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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnmsgn0g | Structured version Visualization version GIF version |
Description: The neutral element of the sign subgroup of the complex numbers. (Contributed by Thierry Arnoux, 1-Nov-2023.) |
Ref | Expression |
---|---|
cnmsgn0g.1 | ⊢ 𝑈 = ((mulGrp‘ℂfld) ↾s {1, -1}) |
Ref | Expression |
---|---|
cnmsgn0g | ⊢ 1 = (0g‘𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnring 21332 | . . 3 ⊢ ℂfld ∈ Ring | |
2 | eqid 2728 | . . . 4 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
3 | 2 | ringmgp 20193 | . . 3 ⊢ (ℂfld ∈ Ring → (mulGrp‘ℂfld) ∈ Mnd) |
4 | 1, 3 | ax-mp 5 | . 2 ⊢ (mulGrp‘ℂfld) ∈ Mnd |
5 | 1ex 11250 | . . 3 ⊢ 1 ∈ V | |
6 | 5 | prid1 4771 | . 2 ⊢ 1 ∈ {1, -1} |
7 | ax-1cn 11206 | . . 3 ⊢ 1 ∈ ℂ | |
8 | neg1cn 12366 | . . 3 ⊢ -1 ∈ ℂ | |
9 | prssi 4829 | . . 3 ⊢ ((1 ∈ ℂ ∧ -1 ∈ ℂ) → {1, -1} ⊆ ℂ) | |
10 | 7, 8, 9 | mp2an 690 | . 2 ⊢ {1, -1} ⊆ ℂ |
11 | cnmsgn0g.1 | . . 3 ⊢ 𝑈 = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
12 | cnfldbas 21297 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
13 | 2, 12 | mgpbas 20094 | . . 3 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
14 | cnfld1 21335 | . . . 4 ⊢ 1 = (1r‘ℂfld) | |
15 | 2, 14 | ringidval 20137 | . . 3 ⊢ 1 = (0g‘(mulGrp‘ℂfld)) |
16 | 11, 13, 15 | ress0g 18731 | . 2 ⊢ (((mulGrp‘ℂfld) ∈ Mnd ∧ 1 ∈ {1, -1} ∧ {1, -1} ⊆ ℂ) → 1 = (0g‘𝑈)) |
17 | 4, 6, 10, 16 | mp3an 1457 | 1 ⊢ 1 = (0g‘𝑈) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 ⊆ wss 3949 {cpr 4634 ‘cfv 6553 (class class class)co 7426 ℂcc 11146 1c1 11149 -cneg 11485 ↾s cress 17218 0gc0g 17430 Mndcmnd 18703 mulGrpcmgp 20088 Ringcrg 20187 ℂfldccnfld 21293 |
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 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-addf 11227 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-7 12320 df-8 12321 df-9 12322 df-n0 12513 df-z 12599 df-dec 12718 df-uz 12863 df-fz 13527 df-struct 17125 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-mulr 17256 df-starv 17257 df-tset 17261 df-ple 17262 df-ds 17264 df-unif 17265 df-0g 17432 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-grp 18907 df-cmn 19751 df-mgp 20089 df-ur 20136 df-ring 20189 df-cring 20190 df-cnfld 21294 |
This theorem is referenced by: evpmsubg 32897 |
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