106 df-cmn 18907 df-abl 18908 |
This theorem is referenced by: (None) |
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