![]() |
Mathbox for Steve Rodriguez |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > seff | Structured version Visualization version GIF version |
Description: Let set π be the real or complex numbers. Then the exponential function restricted to π is a mapping from π to π. (Contributed by Steve Rodriguez, 6-Nov-2015.) |
Ref | Expression |
---|---|
seff.s | β’ (π β π β {β, β}) |
Ref | Expression |
---|---|
seff | β’ (π β (exp βΎ π):πβΆπ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seff.s | . 2 β’ (π β π β {β, β}) | |
2 | elpri 4612 | . 2 β’ (π β {β, β} β (π = β β¨ π = β)) | |
3 | reeff1 16010 | . . . . . 6 β’ (exp βΎ β):ββ1-1ββ+ | |
4 | f1f 6742 | . . . . . 6 β’ ((exp βΎ β):ββ1-1ββ+ β (exp βΎ β):ββΆβ+) | |
5 | rpssre 12930 | . . . . . . 7 β’ β+ β β | |
6 | fss 6689 | . . . . . . 7 β’ (((exp βΎ β):ββΆβ+ β§ β+ β β) β (exp βΎ β):ββΆβ) | |
7 | 5, 6 | mpan2 690 | . . . . . 6 β’ ((exp βΎ β):ββΆβ+ β (exp βΎ β):ββΆβ) |
8 | 3, 4, 7 | mp2b 10 | . . . . 5 β’ (exp βΎ β):ββΆβ |
9 | feq23 6656 | . . . . . 6 β’ ((π = β β§ π = β) β ((exp βΎ β):πβΆπ β (exp βΎ β):ββΆβ)) | |
10 | 9 | anidms 568 | . . . . 5 β’ (π = β β ((exp βΎ β):πβΆπ β (exp βΎ β):ββΆβ)) |
11 | 8, 10 | mpbiri 258 | . . . 4 β’ (π = β β (exp βΎ β):πβΆπ) |
12 | reseq2 5936 | . . . . 5 β’ (π = β β (exp βΎ π) = (exp βΎ β)) | |
13 | 12 | feq1d 6657 | . . . 4 β’ (π = β β ((exp βΎ π):πβΆπ β (exp βΎ β):πβΆπ)) |
14 | 11, 13 | mpbird 257 | . . 3 β’ (π = β β (exp βΎ π):πβΆπ) |
15 | eff 15972 | . . . . . 6 β’ exp:ββΆβ | |
16 | frel 6677 | . . . . . . . . 9 β’ (exp:ββΆβ β Rel exp) | |
17 | resdm 5986 | . . . . . . . . 9 β’ (Rel exp β (exp βΎ dom exp) = exp) | |
18 | 15, 16, 17 | mp2b 10 | . . . . . . . 8 β’ (exp βΎ dom exp) = exp |
19 | 15 | fdmi 6684 | . . . . . . . . 9 β’ dom exp = β |
20 | 19 | reseq2i 5938 | . . . . . . . 8 β’ (exp βΎ dom exp) = (exp βΎ β) |
21 | 18, 20 | eqtr3i 2763 | . . . . . . 7 β’ exp = (exp βΎ β) |
22 | 21 | feq1i 6663 | . . . . . 6 β’ (exp:ββΆβ β (exp βΎ β):ββΆβ) |
23 | 15, 22 | mpbi 229 | . . . . 5 β’ (exp βΎ β):ββΆβ |
24 | feq23 6656 | . . . . . 6 β’ ((π = β β§ π = β) β ((exp βΎ β):πβΆπ β (exp βΎ β):ββΆβ)) | |
25 | 24 | anidms 568 | . . . . 5 β’ (π = β β ((exp βΎ β):πβΆπ β (exp βΎ β):ββΆβ)) |
26 | 23, 25 | mpbiri 258 | . . . 4 β’ (π = β β (exp βΎ β):πβΆπ) |
27 | reseq2 5936 | . . . . 5 β’ (π = β β (exp βΎ π) = (exp βΎ β)) | |
28 | 27 | feq1d 6657 | . . . 4 β’ (π = β β ((exp βΎ π):πβΆπ β (exp βΎ β):πβΆπ)) |
29 | 26, 28 | mpbird 257 | . . 3 β’ (π = β β (exp βΎ π):πβΆπ) |
30 | 14, 29 | jaoi 856 | . 2 β’ ((π = β β¨ π = β) β (exp βΎ π):πβΆπ) |
31 | 1, 2, 30 | 3syl 18 | 1 β’ (π β (exp βΎ π):πβΆπ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β¨ wo 846 = wceq 1542 β wcel 2107 β wss 3914 {cpr 4592 dom cdm 5637 βΎ cres 5639 Rel wrel 5642 βΆwf 6496 β1-1βwf1 6497 βcc 11057 βcr 11058 β+crp 12923 expce 15952 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-inf2 9585 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 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 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-pm 8774 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-sup 9386 df-inf 9387 df-oi 9454 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-n0 12422 df-z 12508 df-uz 12772 df-rp 12924 df-ico 13279 df-fz 13434 df-fzo 13577 df-fl 13706 df-seq 13916 df-exp 13977 df-fac 14183 df-bc 14212 df-hash 14240 df-shft 14961 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-limsup 15362 df-clim 15379 df-rlim 15380 df-sum 15580 df-ef 15958 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |