![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
Mirrors > Home > MPE Home > Th. List > reefgim | Structured version Unicode 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 |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Ref | Expression |
---|---|
reefgim |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resubdrg 18147 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | 1 | simpli 458 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() |
3 | df-refld 18144 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() | |
4 | 3 | subrgbas 16980 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | 2, 4 | ax-mp 5 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | eqid 2451 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 6 | rpmsubg 17985 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
8 | reefgim.1 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | cnex 9464 |
. . . . . . . . 9
![]() ![]() ![]() ![]() | |
10 | difexg 4538 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
11 | 9, 10 | ax-mp 5 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
12 | rpcn 11100 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
13 | rpne0 11107 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | eldifsn 4098 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
15 | 12, 13, 14 | sylanbrc 664 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
16 | 15 | ssriv 3458 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
17 | ressabs 14338 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
18 | 11, 16, 17 | mp2an 672 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
19 | 8, 18 | eqtr4i 2483 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | 19 | subgbas 15787 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | 7, 20 | ax-mp 5 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | replusg 18149 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
23 | eqid 2451 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | cnfldmul 17933 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
25 | 23, 24 | mgpplusg 16700 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 8, 25 | ressplusg 14382 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 7, 26 | ax-mp 5 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 3 | subrgrng 16974 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 2, 28 | ax-mp 5 |
. . . . 5
![]() ![]() ![]() |
30 | rnggrp 16756 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
31 | 29, 30 | mp1i 12 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32 | 19 | subggrp 15786 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
33 | 7, 32 | mp1i 12 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
34 | reeff1o 22028 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
35 | f1of 5739 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
36 | 34, 35 | mp1i 12 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
37 | recn 9473 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
38 | recn 9473 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
39 | efadd 13481 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
40 | 37, 38, 39 | syl2an 477 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
41 | readdcl 9466 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
42 | fvres 5803 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
43 | 41, 42 | syl 16 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
44 | fvres 5803 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
45 | fvres 5803 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
46 | 44, 45 | oveqan12d 6209 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
47 | 40, 43, 46 | 3eqtr4d 2502 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
48 | 47 | adantl 466 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
49 | 5, 21, 22, 27, 31, 33, 36, 48 | isghmd 15858 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
50 | 49 | trud 1379 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
51 | 5, 21 | isgim 15892 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
52 | 50, 34, 51 | mpbir2an 911 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1952 ax-ext 2430 ax-rep 4501 ax-sep 4511 ax-nul 4519 ax-pow 4568 ax-pr 4629 ax-un 6472 ax-inf2 7948 ax-cnex 9439 ax-resscn 9440 ax-1cn 9441 ax-icn 9442 ax-addcl 9443 ax-addrcl 9444 ax-mulcl 9445 ax-mulrcl 9446 ax-mulcom 9447 ax-addass 9448 ax-mulass 9449 ax-distr 9450 ax-i2m1 9451 ax-1ne0 9452 ax-1rid 9453 ax-rnegex 9454 ax-rrecex 9455 ax-cnre 9456 ax-pre-lttri 9457 ax-pre-lttrn 9458 ax-pre-ltadd 9459 ax-pre-mulgt0 9460 ax-pre-sup 9461 ax-addf 9462 ax-mulf 9463 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-fal 1376 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-nel 2647 df-ral 2800 df-rex 2801 df-reu 2802 df-rmo 2803 df-rab 2804 df-v 3070 df-sbc 3285 df-csb 3387 df-dif 3429 df-un 3431 df-in 3433 df-ss 3440 df-pss 3442 df-nul 3736 df-if 3890 df-pw 3960 df-sn 3976 df-pr 3978 df-tp 3980 df-op 3982 df-uni 4190 df-int 4227 df-iun 4271 df-iin 4272 df-br 4391 df-opab 4449 df-mpt 4450 df-tr 4484 df-eprel 4730 df-id 4734 df-po 4739 df-so 4740 df-fr 4777 df-se 4778 df-we 4779 df-ord 4820 df-on 4821 df-lim 4822 df-suc 4823 df-xp 4944 df-rel 4945 df-cnv 4946 df-co 4947 df-dm 4948 df-rn 4949 df-res 4950 df-ima 4951 df-iota 5479 df-fun 5518 df-fn 5519 df-f 5520 df-f1 5521 df-fo 5522 df-f1o 5523 df-fv 5524 df-isom 5525 df-riota 6151 df-ov 6193 df-oprab 6194 df-mpt2 6195 df-of 6420 df-om 6577 df-1st 6677 df-2nd 6678 df-supp 6791 df-tpos 6845 df-recs 6932 df-rdg 6966 df-1o 7020 df-2o 7021 df-oadd 7024 df-er 7201 df-map 7316 df-pm 7317 df-ixp 7364 df-en 7411 df-dom 7412 df-sdom 7413 df-fin 7414 df-fsupp 7722 df-fi 7762 df-sup 7792 df-oi 7825 df-card 8210 df-cda 8438 df-pnf 9521 df-mnf 9522 df-xr 9523 df-ltxr 9524 df-le 9525 df-sub 9698 df-neg 9699 df-div 10095 df-nn 10424 df-2 10481 df-3 10482 df-4 10483 df-5 10484 df-6 10485 df-7 10486 df-8 10487 df-9 10488 df-10 10489 df-n0 10681 df-z 10748 df-dec 10857 df-uz 10963 df-q 11055 df-rp 11093 df-xneg 11190 df-xadd 11191 df-xmul 11192 df-ioo 11405 df-ico 11407 df-icc 11408 df-fz 11539 df-fzo 11650 df-fl 11743 df-seq 11908 df-exp 11967 df-fac 12153 df-bc 12180 df-hash 12205 df-shft 12658 df-cj 12690 df-re 12691 df-im 12692 df-sqr 12826 df-abs 12827 df-limsup 13051 df-clim 13068 df-rlim 13069 df-sum 13266 df-ef 13455 df-struct 14278 df-ndx 14279 df-slot 14280 df-base 14281 df-sets 14282 df-ress 14283 df-plusg 14353 df-mulr 14354 df-starv 14355 df-sca 14356 df-vsca 14357 df-ip 14358 df-tset 14359 df-ple 14360 df-ds 14362 df-unif 14363 df-hom 14364 df-cco 14365 df-rest 14463 df-topn 14464 df-0g 14482 df-gsum 14483 df-topgen 14484 df-pt 14485 df-prds 14488 df-xrs 14542 df-qtop 14547 df-imas 14548 df-xps 14550 df-mre 14626 df-mrc 14627 df-acs 14629 df-mnd 15517 df-submnd 15567 df-grp 15647 df-minusg 15648 df-mulg 15650 df-subg 15780 df-ghm 15847 df-gim 15889 df-cntz 15937 df-cmn 16383 df-abl 16384 df-mgp 16697 df-ur 16709 df-rng 16753 df-cring 16754 df-oppr 16821 df-dvdsr 16839 df-unit 16840 df-invr 16870 df-dvr 16881 df-drng 16940 df-subrg 16969 df-psmet 17918 df-xmet 17919 df-met 17920 df-bl 17921 df-mopn 17922 df-fbas 17923 df-fg 17924 df-cnfld 17928 df-refld 18144 df-top 18619 df-bases 18621 df-topon 18622 df-topsp 18623 df-cld 18739 df-ntr 18740 df-cls 18741 df-nei 18818 df-lp 18856 df-perf 18857 df-cn 18947 df-cnp 18948 df-haus 19035 df-tx 19251 df-hmeo 19444 df-fil 19535 df-fm 19627 df-flim 19628 df-flf 19629 df-xms 20011 df-ms 20012 df-tms 20013 df-cncf 20570 df-limc 21457 df-dv 21458 |
This theorem is referenced by: reloggim 22163 |
Copyright terms: Public domain | W3C validator |