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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 4579 | . 2 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 = ℝ ∨ 𝑆 = ℂ)) | |
3 | reeff1 15461 | . . . . . 6 ⊢ (exp ↾ ℝ):ℝ–1-1→ℝ+ | |
4 | f1f 6568 | . . . . . 6 ⊢ ((exp ↾ ℝ):ℝ–1-1→ℝ+ → (exp ↾ ℝ):ℝ⟶ℝ+) | |
5 | rpssre 12384 | . . . . . . 7 ⊢ ℝ+ ⊆ ℝ | |
6 | fss 6520 | . . . . . . 7 ⊢ (((exp ↾ ℝ):ℝ⟶ℝ+ ∧ ℝ+ ⊆ ℝ) → (exp ↾ ℝ):ℝ⟶ℝ) | |
7 | 5, 6 | mpan2 687 | . . . . . 6 ⊢ ((exp ↾ ℝ):ℝ⟶ℝ+ → (exp ↾ ℝ):ℝ⟶ℝ) |
8 | 3, 4, 7 | mp2b 10 | . . . . 5 ⊢ (exp ↾ ℝ):ℝ⟶ℝ |
9 | feq23 6491 | . . . . . 6 ⊢ ((𝑆 = ℝ ∧ 𝑆 = ℝ) → ((exp ↾ ℝ):𝑆⟶𝑆 ↔ (exp ↾ ℝ):ℝ⟶ℝ)) | |
10 | 9 | anidms 567 | . . . . 5 ⊢ (𝑆 = ℝ → ((exp ↾ ℝ):𝑆⟶𝑆 ↔ (exp ↾ ℝ):ℝ⟶ℝ)) |
11 | 8, 10 | mpbiri 259 | . . . 4 ⊢ (𝑆 = ℝ → (exp ↾ ℝ):𝑆⟶𝑆) |
12 | reseq2 5841 | . . . . 5 ⊢ (𝑆 = ℝ → (exp ↾ 𝑆) = (exp ↾ ℝ)) | |
13 | 12 | feq1d 6492 | . . . 4 ⊢ (𝑆 = ℝ → ((exp ↾ 𝑆):𝑆⟶𝑆 ↔ (exp ↾ ℝ):𝑆⟶𝑆)) |
14 | 11, 13 | mpbird 258 | . . 3 ⊢ (𝑆 = ℝ → (exp ↾ 𝑆):𝑆⟶𝑆) |
15 | eff 15423 | . . . . . 6 ⊢ exp:ℂ⟶ℂ | |
16 | frel 6512 | . . . . . . . . 9 ⊢ (exp:ℂ⟶ℂ → Rel exp) | |
17 | resdm 5890 | . . . . . . . . 9 ⊢ (Rel exp → (exp ↾ dom exp) = exp) | |
18 | 15, 16, 17 | mp2b 10 | . . . . . . . 8 ⊢ (exp ↾ dom exp) = exp |
19 | 15 | fdmi 6517 | . . . . . . . . 9 ⊢ dom exp = ℂ |
20 | 19 | reseq2i 5843 | . . . . . . . 8 ⊢ (exp ↾ dom exp) = (exp ↾ ℂ) |
21 | 18, 20 | eqtr3i 2843 | . . . . . . 7 ⊢ exp = (exp ↾ ℂ) |
22 | 21 | feq1i 6498 | . . . . . 6 ⊢ (exp:ℂ⟶ℂ ↔ (exp ↾ ℂ):ℂ⟶ℂ) |
23 | 15, 22 | mpbi 231 | . . . . 5 ⊢ (exp ↾ ℂ):ℂ⟶ℂ |
24 | feq23 6491 | . . . . . 6 ⊢ ((𝑆 = ℂ ∧ 𝑆 = ℂ) → ((exp ↾ ℂ):𝑆⟶𝑆 ↔ (exp ↾ ℂ):ℂ⟶ℂ)) | |
25 | 24 | anidms 567 | . . . . 5 ⊢ (𝑆 = ℂ → ((exp ↾ ℂ):𝑆⟶𝑆 ↔ (exp ↾ ℂ):ℂ⟶ℂ)) |
26 | 23, 25 | mpbiri 259 | . . . 4 ⊢ (𝑆 = ℂ → (exp ↾ ℂ):𝑆⟶𝑆) |
27 | reseq2 5841 | . . . . 5 ⊢ (𝑆 = ℂ → (exp ↾ 𝑆) = (exp ↾ ℂ)) | |
28 | 27 | feq1d 6492 | . . . 4 ⊢ (𝑆 = ℂ → ((exp ↾ 𝑆):𝑆⟶𝑆 ↔ (exp ↾ ℂ):𝑆⟶𝑆)) |
29 | 26, 28 | mpbird 258 | . . 3 ⊢ (𝑆 = ℂ → (exp ↾ 𝑆):𝑆⟶𝑆) |
30 | 14, 29 | jaoi 851 | . 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 207 ∨ wo 841 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 {cpr 4559 dom cdm 5548 ↾ cres 5550 Rel wrel 5553 ⟶wf 6344 –1-1→wf1 6345 ℂcc 10523 ℝcr 10524 ℝ+crp 12377 expce 15403 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-ico 12732 df-fz 12881 df-fzo 13022 df-fl 13150 df-seq 13358 df-exp 13418 df-fac 13622 df-bc 13651 df-hash 13679 df-shft 14414 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-limsup 14816 df-clim 14833 df-rlim 14834 df-sum 15031 df-ef 15409 |
This theorem is referenced by: (None) |
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