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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 21279 | . . 3 ⊢ ℂfld ∈ Ring | |
2 | eqid 2726 | . . . 4 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
3 | 2 | ringmgp 20144 | . . 3 ⊢ (ℂfld ∈ Ring → (mulGrp‘ℂfld) ∈ Mnd) |
4 | 1, 3 | ax-mp 5 | . 2 ⊢ (mulGrp‘ℂfld) ∈ Mnd |
5 | 1ex 11214 | . . 3 ⊢ 1 ∈ V | |
6 | 5 | prid1 4761 | . 2 ⊢ 1 ∈ {1, -1} |
7 | ax-1cn 11170 | . . 3 ⊢ 1 ∈ ℂ | |
8 | neg1cn 12330 | . . 3 ⊢ -1 ∈ ℂ | |
9 | prssi 4819 | . . 3 ⊢ ((1 ∈ ℂ ∧ -1 ∈ ℂ) → {1, -1} ⊆ ℂ) | |
10 | 7, 8, 9 | mp2an 689 | . 2 ⊢ {1, -1} ⊆ ℂ |
11 | cnmsgn0g.1 | . . 3 ⊢ 𝑈 = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
12 | cnfldbas 21244 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
13 | 2, 12 | mgpbas 20045 | . . 3 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
14 | cnfld1 21282 | . . . 4 ⊢ 1 = (1r‘ℂfld) | |
15 | 2, 14 | ringidval 20088 | . . 3 ⊢ 1 = (0g‘(mulGrp‘ℂfld)) |
16 | 11, 13, 15 | ress0g 18695 | . 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 3943 {cpr 4625 ‘cfv 6537 (class class class)co 7405 ℂcc 11110 1c1 11113 -cneg 11449 ↾s cress 17182 0gc0g 17394 Mndcmnd 18667 mulGrpcmgp 20039 Ringcrg 20138 ℂfldccnfld 21240 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-addf 11191 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13491 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-0g 17396 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-cmn 19702 df-mgp 20040 df-ur 20087 df-ring 20140 df-cring 20141 df-cnfld 21241 |
This theorem is referenced by: evpmsubg 32812 |
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