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Theorem circgrp 26526

Description: The circle group 𝑇 is an Abelian group. (Contributed by Paul Chapman, 25-Mar-2008.) (Revised by Mario Carneiro, 13-May-2014.) (Revised 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
circgrp	⊢ 𝑇 ∈ Abel

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

Step	Hyp	Ref	Expression
1		oveq2 7426	. . . . 5 ⊢ (𝑥 = 𝑦 → (i · 𝑥) = (i · 𝑦))
2	1	fveq2d 6899	. . . 4 ⊢ (𝑥 = 𝑦 → (exp‘(i · 𝑥)) = (exp‘(i · 𝑦)))
3	2	cbvmptv 5262	. . 3 ⊢ (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥))) = (𝑦 ∈ ℝ ↦ (exp‘(i · 𝑦)))
4		circgrp.2	. . . 4 ⊢ 𝑇 = ((mulGrp‘ℂ_fld) ↾_s 𝐶)
5		circgrp.1	. . . . . . . 8 ⊢ 𝐶 = (^◡abs “ {1})
6	3, 5	efifo 26521	. . . . . . 7 ⊢ (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥))):ℝ–onto→𝐶
7		forn 6812	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥))):ℝ–onto→𝐶 → ran (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥))) = 𝐶)
8	6, 7	ax-mp 5	. . . . . 6 ⊢ ran (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥))) = 𝐶
9	8	eqcomi 2734	. . . . 5 ⊢ 𝐶 = ran (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥)))
10	9	oveq2i 7429	. . . 4 ⊢ ((mulGrp‘ℂ_fld) ↾_s 𝐶) = ((mulGrp‘ℂ_fld) ↾_s ran (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥))))
11	4, 10	eqtri 2753	. . 3 ⊢ 𝑇 = ((mulGrp‘ℂ_fld) ↾_s ran (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥))))
12		ax-icn 11198	. . . 4 ⊢ i ∈ ℂ
13	12	a1i 11	. . 3 ⊢ (⊤ → i ∈ ℂ)
14		resubdrg 21554	. . . . . 6 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) ∧ ℝ_fld ∈ DivRing)
15	14	simpli 482	. . . . 5 ⊢ ℝ ∈ (SubRing‘ℂ_fld)
16		subrgsubg 20525	. . . . 5 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) → ℝ ∈ (SubGrp‘ℂ_fld))
17	15, 16	ax-mp 5	. . . 4 ⊢ ℝ ∈ (SubGrp‘ℂ_fld)
18	17	a1i 11	. . 3 ⊢ (⊤ → ℝ ∈ (SubGrp‘ℂ_fld))
19	3, 11, 13, 18	efabl 26524	. 2 ⊢ (⊤ → 𝑇 ∈ Abel)
20	19	mptru 1540	1 ⊢ 𝑇 ∈ Abel

Colors of variables: wff setvar class

Syntax hints: = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 {csn 4630 ↦ cmpt 5232 ^◡ccnv 5677 ran crn 5679 “ cima 5681 –onto→wfo 6546 ‘cfv 6548 (class class class)co 7418 ℂcc 11137 ℝcr 11138 1c1 11140 ici 11141 · cmul 11144 abscabs 15215 expce 16039 ↾_s cress 17210 SubGrpcsubg 19081 Abelcabl 19745 mulGrpcmgp 20083 SubRingcsubrg 20515 DivRingcdr 20633 ℂ_fldccnfld 21293 ℝ_fldcrefld 21550

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 7740 ax-inf2 9665 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 ax-addf 11218 ax-mulf 11219

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-int 4951 df-iun 4999 df-iin 5000 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-se 5634 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 6306 df-ord 6373 df-on 6374 df-lim 6375 df-suc 6376 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7374 df-ov 7421 df-oprab 7422 df-mpo 7423 df-of 7684 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8165 df-tpos 8231 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-2o 8487 df-er 8724 df-map 8846 df-pm 8847 df-ixp 8916 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-fsupp 9387 df-fi 9435 df-sup 9466 df-inf 9467 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-ioo 13361 df-ioc 13362 df-ico 13363 df-icc 13364 df-fz 13518 df-fzo 13661 df-fl 13791 df-mod 13869 df-seq 14001 df-exp 14061 df-fac 14267 df-bc 14296 df-hash 14324 df-shft 15048 df-cj 15080 df-re 15081 df-im 15082 df-sqrt 15216 df-abs 15217 df-limsup 15449 df-clim 15466 df-rlim 15467 df-sum 15667 df-ef 16045 df-sin 16047 df-cos 16048 df-pi 16050 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17182 df-ress 17211 df-plusg 17247 df-mulr 17248 df-starv 17249 df-sca 17250 df-vsca 17251 df-ip 17252 df-tset 17253 df-ple 17254 df-ds 17256 df-unif 17257 df-hom 17258 df-cco 17259 df-rest 17405 df-topn 17406 df-0g 17424 df-gsum 17425 df-topgen 17426 df-pt 17427 df-prds 17430 df-xrs 17485 df-qtop 17490 df-imas 17491 df-xps 17493 df-mre 17567 df-mrc 17568 df-acs 17570 df-mgm 18601 df-sgrp 18680 df-mnd 18696 df-submnd 18742 df-grp 18899 df-minusg 18900 df-mulg 19030 df-subg 19084 df-cntz 19277 df-cmn 19746 df-abl 19747 df-mgp 20084 df-rng 20102 df-ur 20131 df-ring 20184 df-cring 20185 df-oppr 20282 df-dvdsr 20305 df-unit 20306 df-invr 20336 df-dvr 20349 df-subrng 20492 df-subrg 20517 df-drng 20635 df-psmet 21285 df-xmet 21286 df-met 21287 df-bl 21288 df-mopn 21289 df-fbas 21290 df-fg 21291 df-cnfld 21294 df-refld 21551 df-top 22836 df-topon 22853 df-topsp 22875 df-bases 22889 df-cld 22963 df-ntr 22964 df-cls 22965 df-nei 23042 df-lp 23080 df-perf 23081 df-cn 23171 df-cnp 23172 df-haus 23259 df-tx 23506 df-hmeo 23699 df-fil 23790 df-fm 23882 df-flim 23883 df-flf 23884 df-xms 24266 df-ms 24267 df-tms 24268 df-cncf 24838 df-limc 25835 df-dv 25836

This theorem is referenced by: (None)

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