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Theorem circsubm 24520

Description: The circle group 𝑇 is a submonoid of the multiplicative group of ℂ_fld. (Contributed by Thierry Arnoux, 26-Jan-2020.)

**Hypotheses**
Ref	Expression
circgrp.1	⊢ 𝐶 = (^◡abs “ {1})
circgrp.2	⊢ 𝑇 = ((mulGrp‘ℂ_fld) ↾_s 𝐶)

**Assertion**
Ref	Expression
circsubm	⊢ 𝐶 ∈ (SubMnd‘(mulGrp‘ℂ_fld))

Proof of Theorem circsubm Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6801	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (i · 𝑥) = (i · 𝑦))
2	1	fveq2d 6336	. . . . . 6 ⊢ (𝑥 = 𝑦 → (exp‘(i · 𝑥)) = (exp‘(i · 𝑦)))
3	2	cbvmptv 4884	. . . . 5 ⊢ (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥))) = (𝑦 ∈ ℝ ↦ (exp‘(i · 𝑦)))
4		circgrp.1	. . . . 5 ⊢ 𝐶 = (^◡abs “ {1})
5	3, 4	efifo 24514	. . . 4 ⊢ (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥))):ℝ–onto→𝐶
6		forn 6259	. . . 4 ⊢ ((𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥))):ℝ–onto→𝐶 → ran (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥))) = 𝐶)
7	5, 6	ax-mp 5	. . 3 ⊢ ran (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥))) = 𝐶
8	7	eqcomi 2780	. 2 ⊢ 𝐶 = ran (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥)))
9	8	oveq2i 6804	. . . 4 ⊢ ((mulGrp‘ℂ_fld) ↾_s 𝐶) = ((mulGrp‘ℂ_fld) ↾_s ran (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥))))
10		ax-icn 10197	. . . . 5 ⊢ i ∈ ℂ
11	10	a1i 11	. . . 4 ⊢ (⊤ → i ∈ ℂ)
12		resubdrg 20171	. . . . . . 7 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) ∧ ℝ_fld ∈ DivRing)
13	12	simpli 470	. . . . . 6 ⊢ ℝ ∈ (SubRing‘ℂ_fld)
14		subrgsubg 18996	. . . . . 6 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) → ℝ ∈ (SubGrp‘ℂ_fld))
15	13, 14	ax-mp 5	. . . . 5 ⊢ ℝ ∈ (SubGrp‘ℂ_fld)
16	15	a1i 11	. . . 4 ⊢ (⊤ → ℝ ∈ (SubGrp‘ℂ_fld))
17	3, 9, 11, 16	efsubm 24518	. . 3 ⊢ (⊤ → ran (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥))) ∈ (SubMnd‘(mulGrp‘ℂ_fld)))
18	17	trud 1641	. 2 ⊢ ran (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥))) ∈ (SubMnd‘(mulGrp‘ℂ_fld))
19	8, 18	eqeltri 2846	1 ⊢ 𝐶 ∈ (SubMnd‘(mulGrp‘ℂ_fld))

Colors of variables: wff setvar class

Syntax hints: = wceq 1631 ⊤wtru 1632 ∈ wcel 2145 {csn 4316 ↦ cmpt 4863 ^◡ccnv 5248 ran crn 5250 “ cima 5252 –onto→wfo 6029 ‘cfv 6031 (class class class)co 6793 ℂcc 10136 ℝcr 10137 1c1 10139 ici 10140 · cmul 10143 abscabs 14182 expce 14998 ↾_s cress 16065 SubMndcsubmnd 17542 SubGrpcsubg 17796 mulGrpcmgp 18697 DivRingcdr 18957 SubRingcsubrg 18986 ℂ_fldccnfld 19961 ℝ_fldcrefld 20167

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-inf2 8702 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 ax-pre-sup 10216 ax-addf 10217 ax-mulf 10218

This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-iin 4657 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-isom 6040 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-of 7044 df-om 7213 df-1st 7315 df-2nd 7316 df-supp 7447 df-tpos 7504 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-2o 7714 df-oadd 7717 df-er 7896 df-map 8011 df-pm 8012 df-ixp 8063 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-fsupp 8432 df-fi 8473 df-sup 8504 df-inf 8505 df-oi 8571 df-card 8965 df-cda 9192 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-div 10887 df-nn 11223 df-2 11281 df-3 11282 df-4 11283 df-5 11284 df-6 11285 df-7 11286 df-8 11287 df-9 11288 df-n0 11495 df-z 11580 df-dec 11696 df-uz 11889 df-q 11992 df-rp 12036 df-xneg 12151 df-xadd 12152 df-xmul 12153 df-ioo 12384 df-ioc 12385 df-ico 12386 df-icc 12387 df-fz 12534 df-fzo 12674 df-fl 12801 df-mod 12877 df-seq 13009 df-exp 13068 df-fac 13265 df-bc 13294 df-hash 13322 df-shft 14015 df-cj 14047 df-re 14048 df-im 14049 df-sqrt 14183 df-abs 14184 df-limsup 14410 df-clim 14427 df-rlim 14428 df-sum 14625 df-ef 15004 df-sin 15006 df-cos 15007 df-pi 15009 df-struct 16066 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-ress 16072 df-plusg 16162 df-mulr 16163 df-starv 16164 df-sca 16165 df-vsca 16166 df-ip 16167 df-tset 16168 df-ple 16169 df-ds 16172 df-unif 16173 df-hom 16174 df-cco 16175 df-rest 16291 df-topn 16292 df-0g 16310 df-gsum 16311 df-topgen 16312 df-pt 16313 df-prds 16316 df-xrs 16370 df-qtop 16375 df-imas 16376 df-xps 16378 df-mre 16454 df-mrc 16455 df-acs 16457 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-submnd 17544 df-grp 17633 df-minusg 17634 df-mulg 17749 df-subg 17799 df-cntz 17957 df-cmn 18402 df-abl 18403 df-mgp 18698 df-ur 18710 df-ring 18757 df-cring 18758 df-oppr 18831 df-dvdsr 18849 df-unit 18850 df-invr 18880 df-dvr 18891 df-drng 18959 df-subrg 18988 df-psmet 19953 df-xmet 19954 df-met 19955 df-bl 19956 df-mopn 19957 df-fbas 19958 df-fg 19959 df-cnfld 19962 df-refld 20168 df-top 20919 df-topon 20936 df-topsp 20958 df-bases 20971 df-cld 21044 df-ntr 21045 df-cls 21046 df-nei 21123 df-lp 21161 df-perf 21162 df-cn 21252 df-cnp 21253 df-haus 21340 df-tx 21586 df-hmeo 21779 df-fil 21870 df-fm 21962 df-flim 21963 df-flf 21964 df-xms 22345 df-ms 22346 df-tms 22347 df-cncf 22901 df-limc 23850 df-dv 23851

This theorem is referenced by: (None)

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