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Mirrors > Home > MPE Home > Th. List > reefgim | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
reefgim.1 | ⊢ 𝑃 = ((mulGrp‘ℂfld) ↾s ℝ+) |
Ref | Expression |
---|---|
reefgim | ⊢ (exp ↾ ℝ) ∈ (ℝfld GrpIso 𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rebase 20436 | . . . 4 ⊢ ℝ = (Base‘ℝfld) | |
2 | eqid 2797 | . . . . . 6 ⊢ ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) | |
3 | 2 | rpmsubg 20295 | . . . . 5 ⊢ ℝ+ ∈ (SubGrp‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) |
4 | reefgim.1 | . . . . . . 7 ⊢ 𝑃 = ((mulGrp‘ℂfld) ↾s ℝ+) | |
5 | cnex 10471 | . . . . . . . . 9 ⊢ ℂ ∈ V | |
6 | 5 | difexi 5130 | . . . . . . . 8 ⊢ (ℂ ∖ {0}) ∈ V |
7 | rpcndif0 12262 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ (ℂ ∖ {0})) | |
8 | 7 | ssriv 3899 | . . . . . . . 8 ⊢ ℝ+ ⊆ (ℂ ∖ {0}) |
9 | ressabs 16396 | . . . . . . . 8 ⊢ (((ℂ ∖ {0}) ∈ V ∧ ℝ+ ⊆ (ℂ ∖ {0})) → (((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) ↾s ℝ+) = ((mulGrp‘ℂfld) ↾s ℝ+)) | |
10 | 6, 8, 9 | mp2an 688 | . . . . . . 7 ⊢ (((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) ↾s ℝ+) = ((mulGrp‘ℂfld) ↾s ℝ+) |
11 | 4, 10 | eqtr4i 2824 | . . . . . 6 ⊢ 𝑃 = (((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) ↾s ℝ+) |
12 | 11 | subgbas 18041 | . . . . 5 ⊢ (ℝ+ ∈ (SubGrp‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) → ℝ+ = (Base‘𝑃)) |
13 | 3, 12 | ax-mp 5 | . . . 4 ⊢ ℝ+ = (Base‘𝑃) |
14 | replusg 20440 | . . . 4 ⊢ + = (+g‘ℝfld) | |
15 | eqid 2797 | . . . . . . 7 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
16 | cnfldmul 20237 | . . . . . . 7 ⊢ · = (.r‘ℂfld) | |
17 | 15, 16 | mgpplusg 18937 | . . . . . 6 ⊢ · = (+g‘(mulGrp‘ℂfld)) |
18 | 4, 17 | ressplusg 16445 | . . . . 5 ⊢ (ℝ+ ∈ (SubGrp‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) → · = (+g‘𝑃)) |
19 | 3, 18 | ax-mp 5 | . . . 4 ⊢ · = (+g‘𝑃) |
20 | resubdrg 20438 | . . . . . . 7 ⊢ (ℝ ∈ (SubRing‘ℂfld) ∧ ℝfld ∈ DivRing) | |
21 | 20 | simpli 484 | . . . . . 6 ⊢ ℝ ∈ (SubRing‘ℂfld) |
22 | df-refld 20435 | . . . . . . 7 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
23 | 22 | subrgring 19232 | . . . . . 6 ⊢ (ℝ ∈ (SubRing‘ℂfld) → ℝfld ∈ Ring) |
24 | 21, 23 | ax-mp 5 | . . . . 5 ⊢ ℝfld ∈ Ring |
25 | ringgrp 18996 | . . . . 5 ⊢ (ℝfld ∈ Ring → ℝfld ∈ Grp) | |
26 | 24, 25 | mp1i 13 | . . . 4 ⊢ (⊤ → ℝfld ∈ Grp) |
27 | 11 | subggrp 18040 | . . . . 5 ⊢ (ℝ+ ∈ (SubGrp‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) → 𝑃 ∈ Grp) |
28 | 3, 27 | mp1i 13 | . . . 4 ⊢ (⊤ → 𝑃 ∈ Grp) |
29 | reeff1o 24722 | . . . . 5 ⊢ (exp ↾ ℝ):ℝ–1-1-onto→ℝ+ | |
30 | f1of 6490 | . . . . 5 ⊢ ((exp ↾ ℝ):ℝ–1-1-onto→ℝ+ → (exp ↾ ℝ):ℝ⟶ℝ+) | |
31 | 29, 30 | mp1i 13 | . . . 4 ⊢ (⊤ → (exp ↾ ℝ):ℝ⟶ℝ+) |
32 | recn 10480 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ) | |
33 | recn 10480 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ) | |
34 | efadd 15284 | . . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (exp‘(𝑥 + 𝑦)) = ((exp‘𝑥) · (exp‘𝑦))) | |
35 | 32, 33, 34 | syl2an 595 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (exp‘(𝑥 + 𝑦)) = ((exp‘𝑥) · (exp‘𝑦))) |
36 | readdcl 10473 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + 𝑦) ∈ ℝ) | |
37 | 36 | fvresd 6565 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((exp ↾ ℝ)‘(𝑥 + 𝑦)) = (exp‘(𝑥 + 𝑦))) |
38 | fvres 6564 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ → ((exp ↾ ℝ)‘𝑥) = (exp‘𝑥)) | |
39 | fvres 6564 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → ((exp ↾ ℝ)‘𝑦) = (exp‘𝑦)) | |
40 | 38, 39 | oveqan12d 7042 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (((exp ↾ ℝ)‘𝑥) · ((exp ↾ ℝ)‘𝑦)) = ((exp‘𝑥) · (exp‘𝑦))) |
41 | 35, 37, 40 | 3eqtr4d 2843 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((exp ↾ ℝ)‘(𝑥 + 𝑦)) = (((exp ↾ ℝ)‘𝑥) · ((exp ↾ ℝ)‘𝑦))) |
42 | 41 | adantl 482 | . . . 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 18112 | . . 3 ⊢ (⊤ → (exp ↾ ℝ) ∈ (ℝfld GrpHom 𝑃)) |
44 | 43 | mptru 1532 | . 2 ⊢ (exp ↾ ℝ) ∈ (ℝfld GrpHom 𝑃) |
45 | 1, 13 | isgim 18147 | . 2 ⊢ ((exp ↾ ℝ) ∈ (ℝfld GrpIso 𝑃) ↔ ((exp ↾ ℝ) ∈ (ℝfld GrpHom 𝑃) ∧ (exp ↾ ℝ):ℝ–1-1-onto→ℝ+)) |
46 | 44, 29, 45 | mpbir2an 707 | 1 ⊢ (exp ↾ ℝ) ∈ (ℝfld GrpIso 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1525 ⊤wtru 1526 ∈ wcel 2083 Vcvv 3440 ∖ cdif 3862 ⊆ wss 3865 {csn 4478 ↾ cres 5452 ⟶wf 6228 –1-1-onto→wf1o 6231 ‘cfv 6232 (class class class)co 7023 ℂcc 10388 ℝcr 10389 0cc0 10390 + caddc 10393 · cmul 10395 ℝ+crp 12243 expce 15252 Basecbs 16316 ↾s cress 16317 +gcplusg 16398 Grpcgrp 17865 SubGrpcsubg 18031 GrpHom cghm 18100 GrpIso cgim 18142 mulGrpcmgp 18933 Ringcrg 18991 DivRingcdr 19196 SubRingcsubrg 19225 ℂfldccnfld 20231 ℝfldcrefld 20434 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-inf2 8957 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 ax-pre-sup 10468 ax-addf 10469 ax-mulf 10470 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-fal 1538 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-iin 4834 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-se 5410 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-isom 6241 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-of 7274 df-om 7444 df-1st 7552 df-2nd 7553 df-supp 7689 df-tpos 7750 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-2o 7961 df-oadd 7964 df-er 8146 df-map 8265 df-pm 8266 df-ixp 8318 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-fsupp 8687 df-fi 8728 df-sup 8759 df-inf 8760 df-oi 8827 df-card 9221 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-div 11152 df-nn 11493 df-2 11554 df-3 11555 df-4 11556 df-5 11557 df-6 11558 df-7 11559 df-8 11560 df-9 11561 df-n0 11752 df-z 11836 df-dec 11953 df-uz 12098 df-q 12202 df-rp 12244 df-xneg 12361 df-xadd 12362 df-xmul 12363 df-ioo 12596 df-ico 12598 df-icc 12599 df-fz 12747 df-fzo 12888 df-fl 13016 df-seq 13224 df-exp 13284 df-fac 13488 df-bc 13517 df-hash 13545 df-shft 14264 df-cj 14296 df-re 14297 df-im 14298 df-sqrt 14432 df-abs 14433 df-limsup 14666 df-clim 14683 df-rlim 14684 df-sum 14881 df-ef 15258 df-struct 16318 df-ndx 16319 df-slot 16320 df-base 16322 df-sets 16323 df-ress 16324 df-plusg 16411 df-mulr 16412 df-starv 16413 df-sca 16414 df-vsca 16415 df-ip 16416 df-tset 16417 df-ple 16418 df-ds 16420 df-unif 16421 df-hom 16422 df-cco 16423 df-rest 16529 df-topn 16530 df-0g 16548 df-gsum 16549 df-topgen 16550 df-pt 16551 df-prds 16554 df-xrs 16608 df-qtop 16613 df-imas 16614 df-xps 16616 df-mre 16690 df-mrc 16691 df-acs 16693 df-mgm 17685 df-sgrp 17727 df-mnd 17738 df-submnd 17779 df-grp 17868 df-minusg 17869 df-mulg 17986 df-subg 18034 df-ghm 18101 df-gim 18144 df-cntz 18192 df-cmn 18639 df-abl 18640 df-mgp 18934 df-ur 18946 df-ring 18993 df-cring 18994 df-oppr 19067 df-dvdsr 19085 df-unit 19086 df-invr 19116 df-dvr 19127 df-drng 19198 df-subrg 19227 df-psmet 20223 df-xmet 20224 df-met 20225 df-bl 20226 df-mopn 20227 df-fbas 20228 df-fg 20229 df-cnfld 20232 df-refld 20435 df-top 21190 df-topon 21207 df-topsp 21229 df-bases 21242 df-cld 21315 df-ntr 21316 df-cls 21317 df-nei 21394 df-lp 21432 df-perf 21433 df-cn 21523 df-cnp 21524 df-haus 21611 df-tx 21858 df-hmeo 22051 df-fil 22142 df-fm 22234 df-flim 22235 df-flf 22236 df-xms 22617 df-ms 22618 df-tms 22619 df-cncf 23173 df-limc 24151 df-dv 24152 |
This theorem is referenced by: reloggim 24867 |
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