reefgim - Metamath Proof Explorer

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Theorem reefgim 24249

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 20000	. . . 4 ⊢ ℝ = (Base‘ℝ_fld)
2		eqid 2651	. . . . . 6 ⊢ ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0})) = ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0}))
3	2	rpmsubg 19858	. . . . 5 ⊢ ℝ⁺ ∈ (SubGrp‘((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0})))
4		reefgim.1	. . . . . . 7 ⊢ 𝑃 = ((mulGrp‘ℂ_fld) ↾_s ℝ⁺)
5		cnex 10055	. . . . . . . . 9 ⊢ ℂ ∈ V
6		difexg 4841	. . . . . . . . 9 ⊢ (ℂ ∈ V → (ℂ ∖ {0}) ∈ V)
7	5, 6	ax-mp 5	. . . . . . . 8 ⊢ (ℂ ∖ {0}) ∈ V
8		rpcn 11879	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
9		rpne0 11886	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
10		eldifsn 4350	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
11	8, 9, 10	sylanbrc 699	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ (ℂ ∖ {0}))
12	11	ssriv 3640	. . . . . . . 8 ⊢ ℝ⁺ ⊆ (ℂ ∖ {0})
13		ressabs 15986	. . . . . . . 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 2676	. . . . . 6 ⊢ 𝑃 = (((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0})) ↾_s ℝ⁺)
16	15	subgbas 17645	. . . . 5 ⊢ (ℝ⁺ ∈ (SubGrp‘((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0}))) → ℝ⁺ = (Base‘𝑃))
17	3, 16	ax-mp 5	. . . 4 ⊢ ℝ⁺ = (Base‘𝑃)
18		replusg 20004	. . . 4 ⊢ + = (+_g‘ℝ_fld)
19		eqid 2651	. . . . . . 7 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
20		cnfldmul 19800	. . . . . . 7 ⊢ · = (._r‘ℂ_fld)
21	19, 20	mgpplusg 18539	. . . . . 6 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
22	4, 21	ressplusg 16040	. . . . 5 ⊢ (ℝ⁺ ∈ (SubGrp‘((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0}))) → · = (+_g‘𝑃))
23	3, 22	ax-mp 5	. . . 4 ⊢ · = (+_g‘𝑃)
24		resubdrg 20002	. . . . . . 7 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) ∧ ℝ_fld ∈ DivRing)
25	24	simpli 473	. . . . . 6 ⊢ ℝ ∈ (SubRing‘ℂ_fld)
26		df-refld 19999	. . . . . . 7 ⊢ ℝ_fld = (ℂ_fld ↾_s ℝ)
27	26	subrgring 18831	. . . . . 6 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) → ℝ_fld ∈ Ring)
28	25, 27	ax-mp 5	. . . . 5 ⊢ ℝ_fld ∈ Ring
29		ringgrp 18598	. . . . 5 ⊢ (ℝ_fld ∈ Ring → ℝ_fld ∈ Grp)
30	28, 29	mp1i 13	. . . 4 ⊢ (⊤ → ℝ_fld ∈ Grp)
31	15	subggrp 17644	. . . . 5 ⊢ (ℝ⁺ ∈ (SubGrp‘((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0}))) → 𝑃 ∈ Grp)
32	3, 31	mp1i 13	. . . 4 ⊢ (⊤ → 𝑃 ∈ Grp)
33		reeff1o 24246	. . . . 5 ⊢ (exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺
34		f1of 6175	. . . . 5 ⊢ ((exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺ → (exp ↾ ℝ):ℝ⟶ℝ⁺)
35	33, 34	mp1i 13	. . . 4 ⊢ (⊤ → (exp ↾ ℝ):ℝ⟶ℝ⁺)
36		recn 10064	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
37		recn 10064	. . . . . . 7 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
38		efadd 14868	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (exp‘(𝑥 + 𝑦)) = ((exp‘𝑥) · (exp‘𝑦)))
39	36, 37, 38	syl2an 493	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (exp‘(𝑥 + 𝑦)) = ((exp‘𝑥) · (exp‘𝑦)))
40		readdcl 10057	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + 𝑦) ∈ ℝ)
41		fvres 6245	. . . . . . 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 6245	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → ((exp ↾ ℝ)‘𝑥) = (exp‘𝑥))
44		fvres 6245	. . . . . . 7 ⊢ (𝑦 ∈ ℝ → ((exp ↾ ℝ)‘𝑦) = (exp‘𝑦))
45	43, 44	oveqan12d 6709	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (((exp ↾ ℝ)‘𝑥) · ((exp ↾ ℝ)‘𝑦)) = ((exp‘𝑥) · (exp‘𝑦)))
46	39, 42, 45	3eqtr4d 2695	. . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((exp ↾ ℝ)‘(𝑥 + 𝑦)) = (((exp ↾ ℝ)‘𝑥) · ((exp ↾ ℝ)‘𝑦)))
47	46	adantl 481	. . . 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 17716	. . 3 ⊢ (⊤ → (exp ↾ ℝ) ∈ (ℝ_fld GrpHom 𝑃))
49	48	trud 1533	. 2 ⊢ (exp ↾ ℝ) ∈ (ℝ_fld GrpHom 𝑃)
50	1, 17	isgim 17751	. 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 975	1 ⊢ (exp ↾ ℝ) ∈ (ℝ_fld GrpIso 𝑃)

Colors of variables: wff setvar class

Syntax hints: ∧ wa 383 = wceq 1523 ⊤wtru 1524 ∈ wcel 2030 ≠ wne 2823 Vcvv 3231 ∖ cdif 3604 ⊆ wss 3607 {csn 4210 ↾ cres 5145 ⟶wf 5922 –1-1-onto→wf1o 5925 ‘cfv 5926 (class class class)co 6690 ℂcc 9972 ℝcr 9973 0cc0 9974 + caddc 9977 · cmul 9979 ℝ⁺crp 11870 expce 14836 Basecbs 15904 ↾_s cress 15905 +_gcplusg 15988 Grpcgrp 17469 SubGrpcsubg 17635 GrpHom cghm 17704 GrpIso cgim 17746 mulGrpcmgp 18535 Ringcrg 18593 DivRingcdr 18795 SubRingcsubrg 18824 ℂ_fldccnfld 19794 ℝ_fldcrefld 19998

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 ax-addf 10053 ax-mulf 10054

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-supp 7341 df-tpos 7397 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-er 7787 df-map 7901 df-pm 7902 df-ixp 7951 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-fsupp 8317 df-fi 8358 df-sup 8389 df-inf 8390 df-oi 8456 df-card 8803 df-cda 9028 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-q 11827 df-rp 11871 df-xneg 11984 df-xadd 11985 df-xmul 11986 df-ioo 12217 df-ico 12219 df-icc 12220 df-fz 12365 df-fzo 12505 df-fl 12633 df-seq 12842 df-exp 12901 df-fac 13101 df-bc 13130 df-hash 13158 df-shft 13851 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-limsup 14246 df-clim 14263 df-rlim 14264 df-sum 14461 df-ef 14842 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-starv 16003 df-sca 16004 df-vsca 16005 df-ip 16006 df-tset 16007 df-ple 16008 df-ds 16011 df-unif 16012 df-hom 16013 df-cco 16014 df-rest 16130 df-topn 16131 df-0g 16149 df-gsum 16150 df-topgen 16151 df-pt 16152 df-prds 16155 df-xrs 16209 df-qtop 16214 df-imas 16215 df-xps 16217 df-mre 16293 df-mrc 16294 df-acs 16296 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-submnd 17383 df-grp 17472 df-minusg 17473 df-mulg 17588 df-subg 17638 df-ghm 17705 df-gim 17748 df-cntz 17796 df-cmn 18241 df-abl 18242 df-mgp 18536 df-ur 18548 df-ring 18595 df-cring 18596 df-oppr 18669 df-dvdsr 18687 df-unit 18688 df-invr 18718 df-dvr 18729 df-drng 18797 df-subrg 18826 df-psmet 19786 df-xmet 19787 df-met 19788 df-bl 19789 df-mopn 19790 df-fbas 19791 df-fg 19792 df-cnfld 19795 df-refld 19999 df-top 20747 df-topon 20764 df-topsp 20785 df-bases 20798 df-cld 20871 df-ntr 20872 df-cls 20873 df-nei 20950 df-lp 20988 df-perf 20989 df-cn 21079 df-cnp 21080 df-haus 21167 df-tx 21413 df-hmeo 21606 df-fil 21697 df-fm 21789 df-flim 21790 df-flf 21791 df-xms 22172 df-ms 22173 df-tms 22174 df-cncf 22728 df-limc 23675 df-dv 23676

This theorem is referenced by: reloggim 24390

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