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Theorem circsubm 25737

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 7303	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (i · 𝑥) = (i · 𝑦))
2	1	fveq2d 6796	. . . . . 6 ⊢ (𝑥 = 𝑦 → (exp‘(i · 𝑥)) = (exp‘(i · 𝑦)))
3	2	cbvmptv 5190	. . . . 5 ⊢ (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥))) = (𝑦 ∈ ℝ ↦ (exp‘(i · 𝑦)))
4		circgrp.1	. . . . 5 ⊢ 𝐶 = (^◡abs “ {1})
5	3, 4	efifo 25731	. . . 4 ⊢ (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥))):ℝ–onto→𝐶
6		forn 6709	. . . 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 2742	. 2 ⊢ 𝐶 = ran (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥)))
9	8	oveq2i 7306	. . . 4 ⊢ ((mulGrp‘ℂ_fld) ↾_s 𝐶) = ((mulGrp‘ℂ_fld) ↾_s ran (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥))))
10		ax-icn 10958	. . . . 5 ⊢ i ∈ ℂ
11	10	a1i 11	. . . 4 ⊢ (⊤ → i ∈ ℂ)
12		resubdrg 20841	. . . . . . 7 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) ∧ ℝ_fld ∈ DivRing)
13	12	simpli 483	. . . . . 6 ⊢ ℝ ∈ (SubRing‘ℂ_fld)
14		subrgsubg 20058	. . . . . 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 25735	. . 3 ⊢ (⊤ → ran (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥))) ∈ (SubMnd‘(mulGrp‘ℂ_fld)))
18	17	mptru 1544	. 2 ⊢ ran (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥))) ∈ (SubMnd‘(mulGrp‘ℂ_fld))
19	8, 18	eqeltri 2830	1 ⊢ 𝐶 ∈ (SubMnd‘(mulGrp‘ℂ_fld))

Colors of variables: wff setvar class

Syntax hints: = wceq 1537 ⊤wtru 1538 ∈ wcel 2101 {csn 4564 ↦ cmpt 5160 ^◡ccnv 5590 ran crn 5592 “ cima 5594 –onto→wfo 6445 ‘cfv 6447 (class class class)co 7295 ℂcc 10897 ℝcr 10898 1c1 10900 ici 10901 · cmul 10904 abscabs 14973 expce 15799 ↾_s cress 16969 SubMndcsubmnd 18457 SubGrpcsubg 18777 mulGrpcmgp 19748 DivRingcdr 20019 SubRingcsubrg 20048 ℂ_fldccnfld 20625 ℝ_fldcrefld 20837

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-rep 5212 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-inf2 9427 ax-cnex 10955 ax-resscn 10956 ax-1cn 10957 ax-icn 10958 ax-addcl 10959 ax-addrcl 10960 ax-mulcl 10961 ax-mulrcl 10962 ax-mulcom 10963 ax-addass 10964 ax-mulass 10965 ax-distr 10966 ax-i2m1 10967 ax-1ne0 10968 ax-1rid 10969 ax-rnegex 10970 ax-rrecex 10971 ax-cnre 10972 ax-pre-lttri 10973 ax-pre-lttrn 10974 ax-pre-ltadd 10975 ax-pre-mulgt0 10976 ax-pre-sup 10977 ax-addf 10978 ax-mulf 10979

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3222 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4842 df-int 4883 df-iun 4929 df-iin 4930 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-se 5547 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-isom 6456 df-riota 7252 df-ov 7298 df-oprab 7299 df-mpo 7300 df-of 7553 df-om 7733 df-1st 7851 df-2nd 7852 df-supp 7998 df-tpos 8062 df-frecs 8117 df-wrecs 8148 df-recs 8222 df-rdg 8261 df-1o 8317 df-2o 8318 df-er 8518 df-map 8637 df-pm 8638 df-ixp 8706 df-en 8754 df-dom 8755 df-sdom 8756 df-fin 8757 df-fsupp 9157 df-fi 9198 df-sup 9229 df-inf 9230 df-oi 9297 df-card 9725 df-pnf 11039 df-mnf 11040 df-xr 11041 df-ltxr 11042 df-le 11043 df-sub 11235 df-neg 11236 df-div 11661 df-nn 12002 df-2 12064 df-3 12065 df-4 12066 df-5 12067 df-6 12068 df-7 12069 df-8 12070 df-9 12071 df-n0 12262 df-z 12348 df-dec 12466 df-uz 12611 df-q 12717 df-rp 12759 df-xneg 12876 df-xadd 12877 df-xmul 12878 df-ioo 13111 df-ioc 13112 df-ico 13113 df-icc 13114 df-fz 13268 df-fzo 13411 df-fl 13540 df-mod 13618 df-seq 13750 df-exp 13811 df-fac 14016 df-bc 14045 df-hash 14073 df-shft 14806 df-cj 14838 df-re 14839 df-im 14840 df-sqrt 14974 df-abs 14975 df-limsup 15208 df-clim 15225 df-rlim 15226 df-sum 15426 df-ef 15805 df-sin 15807 df-cos 15808 df-pi 15810 df-struct 16876 df-sets 16893 df-slot 16911 df-ndx 16923 df-base 16941 df-ress 16970 df-plusg 17003 df-mulr 17004 df-starv 17005 df-sca 17006 df-vsca 17007 df-ip 17008 df-tset 17009 df-ple 17010 df-ds 17012 df-unif 17013 df-hom 17014 df-cco 17015 df-rest 17161 df-topn 17162 df-0g 17180 df-gsum 17181 df-topgen 17182 df-pt 17183 df-prds 17186 df-xrs 17241 df-qtop 17246 df-imas 17247 df-xps 17249 df-mre 17323 df-mrc 17324 df-acs 17326 df-mgm 18354 df-sgrp 18403 df-mnd 18414 df-submnd 18459 df-grp 18608 df-minusg 18609 df-mulg 18729 df-subg 18780 df-cntz 18951 df-cmn 19416 df-abl 19417 df-mgp 19749 df-ur 19766 df-ring 19813 df-cring 19814 df-oppr 19890 df-dvdsr 19911 df-unit 19912 df-invr 19942 df-dvr 19953 df-drng 20021 df-subrg 20050 df-psmet 20617 df-xmet 20618 df-met 20619 df-bl 20620 df-mopn 20621 df-fbas 20622 df-fg 20623 df-cnfld 20626 df-refld 20838 df-top 22071 df-topon 22088 df-topsp 22110 df-bases 22124 df-cld 22198 df-ntr 22199 df-cls 22200 df-nei 22277 df-lp 22315 df-perf 22316 df-cn 22406 df-cnp 22407 df-haus 22494 df-tx 22741 df-hmeo 22934 df-fil 23025 df-fm 23117 df-flim 23118 df-flf 23119 df-xms 23501 df-ms 23502 df-tms 23503 df-cncf 24069 df-limc 25058 df-dv 25059

This theorem is referenced by: (None)

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