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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 20752 | . . . 4 ⊢ ℝ = (Base‘ℝfld) | |
2 | eqid 2823 | . . . . . 6 ⊢ ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) | |
3 | 2 | rpmsubg 20611 | . . . . 5 ⊢ ℝ+ ∈ (SubGrp‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) |
4 | reefgim.1 | . . . . . . 7 ⊢ 𝑃 = ((mulGrp‘ℂfld) ↾s ℝ+) | |
5 | cnex 10620 | . . . . . . . . 9 ⊢ ℂ ∈ V | |
6 | 5 | difexi 5234 | . . . . . . . 8 ⊢ (ℂ ∖ {0}) ∈ V |
7 | rpcndif0 12411 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ (ℂ ∖ {0})) | |
8 | 7 | ssriv 3973 | . . . . . . . 8 ⊢ ℝ+ ⊆ (ℂ ∖ {0}) |
9 | ressabs 16565 | . . . . . . . 8 ⊢ (((ℂ ∖ {0}) ∈ V ∧ ℝ+ ⊆ (ℂ ∖ {0})) → (((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) ↾s ℝ+) = ((mulGrp‘ℂfld) ↾s ℝ+)) | |
10 | 6, 8, 9 | mp2an 690 | . . . . . . 7 ⊢ (((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) ↾s ℝ+) = ((mulGrp‘ℂfld) ↾s ℝ+) |
11 | 4, 10 | eqtr4i 2849 | . . . . . 6 ⊢ 𝑃 = (((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) ↾s ℝ+) |
12 | 11 | subgbas 18285 | . . . . 5 ⊢ (ℝ+ ∈ (SubGrp‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) → ℝ+ = (Base‘𝑃)) |
13 | 3, 12 | ax-mp 5 | . . . 4 ⊢ ℝ+ = (Base‘𝑃) |
14 | replusg 20756 | . . . 4 ⊢ + = (+g‘ℝfld) | |
15 | eqid 2823 | . . . . . . 7 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
16 | cnfldmul 20553 | . . . . . . 7 ⊢ · = (.r‘ℂfld) | |
17 | 15, 16 | mgpplusg 19245 | . . . . . 6 ⊢ · = (+g‘(mulGrp‘ℂfld)) |
18 | 4, 17 | ressplusg 16614 | . . . . 5 ⊢ (ℝ+ ∈ (SubGrp‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) → · = (+g‘𝑃)) |
19 | 3, 18 | ax-mp 5 | . . . 4 ⊢ · = (+g‘𝑃) |
20 | resubdrg 20754 | . . . . . . 7 ⊢ (ℝ ∈ (SubRing‘ℂfld) ∧ ℝfld ∈ DivRing) | |
21 | 20 | simpli 486 | . . . . . 6 ⊢ ℝ ∈ (SubRing‘ℂfld) |
22 | df-refld 20751 | . . . . . . 7 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
23 | 22 | subrgring 19540 | . . . . . 6 ⊢ (ℝ ∈ (SubRing‘ℂfld) → ℝfld ∈ Ring) |
24 | 21, 23 | ax-mp 5 | . . . . 5 ⊢ ℝfld ∈ Ring |
25 | ringgrp 19304 | . . . . 5 ⊢ (ℝfld ∈ Ring → ℝfld ∈ Grp) | |
26 | 24, 25 | mp1i 13 | . . . 4 ⊢ (⊤ → ℝfld ∈ Grp) |
27 | 11 | subggrp 18284 | . . . . 5 ⊢ (ℝ+ ∈ (SubGrp‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) → 𝑃 ∈ Grp) |
28 | 3, 27 | mp1i 13 | . . . 4 ⊢ (⊤ → 𝑃 ∈ Grp) |
29 | reeff1o 25037 | . . . . 5 ⊢ (exp ↾ ℝ):ℝ–1-1-onto→ℝ+ | |
30 | f1of 6617 | . . . . 5 ⊢ ((exp ↾ ℝ):ℝ–1-1-onto→ℝ+ → (exp ↾ ℝ):ℝ⟶ℝ+) | |
31 | 29, 30 | mp1i 13 | . . . 4 ⊢ (⊤ → (exp ↾ ℝ):ℝ⟶ℝ+) |
32 | recn 10629 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ) | |
33 | recn 10629 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ) | |
34 | efadd 15449 | . . . . . . 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 10622 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + 𝑦) ∈ ℝ) | |
37 | 36 | fvresd 6692 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((exp ↾ ℝ)‘(𝑥 + 𝑦)) = (exp‘(𝑥 + 𝑦))) |
38 | fvres 6691 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ → ((exp ↾ ℝ)‘𝑥) = (exp‘𝑥)) | |
39 | fvres 6691 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → ((exp ↾ ℝ)‘𝑦) = (exp‘𝑦)) | |
40 | 38, 39 | oveqan12d 7177 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (((exp ↾ ℝ)‘𝑥) · ((exp ↾ ℝ)‘𝑦)) = ((exp‘𝑥) · (exp‘𝑦))) |
41 | 35, 37, 40 | 3eqtr4d 2868 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((exp ↾ ℝ)‘(𝑥 + 𝑦)) = (((exp ↾ ℝ)‘𝑥) · ((exp ↾ ℝ)‘𝑦))) |
42 | 41 | adantl 484 | . . . 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 18369 | . . 3 ⊢ (⊤ → (exp ↾ ℝ) ∈ (ℝfld GrpHom 𝑃)) |
44 | 43 | mptru 1544 | . 2 ⊢ (exp ↾ ℝ) ∈ (ℝfld GrpHom 𝑃) |
45 | 1, 13 | isgim 18404 | . 2 ⊢ ((exp ↾ ℝ) ∈ (ℝfld GrpIso 𝑃) ↔ ((exp ↾ ℝ) ∈ (ℝfld GrpHom 𝑃) ∧ (exp ↾ ℝ):ℝ–1-1-onto→ℝ+)) |
46 | 44, 29, 45 | mpbir2an 709 | 1 ⊢ (exp ↾ ℝ) ∈ (ℝfld GrpIso 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 Vcvv 3496 ∖ cdif 3935 ⊆ wss 3938 {csn 4569 ↾ cres 5559 ⟶wf 6353 –1-1-onto→wf1o 6356 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 ℝcr 10538 0cc0 10539 + caddc 10542 · cmul 10544 ℝ+crp 12392 expce 15417 Basecbs 16485 ↾s cress 16486 +gcplusg 16567 Grpcgrp 18105 SubGrpcsubg 18275 GrpHom cghm 18357 GrpIso cgim 18399 mulGrpcmgp 19241 Ringcrg 19299 DivRingcdr 19504 SubRingcsubrg 19533 ℂfldccnfld 20547 ℝfldcrefld 20750 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-seq 13373 df-exp 13433 df-fac 13637 df-bc 13666 df-hash 13694 df-shft 14428 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-limsup 14830 df-clim 14847 df-rlim 14848 df-sum 15045 df-ef 15423 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-xrs 16777 df-qtop 16782 df-imas 16783 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-grp 18108 df-minusg 18109 df-mulg 18227 df-subg 18278 df-ghm 18358 df-gim 18401 df-cntz 18449 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-dvr 19435 df-drng 19506 df-subrg 19535 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-cnfld 20548 df-refld 20751 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-lp 21746 df-perf 21747 df-cn 21837 df-cnp 21838 df-haus 21925 df-tx 22172 df-hmeo 22365 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-xms 22932 df-ms 22933 df-tms 22934 df-cncf 23488 df-limc 24466 df-dv 24467 |
This theorem is referenced by: reloggim 25184 |
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