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Theorem reefgim 26428

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 21573	. . . 4 ⊢ ℝ = (Base‘ℝ_fld)
2		eqid 2737	. . . . . 6 ⊢ ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0})) = ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0}))
3	2	rpmsubg 21398	. . . . 5 ⊢ ℝ⁺ ∈ (SubGrp‘((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0})))
4		reefgim.1	. . . . . . 7 ⊢ 𝑃 = ((mulGrp‘ℂ_fld) ↾_s ℝ⁺)
5		cnex 11119	. . . . . . . . 9 ⊢ ℂ ∈ V
6	5	difexi 5277	. . . . . . . 8 ⊢ (ℂ ∖ {0}) ∈ V
7		rpcndif0 12938	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ (ℂ ∖ {0}))
8	7	ssriv 3939	. . . . . . . 8 ⊢ ℝ⁺ ⊆ (ℂ ∖ {0})
9		ressabs 17187	. . . . . . . 8 ⊢ (((ℂ ∖ {0}) ∈ V ∧ ℝ⁺ ⊆ (ℂ ∖ {0})) → (((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0})) ↾_s ℝ⁺) = ((mulGrp‘ℂ_fld) ↾_s ℝ⁺))
10	6, 8, 9	mp2an 693	. . . . . . 7 ⊢ (((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0})) ↾_s ℝ⁺) = ((mulGrp‘ℂ_fld) ↾_s ℝ⁺)
11	4, 10	eqtr4i 2763	. . . . . 6 ⊢ 𝑃 = (((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0})) ↾_s ℝ⁺)
12	11	subgbas 19072	. . . . 5 ⊢ (ℝ⁺ ∈ (SubGrp‘((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0}))) → ℝ⁺ = (Base‘𝑃))
13	3, 12	ax-mp 5	. . . 4 ⊢ ℝ⁺ = (Base‘𝑃)
14		replusg 21577	. . . 4 ⊢ + = (+_g‘ℝ_fld)
15		eqid 2737	. . . . . . 7 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
16		cnfldmul 21329	. . . . . . 7 ⊢ · = (._r‘ℂ_fld)
17	15, 16	mgpplusg 20091	. . . . . 6 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
18	4, 17	ressplusg 17223	. . . . 5 ⊢ (ℝ⁺ ∈ (SubGrp‘((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0}))) → · = (+_g‘𝑃))
19	3, 18	ax-mp 5	. . . 4 ⊢ · = (+_g‘𝑃)
20		resubdrg 21575	. . . . . . 7 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) ∧ ℝ_fld ∈ DivRing)
21	20	simpli 483	. . . . . 6 ⊢ ℝ ∈ (SubRing‘ℂ_fld)
22		df-refld 21572	. . . . . . 7 ⊢ ℝ_fld = (ℂ_fld ↾_s ℝ)
23	22	subrgring 20519	. . . . . 6 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) → ℝ_fld ∈ Ring)
24	21, 23	ax-mp 5	. . . . 5 ⊢ ℝ_fld ∈ Ring
25		ringgrp 20185	. . . . 5 ⊢ (ℝ_fld ∈ Ring → ℝ_fld ∈ Grp)
26	24, 25	mp1i 13	. . . 4 ⊢ (⊤ → ℝ_fld ∈ Grp)
27	11	subggrp 19071	. . . . 5 ⊢ (ℝ⁺ ∈ (SubGrp‘((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0}))) → 𝑃 ∈ Grp)
28	3, 27	mp1i 13	. . . 4 ⊢ (⊤ → 𝑃 ∈ Grp)
29		reeff1o 26425	. . . . 5 ⊢ (exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺
30		f1of 6782	. . . . 5 ⊢ ((exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺ → (exp ↾ ℝ):ℝ⟶ℝ⁺)
31	29, 30	mp1i 13	. . . 4 ⊢ (⊤ → (exp ↾ ℝ):ℝ⟶ℝ⁺)
32		recn 11128	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
33		recn 11128	. . . . . . 7 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
34		efadd 16029	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (exp‘(𝑥 + 𝑦)) = ((exp‘𝑥) · (exp‘𝑦)))
35	32, 33, 34	syl2an 597	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (exp‘(𝑥 + 𝑦)) = ((exp‘𝑥) · (exp‘𝑦)))
36		readdcl 11121	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + 𝑦) ∈ ℝ)
37	36	fvresd 6862	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((exp ↾ ℝ)‘(𝑥 + 𝑦)) = (exp‘(𝑥 + 𝑦)))
38		fvres 6861	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → ((exp ↾ ℝ)‘𝑥) = (exp‘𝑥))
39		fvres 6861	. . . . . . 7 ⊢ (𝑦 ∈ ℝ → ((exp ↾ ℝ)‘𝑦) = (exp‘𝑦))
40	38, 39	oveqan12d 7387	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (((exp ↾ ℝ)‘𝑥) · ((exp ↾ ℝ)‘𝑦)) = ((exp‘𝑥) · (exp‘𝑦)))
41	35, 37, 40	3eqtr4d 2782	. . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((exp ↾ ℝ)‘(𝑥 + 𝑦)) = (((exp ↾ ℝ)‘𝑥) · ((exp ↾ ℝ)‘𝑦)))
42	41	adantl 481	. . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → ((exp ↾ ℝ)‘(𝑥 + 𝑦)) = (((exp ↾ ℝ)‘𝑥) · ((exp ↾ ℝ)‘𝑦)))
43	1, 13, 14, 19, 26, 28, 31, 42	isghmd 19166	. . 3 ⊢ (⊤ → (exp ↾ ℝ) ∈ (ℝ_fld GrpHom 𝑃))
44	43	mptru 1549	. 2 ⊢ (exp ↾ ℝ) ∈ (ℝ_fld GrpHom 𝑃)
45	1, 13	isgim 19203	. 2 ⊢ ((exp ↾ ℝ) ∈ (ℝ_fld GrpIso 𝑃) ↔ ((exp ↾ ℝ) ∈ (ℝ_fld GrpHom 𝑃) ∧ (exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺))
46	44, 29, 45	mpbir2an 712	1 ⊢ (exp ↾ ℝ) ∈ (ℝ_fld GrpIso 𝑃)

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1542 ⊤wtru 1543 ∈ wcel 2114 Vcvv 3442 ∖ cdif 3900 ⊆ wss 3903 {csn 4582 ↾ cres 5634 ⟶wf 6496 –1-1-onto→wf1o 6499 ‘cfv 6500 (class class class)co 7368 ℂcc 11036 ℝcr 11037 0cc0 11038 + caddc 11041 · cmul 11043 ℝ⁺crp 12917 expce 15996 Basecbs 17148 ↾_s cress 17169 +_gcplusg 17189 Grpcgrp 18875 SubGrpcsubg 19062 GrpHom cghm 19153 GrpIso cgim 19198 mulGrpcmgp 20087 Ringcrg 20180 SubRingcsubrg 20514 DivRingcdr 20674 ℂ_fldccnfld 21321 ℝ_fldcrefld 21571

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 ax-mulf 11118

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-tpos 8178 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-er 8645 df-map 8777 df-pm 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-fi 9326 df-sup 9357 df-inf 9358 df-oi 9427 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13277 df-ico 13279 df-icc 13280 df-fz 13436 df-fzo 13583 df-fl 13724 df-seq 13937 df-exp 13997 df-fac 14209 df-bc 14238 df-hash 14266 df-shft 15002 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-limsup 15406 df-clim 15423 df-rlim 15424 df-sum 15622 df-ef 16002 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-rest 17354 df-topn 17355 df-0g 17373 df-gsum 17374 df-topgen 17375 df-pt 17376 df-prds 17379 df-xrs 17435 df-qtop 17440 df-imas 17441 df-xps 17443 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-grp 18878 df-minusg 18879 df-mulg 19010 df-subg 19065 df-ghm 19154 df-gim 19200 df-cntz 19258 df-cmn 19723 df-abl 19724 df-mgp 20088 df-rng 20100 df-ur 20129 df-ring 20182 df-cring 20183 df-oppr 20285 df-dvdsr 20305 df-unit 20306 df-invr 20336 df-dvr 20349 df-subrng 20491 df-subrg 20515 df-drng 20676 df-psmet 21313 df-xmet 21314 df-met 21315 df-bl 21316 df-mopn 21317 df-fbas 21318 df-fg 21319 df-cnfld 21322 df-refld 21572 df-top 22850 df-topon 22867 df-topsp 22889 df-bases 22902 df-cld 22975 df-ntr 22976 df-cls 22977 df-nei 23054 df-lp 23092 df-perf 23093 df-cn 23183 df-cnp 23184 df-haus 23271 df-tx 23518 df-hmeo 23711 df-fil 23802 df-fm 23894 df-flim 23895 df-flf 23896 df-xms 24276 df-ms 24277 df-tms 24278 df-cncf 24839 df-limc 25835 df-dv 25836

This theorem is referenced by: reloggim 26576

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