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Mirrors > Home > MPE Home > Th. List > Mathboxes > seff | Structured version Visualization version GIF version |
Description: Let set 𝑆 be the real or complex numbers. Then the exponential function restricted to 𝑆 is a mapping from 𝑆 to 𝑆. (Contributed by Steve Rodriguez, 6-Nov-2015.) |
Ref | Expression |
---|---|
seff.s | ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
Ref | Expression |
---|---|
seff | ⊢ (𝜑 → (exp ↾ 𝑆):𝑆⟶𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seff.s | . 2 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
2 | elpri 4653 | . 2 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 = ℝ ∨ 𝑆 = ℂ)) | |
3 | reeff1 16152 | . . . . . 6 ⊢ (exp ↾ ℝ):ℝ–1-1→ℝ+ | |
4 | f1f 6804 | . . . . . 6 ⊢ ((exp ↾ ℝ):ℝ–1-1→ℝ+ → (exp ↾ ℝ):ℝ⟶ℝ+) | |
5 | rpssre 13039 | . . . . . . 7 ⊢ ℝ+ ⊆ ℝ | |
6 | fss 6752 | . . . . . . 7 ⊢ (((exp ↾ ℝ):ℝ⟶ℝ+ ∧ ℝ+ ⊆ ℝ) → (exp ↾ ℝ):ℝ⟶ℝ) | |
7 | 5, 6 | mpan2 691 | . . . . . 6 ⊢ ((exp ↾ ℝ):ℝ⟶ℝ+ → (exp ↾ ℝ):ℝ⟶ℝ) |
8 | 3, 4, 7 | mp2b 10 | . . . . 5 ⊢ (exp ↾ ℝ):ℝ⟶ℝ |
9 | feq23 6719 | . . . . . 6 ⊢ ((𝑆 = ℝ ∧ 𝑆 = ℝ) → ((exp ↾ ℝ):𝑆⟶𝑆 ↔ (exp ↾ ℝ):ℝ⟶ℝ)) | |
10 | 9 | anidms 566 | . . . . 5 ⊢ (𝑆 = ℝ → ((exp ↾ ℝ):𝑆⟶𝑆 ↔ (exp ↾ ℝ):ℝ⟶ℝ)) |
11 | 8, 10 | mpbiri 258 | . . . 4 ⊢ (𝑆 = ℝ → (exp ↾ ℝ):𝑆⟶𝑆) |
12 | reseq2 5994 | . . . . 5 ⊢ (𝑆 = ℝ → (exp ↾ 𝑆) = (exp ↾ ℝ)) | |
13 | 12 | feq1d 6720 | . . . 4 ⊢ (𝑆 = ℝ → ((exp ↾ 𝑆):𝑆⟶𝑆 ↔ (exp ↾ ℝ):𝑆⟶𝑆)) |
14 | 11, 13 | mpbird 257 | . . 3 ⊢ (𝑆 = ℝ → (exp ↾ 𝑆):𝑆⟶𝑆) |
15 | eff 16113 | . . . . . 6 ⊢ exp:ℂ⟶ℂ | |
16 | frel 6741 | . . . . . . . . 9 ⊢ (exp:ℂ⟶ℂ → Rel exp) | |
17 | resdm 6045 | . . . . . . . . 9 ⊢ (Rel exp → (exp ↾ dom exp) = exp) | |
18 | 15, 16, 17 | mp2b 10 | . . . . . . . 8 ⊢ (exp ↾ dom exp) = exp |
19 | 15 | fdmi 6747 | . . . . . . . . 9 ⊢ dom exp = ℂ |
20 | 19 | reseq2i 5996 | . . . . . . . 8 ⊢ (exp ↾ dom exp) = (exp ↾ ℂ) |
21 | 18, 20 | eqtr3i 2764 | . . . . . . 7 ⊢ exp = (exp ↾ ℂ) |
22 | 21 | feq1i 6727 | . . . . . 6 ⊢ (exp:ℂ⟶ℂ ↔ (exp ↾ ℂ):ℂ⟶ℂ) |
23 | 15, 22 | mpbi 230 | . . . . 5 ⊢ (exp ↾ ℂ):ℂ⟶ℂ |
24 | feq23 6719 | . . . . . 6 ⊢ ((𝑆 = ℂ ∧ 𝑆 = ℂ) → ((exp ↾ ℂ):𝑆⟶𝑆 ↔ (exp ↾ ℂ):ℂ⟶ℂ)) | |
25 | 24 | anidms 566 | . . . . 5 ⊢ (𝑆 = ℂ → ((exp ↾ ℂ):𝑆⟶𝑆 ↔ (exp ↾ ℂ):ℂ⟶ℂ)) |
26 | 23, 25 | mpbiri 258 | . . . 4 ⊢ (𝑆 = ℂ → (exp ↾ ℂ):𝑆⟶𝑆) |
27 | reseq2 5994 | . . . . 5 ⊢ (𝑆 = ℂ → (exp ↾ 𝑆) = (exp ↾ ℂ)) | |
28 | 27 | feq1d 6720 | . . . 4 ⊢ (𝑆 = ℂ → ((exp ↾ 𝑆):𝑆⟶𝑆 ↔ (exp ↾ ℂ):𝑆⟶𝑆)) |
29 | 26, 28 | mpbird 257 | . . 3 ⊢ (𝑆 = ℂ → (exp ↾ 𝑆):𝑆⟶𝑆) |
30 | 14, 29 | jaoi 857 | . 2 ⊢ ((𝑆 = ℝ ∨ 𝑆 = ℂ) → (exp ↾ 𝑆):𝑆⟶𝑆) |
31 | 1, 2, 30 | 3syl 18 | 1 ⊢ (𝜑 → (exp ↾ 𝑆):𝑆⟶𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∨ wo 847 = wceq 1536 ∈ wcel 2105 ⊆ wss 3962 {cpr 4632 dom cdm 5688 ↾ cres 5690 Rel wrel 5693 ⟶wf 6558 –1-1→wf1 6559 ℂcc 11150 ℝcr 11151 ℝ+crp 13031 expce 16093 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-pm 8867 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-inf 9480 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-rp 13032 df-ico 13389 df-fz 13544 df-fzo 13691 df-fl 13828 df-seq 14039 df-exp 14099 df-fac 14309 df-bc 14338 df-hash 14366 df-shft 15102 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-limsup 15503 df-clim 15520 df-rlim 15521 df-sum 15719 df-ef 16099 |
This theorem is referenced by: (None) |
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