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Mirrors > Home > MPE Home > Th. List > cnmsgngrp | Structured version Visualization version GIF version |
Description: The group of signs under multiplication. (Contributed by Stefan O'Rear, 28-Aug-2015.) |
Ref | Expression |
---|---|
cnmsgngrp.u | ⊢ 𝑈 = ((mulGrp‘ℂfld) ↾s {1, -1}) |
Ref | Expression |
---|---|
cnmsgngrp | ⊢ 𝑈 ∈ Grp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2825 | . . 3 ⊢ ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) | |
2 | 1 | cnmsgnsubg 20289 | . 2 ⊢ {1, -1} ∈ (SubGrp‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) |
3 | cnmsgngrp.u | . . . 4 ⊢ 𝑈 = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
4 | cnex 10340 | . . . . . 6 ⊢ ℂ ∈ V | |
5 | difss 3966 | . . . . . 6 ⊢ (ℂ ∖ {0}) ⊆ ℂ | |
6 | 4, 5 | ssexi 5030 | . . . . 5 ⊢ (ℂ ∖ {0}) ∈ V |
7 | ax-1cn 10317 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
8 | ax-1ne0 10328 | . . . . . . 7 ⊢ 1 ≠ 0 | |
9 | eldifsn 4538 | . . . . . . 7 ⊢ (1 ∈ (ℂ ∖ {0}) ↔ (1 ∈ ℂ ∧ 1 ≠ 0)) | |
10 | 7, 8, 9 | mpbir2an 702 | . . . . . 6 ⊢ 1 ∈ (ℂ ∖ {0}) |
11 | neg1cn 11479 | . . . . . . 7 ⊢ -1 ∈ ℂ | |
12 | neg1ne0 11481 | . . . . . . 7 ⊢ -1 ≠ 0 | |
13 | eldifsn 4538 | . . . . . . 7 ⊢ (-1 ∈ (ℂ ∖ {0}) ↔ (-1 ∈ ℂ ∧ -1 ≠ 0)) | |
14 | 11, 12, 13 | mpbir2an 702 | . . . . . 6 ⊢ -1 ∈ (ℂ ∖ {0}) |
15 | prssi 4572 | . . . . . 6 ⊢ ((1 ∈ (ℂ ∖ {0}) ∧ -1 ∈ (ℂ ∖ {0})) → {1, -1} ⊆ (ℂ ∖ {0})) | |
16 | 10, 14, 15 | mp2an 683 | . . . . 5 ⊢ {1, -1} ⊆ (ℂ ∖ {0}) |
17 | ressabs 16310 | . . . . 5 ⊢ (((ℂ ∖ {0}) ∈ V ∧ {1, -1} ⊆ (ℂ ∖ {0})) → (((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1})) | |
18 | 6, 16, 17 | mp2an 683 | . . . 4 ⊢ (((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) |
19 | 3, 18 | eqtr4i 2852 | . . 3 ⊢ 𝑈 = (((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) ↾s {1, -1}) |
20 | 19 | subggrp 17955 | . 2 ⊢ ({1, -1} ∈ (SubGrp‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) → 𝑈 ∈ Grp) |
21 | 2, 20 | ax-mp 5 | 1 ⊢ 𝑈 ∈ Grp |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1656 ∈ wcel 2164 ≠ wne 2999 Vcvv 3414 ∖ cdif 3795 ⊆ wss 3798 {csn 4399 {cpr 4401 ‘cfv 6127 (class class class)co 6910 ℂcc 10257 0cc0 10259 1c1 10260 -cneg 10593 ↾s cress 16230 Grpcgrp 17783 SubGrpcsubg 17946 mulGrpcmgp 18850 ℂfldccnfld 20113 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-addf 10338 ax-mulf 10339 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-tpos 7622 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-dec 11829 df-uz 11976 df-fz 12627 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-starv 16327 df-tset 16331 df-ple 16332 df-ds 16334 df-unif 16335 df-0g 16462 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-grp 17786 df-minusg 17787 df-subg 17949 df-cmn 18555 df-abl 18556 df-mgp 18851 df-ur 18863 df-ring 18910 df-cring 18911 df-oppr 18984 df-dvdsr 19002 df-unit 19003 df-invr 19033 df-dvr 19044 df-drng 19112 df-cnfld 20114 |
This theorem is referenced by: psgnghm 20292 |
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