reefgim - Metamath Proof Explorer

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Theorem reefgim 23953

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 19719	. . . 4 ⊢ ℝ = (Base‘ℝ_fld)
2		eqid 2609	. . . . . 6 ⊢ ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0})) = ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0}))
3	2	rpmsubg 19578	. . . . 5 ⊢ ℝ⁺ ∈ (SubGrp‘((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0})))
4		reefgim.1	. . . . . . 7 ⊢ 𝑃 = ((mulGrp‘ℂ_fld) ↾_s ℝ⁺)
5		cnex 9874	. . . . . . . . 9 ⊢ ℂ ∈ V
6		difexg 4730	. . . . . . . . 9 ⊢ (ℂ ∈ V → (ℂ ∖ {0}) ∈ V)
7	5, 6	ax-mp 5	. . . . . . . 8 ⊢ (ℂ ∖ {0}) ∈ V
8		rpcn 11676	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
9		rpne0 11683	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
10		eldifsn 4259	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
11	8, 9, 10	sylanbrc 694	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ (ℂ ∖ {0}))
12	11	ssriv 3571	. . . . . . . 8 ⊢ ℝ⁺ ⊆ (ℂ ∖ {0})
13		ressabs 15715	. . . . . . . 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 703	. . . . . . 7 ⊢ (((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0})) ↾_s ℝ⁺) = ((mulGrp‘ℂ_fld) ↾_s ℝ⁺)
15	4, 14	eqtr4i 2634	. . . . . 6 ⊢ 𝑃 = (((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0})) ↾_s ℝ⁺)
16	15	subgbas 17370	. . . . 5 ⊢ (ℝ⁺ ∈ (SubGrp‘((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0}))) → ℝ⁺ = (Base‘𝑃))
17	3, 16	ax-mp 5	. . . 4 ⊢ ℝ⁺ = (Base‘𝑃)
18		replusg 19723	. . . 4 ⊢ + = (+_g‘ℝ_fld)
19		eqid 2609	. . . . . . 7 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
20		cnfldmul 19522	. . . . . . 7 ⊢ · = (._r‘ℂ_fld)
21	19, 20	mgpplusg 18265	. . . . . 6 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
22	4, 21	ressplusg 15767	. . . . 5 ⊢ (ℝ⁺ ∈ (SubGrp‘((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0}))) → · = (+_g‘𝑃))
23	3, 22	ax-mp 5	. . . 4 ⊢ · = (+_g‘𝑃)
24		resubdrg 19721	. . . . . . 7 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) ∧ ℝ_fld ∈ DivRing)
25	24	simpli 472	. . . . . 6 ⊢ ℝ ∈ (SubRing‘ℂ_fld)
26		df-refld 19718	. . . . . . 7 ⊢ ℝ_fld = (ℂ_fld ↾_s ℝ)
27	26	subrgring 18555	. . . . . 6 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) → ℝ_fld ∈ Ring)
28	25, 27	ax-mp 5	. . . . 5 ⊢ ℝ_fld ∈ Ring
29		ringgrp 18324	. . . . 5 ⊢ (ℝ_fld ∈ Ring → ℝ_fld ∈ Grp)
30	28, 29	mp1i 13	. . . 4 ⊢ (⊤ → ℝ_fld ∈ Grp)
31	15	subggrp 17369	. . . . 5 ⊢ (ℝ⁺ ∈ (SubGrp‘((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0}))) → 𝑃 ∈ Grp)
32	3, 31	mp1i 13	. . . 4 ⊢ (⊤ → 𝑃 ∈ Grp)
33		reeff1o 23950	. . . . 5 ⊢ (exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺
34		f1of 6035	. . . . 5 ⊢ ((exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺ → (exp ↾ ℝ):ℝ⟶ℝ⁺)
35	33, 34	mp1i 13	. . . 4 ⊢ (⊤ → (exp ↾ ℝ):ℝ⟶ℝ⁺)
36		recn 9883	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
37		recn 9883	. . . . . . 7 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
38		efadd 14612	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (exp‘(𝑥 + 𝑦)) = ((exp‘𝑥) · (exp‘𝑦)))
39	36, 37, 38	syl2an 492	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (exp‘(𝑥 + 𝑦)) = ((exp‘𝑥) · (exp‘𝑦)))
40		readdcl 9876	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + 𝑦) ∈ ℝ)
41		fvres 6102	. . . . . . 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 6102	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → ((exp ↾ ℝ)‘𝑥) = (exp‘𝑥))
44		fvres 6102	. . . . . . 7 ⊢ (𝑦 ∈ ℝ → ((exp ↾ ℝ)‘𝑦) = (exp‘𝑦))
45	43, 44	oveqan12d 6546	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (((exp ↾ ℝ)‘𝑥) · ((exp ↾ ℝ)‘𝑦)) = ((exp‘𝑥) · (exp‘𝑦)))
46	39, 42, 45	3eqtr4d 2653	. . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((exp ↾ ℝ)‘(𝑥 + 𝑦)) = (((exp ↾ ℝ)‘𝑥) · ((exp ↾ ℝ)‘𝑦)))
47	46	adantl 480	. . . 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 17441	. . 3 ⊢ (⊤ → (exp ↾ ℝ) ∈ (ℝ_fld GrpHom 𝑃))
49	48	trud 1483	. 2 ⊢ (exp ↾ ℝ) ∈ (ℝ_fld GrpHom 𝑃)
50	1, 17	isgim 17476	. 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 956	1 ⊢ (exp ↾ ℝ) ∈ (ℝ_fld GrpIso 𝑃)

Colors of variables: wff setvar class

Syntax hints: ∧ wa 382 = wceq 1474 ⊤wtru 1475 ∈ wcel 1976 ≠ wne 2779 Vcvv 3172 ∖ cdif 3536 ⊆ wss 3539 {csn 4124 ↾ cres 5030 ⟶wf 5786 –1-1-onto→wf1o 5789 ‘cfv 5790 (class class class)co 6527 ℂcc 9791 ℝcr 9792 0cc0 9793 + caddc 9796 · cmul 9798 ℝ⁺crp 11667 expce 14580 Basecbs 15644 ↾_s cress 15645 +_gcplusg 15717 Grpcgrp 17194 SubGrpcsubg 17360 GrpHom cghm 17429 GrpIso cgim 17471 mulGrpcmgp 18261 Ringcrg 18319 DivRingcdr 18519 SubRingcsubrg 18548 ℂ_fldccnfld 19516 ℝ_fldcrefld 19717

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1712 ax-4 1727 ax-5 1826 ax-6 1874 ax-7 1921 ax-8 1978 ax-9 1985 ax-10 2005 ax-11 2020 ax-12 2033 ax-13 2233 ax-ext 2589 ax-rep 4693 ax-sep 4703 ax-nul 4712 ax-pow 4764 ax-pr 4828 ax-un 6825 ax-inf2 8399 ax-cnex 9849 ax-resscn 9850 ax-1cn 9851 ax-icn 9852 ax-addcl 9853 ax-addrcl 9854 ax-mulcl 9855 ax-mulrcl 9856 ax-mulcom 9857 ax-addass 9858 ax-mulass 9859 ax-distr 9860 ax-i2m1 9861 ax-1ne0 9862 ax-1rid 9863 ax-rnegex 9864 ax-rrecex 9865 ax-cnre 9866 ax-pre-lttri 9867 ax-pre-lttrn 9868 ax-pre-ltadd 9869 ax-pre-mulgt0 9870 ax-pre-sup 9871 ax-addf 9872 ax-mulf 9873

This theorem depends on definitions: df-bi 195 df-or 383 df-an 384 df-3or 1031 df-3an 1032 df-tru 1477 df-fal 1480 df-ex 1695 df-nf 1700 df-sb 1867 df-eu 2461 df-mo 2462 df-clab 2596 df-cleq 2602 df-clel 2605 df-nfc 2739 df-ne 2781 df-nel 2782 df-ral 2900 df-rex 2901 df-reu 2902 df-rmo 2903 df-rab 2904 df-v 3174 df-sbc 3402 df-csb 3499 df-dif 3542 df-un 3544 df-in 3546 df-ss 3553 df-pss 3555 df-nul 3874 df-if 4036 df-pw 4109 df-sn 4125 df-pr 4127 df-tp 4129 df-op 4131 df-uni 4367 df-int 4405 df-iun 4451 df-iin 4452 df-br 4578 df-opab 4638 df-mpt 4639 df-tr 4675 df-eprel 4939 df-id 4943 df-po 4949 df-so 4950 df-fr 4987 df-se 4988 df-we 4989 df-xp 5034 df-rel 5035 df-cnv 5036 df-co 5037 df-dm 5038 df-rn 5039 df-res 5040 df-ima 5041 df-pred 5583 df-ord 5629 df-on 5630 df-lim 5631 df-suc 5632 df-iota 5754 df-fun 5792 df-fn 5793 df-f 5794 df-f1 5795 df-fo 5796 df-f1o 5797 df-fv 5798 df-isom 5799 df-riota 6489 df-ov 6530 df-oprab 6531 df-mpt2 6532 df-of 6773 df-om 6936 df-1st 7037 df-2nd 7038 df-supp 7161 df-tpos 7217 df-wrecs 7272 df-recs 7333 df-rdg 7371 df-1o 7425 df-2o 7426 df-oadd 7429 df-er 7607 df-map 7724 df-pm 7725 df-ixp 7773 df-en 7820 df-dom 7821 df-sdom 7822 df-fin 7823 df-fsupp 8137 df-fi 8178 df-sup 8209 df-inf 8210 df-oi 8276 df-card 8626 df-cda 8851 df-pnf 9933 df-mnf 9934 df-xr 9935 df-ltxr 9936 df-le 9937 df-sub 10120 df-neg 10121 df-div 10537 df-nn 10871 df-2 10929 df-3 10930 df-4 10931 df-5 10932 df-6 10933 df-7 10934 df-8 10935 df-9 10936 df-n0 11143 df-z 11214 df-dec 11329 df-uz 11523 df-q 11624 df-rp 11668 df-xneg 11781 df-xadd 11782 df-xmul 11783 df-ioo 12009 df-ico 12011 df-icc 12012 df-fz 12156 df-fzo 12293 df-fl 12413 df-seq 12622 df-exp 12681 df-fac 12881 df-bc 12910 df-hash 12938 df-shft 13604 df-cj 13636 df-re 13637 df-im 13638 df-sqrt 13772 df-abs 13773 df-limsup 13999 df-clim 14016 df-rlim 14017 df-sum 14214 df-ef 14586 df-struct 15646 df-ndx 15647 df-slot 15648 df-base 15649 df-sets 15650 df-ress 15651 df-plusg 15730 df-mulr 15731 df-starv 15732 df-sca 15733 df-vsca 15734 df-ip 15735 df-tset 15736 df-ple 15737 df-ds 15740 df-unif 15741 df-hom 15742 df-cco 15743 df-rest 15855 df-topn 15856 df-0g 15874 df-gsum 15875 df-topgen 15876 df-pt 15877 df-prds 15880 df-xrs 15934 df-qtop 15939 df-imas 15940 df-xps 15942 df-mre 16018 df-mrc 16019 df-acs 16021 df-mgm 17014 df-sgrp 17056 df-mnd 17067 df-submnd 17108 df-grp 17197 df-minusg 17198 df-mulg 17313 df-subg 17363 df-ghm 17430 df-gim 17473 df-cntz 17522 df-cmn 17967 df-abl 17968 df-mgp 18262 df-ur 18274 df-ring 18321 df-cring 18322 df-oppr 18395 df-dvdsr 18413 df-unit 18414 df-invr 18444 df-dvr 18455 df-drng 18521 df-subrg 18550 df-psmet 19508 df-xmet 19509 df-met 19510 df-bl 19511 df-mopn 19512 df-fbas 19513 df-fg 19514 df-cnfld 19517 df-refld 19718 df-top 20469 df-bases 20470 df-topon 20471 df-topsp 20472 df-cld 20581 df-ntr 20582 df-cls 20583 df-nei 20660 df-lp 20698 df-perf 20699 df-cn 20789 df-cnp 20790 df-haus 20877 df-tx 21123 df-hmeo 21316 df-fil 21408 df-fm 21500 df-flim 21501 df-flf 21502 df-xms 21883 df-ms 21884 df-tms 21885 df-cncf 22437 df-limc 23381 df-dv 23382

This theorem is referenced by: reloggim 24094

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