reefgim - Metamath Proof Explorer

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Theorem reefgim 24204

Description: The exponential function is a group isomorphism from the group of reals under addition to the group of positive reals under multiplication. (Contributed by Mario Carneiro, 21-Jun-2015.) (Revised by Thierry Arnoux, 30-Jun-2019.)

**Hypothesis**
Ref	Expression
reefgim.1	⊢ 𝑃 = ((mulGrp‘ℂ_fld) ↾_s ℝ⁺)

**Assertion**
Ref	Expression
reefgim	⊢ (exp ↾ ℝ) ∈ (ℝ_fld GrpIso 𝑃)

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

Step	Hyp	Ref	Expression
1		rebase 19952	. . . 4 ⊢ ℝ = (Base‘ℝ_fld)
2		eqid 2622	. . . . . 6 ⊢ ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0})) = ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0}))
3	2	rpmsubg 19810	. . . . 5 ⊢ ℝ⁺ ∈ (SubGrp‘((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0})))
4		reefgim.1	. . . . . . 7 ⊢ 𝑃 = ((mulGrp‘ℂ_fld) ↾_s ℝ⁺)
5		cnex 10017	. . . . . . . . 9 ⊢ ℂ ∈ V
6		difexg 4808	. . . . . . . . 9 ⊢ (ℂ ∈ V → (ℂ ∖ {0}) ∈ V)
7	5, 6	ax-mp 5	. . . . . . . 8 ⊢ (ℂ ∖ {0}) ∈ V
8		rpcn 11841	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
9		rpne0 11848	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
10		eldifsn 4317	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
11	8, 9, 10	sylanbrc 698	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ (ℂ ∖ {0}))
12	11	ssriv 3607	. . . . . . . 8 ⊢ ℝ⁺ ⊆ (ℂ ∖ {0})
13		ressabs 15939	. . . . . . . 8 ⊢ (((ℂ ∖ {0}) ∈ V ∧ ℝ⁺ ⊆ (ℂ ∖ {0})) → (((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0})) ↾_s ℝ⁺) = ((mulGrp‘ℂ_fld) ↾_s ℝ⁺))
14	7, 12, 13	mp2an 708	. . . . . . 7 ⊢ (((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0})) ↾_s ℝ⁺) = ((mulGrp‘ℂ_fld) ↾_s ℝ⁺)
15	4, 14	eqtr4i 2647	. . . . . 6 ⊢ 𝑃 = (((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0})) ↾_s ℝ⁺)
16	15	subgbas 17598	. . . . 5 ⊢ (ℝ⁺ ∈ (SubGrp‘((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0}))) → ℝ⁺ = (Base‘𝑃))
17	3, 16	ax-mp 5	. . . 4 ⊢ ℝ⁺ = (Base‘𝑃)
18		replusg 19956	. . . 4 ⊢ + = (+_g‘ℝ_fld)
19		eqid 2622	. . . . . . 7 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
20		cnfldmul 19752	. . . . . . 7 ⊢ · = (._r‘ℂ_fld)
21	19, 20	mgpplusg 18493	. . . . . 6 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
22	4, 21	ressplusg 15993	. . . . 5 ⊢ (ℝ⁺ ∈ (SubGrp‘((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0}))) → · = (+_g‘𝑃))
23	3, 22	ax-mp 5	. . . 4 ⊢ · = (+_g‘𝑃)
24		resubdrg 19954	. . . . . . 7 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) ∧ ℝ_fld ∈ DivRing)
25	24	simpli 474	. . . . . 6 ⊢ ℝ ∈ (SubRing‘ℂ_fld)
26		df-refld 19951	. . . . . . 7 ⊢ ℝ_fld = (ℂ_fld ↾_s ℝ)
27	26	subrgring 18783	. . . . . 6 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) → ℝ_fld ∈ Ring)
28	25, 27	ax-mp 5	. . . . 5 ⊢ ℝ_fld ∈ Ring
29		ringgrp 18552	. . . . 5 ⊢ (ℝ_fld ∈ Ring → ℝ_fld ∈ Grp)
30	28, 29	mp1i 13	. . . 4 ⊢ (⊤ → ℝ_fld ∈ Grp)
31	15	subggrp 17597	. . . . 5 ⊢ (ℝ⁺ ∈ (SubGrp‘((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0}))) → 𝑃 ∈ Grp)
32	3, 31	mp1i 13	. . . 4 ⊢ (⊤ → 𝑃 ∈ Grp)
33		reeff1o 24201	. . . . 5 ⊢ (exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺
34		f1of 6137	. . . . 5 ⊢ ((exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺ → (exp ↾ ℝ):ℝ⟶ℝ⁺)
35	33, 34	mp1i 13	. . . 4 ⊢ (⊤ → (exp ↾ ℝ):ℝ⟶ℝ⁺)
36		recn 10026	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
37		recn 10026	. . . . . . 7 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
38		efadd 14824	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (exp‘(𝑥 + 𝑦)) = ((exp‘𝑥) · (exp‘𝑦)))
39	36, 37, 38	syl2an 494	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (exp‘(𝑥 + 𝑦)) = ((exp‘𝑥) · (exp‘𝑦)))
40		readdcl 10019	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + 𝑦) ∈ ℝ)
41		fvres 6207	. . . . . . 7 ⊢ ((𝑥 + 𝑦) ∈ ℝ → ((exp ↾ ℝ)‘(𝑥 + 𝑦)) = (exp‘(𝑥 + 𝑦)))
42	40, 41	syl 17	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((exp ↾ ℝ)‘(𝑥 + 𝑦)) = (exp‘(𝑥 + 𝑦)))
43		fvres 6207	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → ((exp ↾ ℝ)‘𝑥) = (exp‘𝑥))
44		fvres 6207	. . . . . . 7 ⊢ (𝑦 ∈ ℝ → ((exp ↾ ℝ)‘𝑦) = (exp‘𝑦))
45	43, 44	oveqan12d 6669	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (((exp ↾ ℝ)‘𝑥) · ((exp ↾ ℝ)‘𝑦)) = ((exp‘𝑥) · (exp‘𝑦)))
46	39, 42, 45	3eqtr4d 2666	. . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((exp ↾ ℝ)‘(𝑥 + 𝑦)) = (((exp ↾ ℝ)‘𝑥) · ((exp ↾ ℝ)‘𝑦)))
47	46	adantl 482	. . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → ((exp ↾ ℝ)‘(𝑥 + 𝑦)) = (((exp ↾ ℝ)‘𝑥) · ((exp ↾ ℝ)‘𝑦)))
48	1, 17, 18, 23, 30, 32, 35, 47	isghmd 17669	. . 3 ⊢ (⊤ → (exp ↾ ℝ) ∈ (ℝ_fld GrpHom 𝑃))
49	48	trud 1493	. 2 ⊢ (exp ↾ ℝ) ∈ (ℝ_fld GrpHom 𝑃)
50	1, 17	isgim 17704	. 2 ⊢ ((exp ↾ ℝ) ∈ (ℝ_fld GrpIso 𝑃) ↔ ((exp ↾ ℝ) ∈ (ℝ_fld GrpHom 𝑃) ∧ (exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺))
51	49, 33, 50	mpbir2an 955	1 ⊢ (exp ↾ ℝ) ∈ (ℝ_fld GrpIso 𝑃)

Colors of variables: wff setvar class

Syntax hints: ∧ wa 384 = wceq 1483 ⊤wtru 1484 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ∖ cdif 3571 ⊆ wss 3574 {csn 4177 ↾ cres 5116 ⟶wf 5884 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 + caddc 9939 · cmul 9941 ℝ⁺crp 11832 expce 14792 Basecbs 15857 ↾_s cress 15858 +_gcplusg 15941 Grpcgrp 17422 SubGrpcsubg 17588 GrpHom cghm 17657 GrpIso cgim 17699 mulGrpcmgp 18489 Ringcrg 18547 DivRingcdr 18747 SubRingcsubrg 18776 ℂ_fldccnfld 19746 ℝ_fldcrefld 19950

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ef 14798 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-mulg 17541 df-subg 17591 df-ghm 17658 df-gim 17701 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-drng 18749 df-subrg 18778 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-refld 19951 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631

This theorem is referenced by: reloggim 24345

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