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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 2725 | . . 3 ⊢ ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) | |
2 | 1 | cnmsgnsubg 21531 | . 2 ⊢ {1, -1} ∈ (SubGrp‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) |
3 | cnmsgngrp.u | . . . 4 ⊢ 𝑈 = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
4 | cnex 11226 | . . . . . 6 ⊢ ℂ ∈ V | |
5 | 4 | difexi 5331 | . . . . 5 ⊢ (ℂ ∖ {0}) ∈ V |
6 | ax-1cn 11203 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
7 | ax-1ne0 11214 | . . . . . . 7 ⊢ 1 ≠ 0 | |
8 | eldifsn 4792 | . . . . . . 7 ⊢ (1 ∈ (ℂ ∖ {0}) ↔ (1 ∈ ℂ ∧ 1 ≠ 0)) | |
9 | 6, 7, 8 | mpbir2an 709 | . . . . . 6 ⊢ 1 ∈ (ℂ ∖ {0}) |
10 | neg1cn 12364 | . . . . . . 7 ⊢ -1 ∈ ℂ | |
11 | neg1ne0 12366 | . . . . . . 7 ⊢ -1 ≠ 0 | |
12 | eldifsn 4792 | . . . . . . 7 ⊢ (-1 ∈ (ℂ ∖ {0}) ↔ (-1 ∈ ℂ ∧ -1 ≠ 0)) | |
13 | 10, 11, 12 | mpbir2an 709 | . . . . . 6 ⊢ -1 ∈ (ℂ ∖ {0}) |
14 | prssi 4826 | . . . . . 6 ⊢ ((1 ∈ (ℂ ∖ {0}) ∧ -1 ∈ (ℂ ∖ {0})) → {1, -1} ⊆ (ℂ ∖ {0})) | |
15 | 9, 13, 14 | mp2an 690 | . . . . 5 ⊢ {1, -1} ⊆ (ℂ ∖ {0}) |
16 | ressabs 17238 | . . . . 5 ⊢ (((ℂ ∖ {0}) ∈ V ∧ {1, -1} ⊆ (ℂ ∖ {0})) → (((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1})) | |
17 | 5, 15, 16 | mp2an 690 | . . . 4 ⊢ (((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) |
18 | 3, 17 | eqtr4i 2756 | . . 3 ⊢ 𝑈 = (((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) ↾s {1, -1}) |
19 | 18 | subggrp 19097 | . 2 ⊢ ({1, -1} ∈ (SubGrp‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) → 𝑈 ∈ Grp) |
20 | 2, 19 | ax-mp 5 | 1 ⊢ 𝑈 ∈ Grp |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 ≠ wne 2929 Vcvv 3461 ∖ cdif 3941 ⊆ wss 3944 {csn 4630 {cpr 4632 ‘cfv 6549 (class class class)co 7419 ℂcc 11143 0cc0 11145 1c1 11146 -cneg 11482 ↾s cress 17217 Grpcgrp 18903 SubGrpcsubg 19088 mulGrpcmgp 20091 ℂfldccnfld 21301 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 ax-addf 11224 ax-mulf 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-div 11909 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-fz 13525 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17189 df-ress 17218 df-plusg 17254 df-mulr 17255 df-starv 17256 df-tset 17260 df-ple 17261 df-ds 17263 df-unif 17264 df-0g 17431 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-grp 18906 df-minusg 18907 df-subg 19091 df-cmn 19754 df-abl 19755 df-mgp 20092 df-rng 20110 df-ur 20139 df-ring 20192 df-cring 20193 df-oppr 20290 df-dvdsr 20313 df-unit 20314 df-invr 20344 df-dvr 20357 df-drng 20643 df-cnfld 21302 |
This theorem is referenced by: psgnghm 21534 evpmsubg 32965 |
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