![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 20839. (Contributed by Steve Rodriguez, 3-Dec-2006.) (Revised by AV, 26-Aug-2021.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cnaddabl.g | β’ πΊ = {β¨(Baseβndx), ββ©, β¨(+gβndx), + β©} |
Ref | Expression |
---|---|
cnaddinv | β’ (π΄ β β β ((invgβπΊ)βπ΄) = -π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negid 11455 | . 2 β’ (π΄ β β β (π΄ + -π΄) = 0) | |
2 | cnaddabl.g | . . . . 5 β’ πΊ = {β¨(Baseβndx), ββ©, β¨(+gβndx), + β©} | |
3 | 2 | cnaddabl 19654 | . . . 4 β’ πΊ β Abel |
4 | ablgrp 19574 | . . . 4 β’ (πΊ β Abel β πΊ β Grp) | |
5 | 3, 4 | ax-mp 5 | . . 3 β’ πΊ β Grp |
6 | id 22 | . . 3 β’ (π΄ β β β π΄ β β) | |
7 | negcl 11408 | . . 3 β’ (π΄ β β β -π΄ β β) | |
8 | cnex 11139 | . . . . 5 β’ β β V | |
9 | 2 | grpbase 17174 | . . . . 5 β’ (β β V β β = (BaseβπΊ)) |
10 | 8, 9 | ax-mp 5 | . . . 4 β’ β = (BaseβπΊ) |
11 | addex 12920 | . . . . 5 β’ + β V | |
12 | 2 | grpplusg 17176 | . . . . 5 β’ ( + β V β + = (+gβπΊ)) |
13 | 11, 12 | ax-mp 5 | . . . 4 β’ + = (+gβπΊ) |
14 | 2 | cnaddid 19655 | . . . . 5 β’ (0gβπΊ) = 0 |
15 | 14 | eqcomi 2746 | . . . 4 β’ 0 = (0gβπΊ) |
16 | eqid 2737 | . . . 4 β’ (invgβπΊ) = (invgβπΊ) | |
17 | 10, 13, 15, 16 | grpinvid1 18809 | . . 3 β’ ((πΊ β Grp β§ π΄ β β β§ -π΄ β β) β (((invgβπΊ)βπ΄) = -π΄ β (π΄ + -π΄) = 0)) |
18 | 5, 6, 7, 17 | mp3an2i 1467 | . 2 β’ (π΄ β β β (((invgβπΊ)βπ΄) = -π΄ β (π΄ + -π΄) = 0)) |
19 | 1, 18 | mpbird 257 | 1 β’ (π΄ β β β ((invgβπΊ)βπ΄) = -π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1542 β wcel 2107 Vcvv 3448 {cpr 4593 β¨cop 4597 βcfv 6501 (class class class)co 7362 βcc 11056 0cc0 11058 + caddc 11061 -cneg 11393 ndxcnx 17072 Basecbs 17090 +gcplusg 17140 0gc0g 17328 Grpcgrp 18755 invgcminusg 18756 Abelcabl 19570 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-addf 11137 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-2 12223 df-n0 12421 df-z 12507 df-uz 12771 df-fz 13432 df-struct 17026 df-slot 17061 df-ndx 17073 df-base 17091 df-plusg 17153 df-0g 17330 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-grp 18758 df-minusg 18759 df-cmn 19571 df-abl 19572 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |