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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 4145 | . 2 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 = ℝ ∨ 𝑆 = ℂ)) | |
3 | reeff1 14689 | . . . . . 6 ⊢ (exp ↾ ℝ):ℝ–1-1→ℝ+ | |
4 | f1f 6014 | . . . . . 6 ⊢ ((exp ↾ ℝ):ℝ–1-1→ℝ+ → (exp ↾ ℝ):ℝ⟶ℝ+) | |
5 | rpssre 11719 | . . . . . . 7 ⊢ ℝ+ ⊆ ℝ | |
6 | fss 5969 | . . . . . . 7 ⊢ (((exp ↾ ℝ):ℝ⟶ℝ+ ∧ ℝ+ ⊆ ℝ) → (exp ↾ ℝ):ℝ⟶ℝ) | |
7 | 5, 6 | mpan2 703 | . . . . . 6 ⊢ ((exp ↾ ℝ):ℝ⟶ℝ+ → (exp ↾ ℝ):ℝ⟶ℝ) |
8 | 3, 4, 7 | mp2b 10 | . . . . 5 ⊢ (exp ↾ ℝ):ℝ⟶ℝ |
9 | feq23 5942 | . . . . . 6 ⊢ ((𝑆 = ℝ ∧ 𝑆 = ℝ) → ((exp ↾ ℝ):𝑆⟶𝑆 ↔ (exp ↾ ℝ):ℝ⟶ℝ)) | |
10 | 9 | anidms 675 | . . . . 5 ⊢ (𝑆 = ℝ → ((exp ↾ ℝ):𝑆⟶𝑆 ↔ (exp ↾ ℝ):ℝ⟶ℝ)) |
11 | 8, 10 | mpbiri 247 | . . . 4 ⊢ (𝑆 = ℝ → (exp ↾ ℝ):𝑆⟶𝑆) |
12 | reseq2 5312 | . . . . 5 ⊢ (𝑆 = ℝ → (exp ↾ 𝑆) = (exp ↾ ℝ)) | |
13 | 12 | feq1d 5943 | . . . 4 ⊢ (𝑆 = ℝ → ((exp ↾ 𝑆):𝑆⟶𝑆 ↔ (exp ↾ ℝ):𝑆⟶𝑆)) |
14 | 11, 13 | mpbird 246 | . . 3 ⊢ (𝑆 = ℝ → (exp ↾ 𝑆):𝑆⟶𝑆) |
15 | eff 14651 | . . . . . 6 ⊢ exp:ℂ⟶ℂ | |
16 | frel 5963 | . . . . . . . . 9 ⊢ (exp:ℂ⟶ℂ → Rel exp) | |
17 | resdm 5361 | . . . . . . . . 9 ⊢ (Rel exp → (exp ↾ dom exp) = exp) | |
18 | 15, 16, 17 | mp2b 10 | . . . . . . . 8 ⊢ (exp ↾ dom exp) = exp |
19 | 15 | fdmi 5965 | . . . . . . . . 9 ⊢ dom exp = ℂ |
20 | 19 | reseq2i 5314 | . . . . . . . 8 ⊢ (exp ↾ dom exp) = (exp ↾ ℂ) |
21 | 18, 20 | eqtr3i 2634 | . . . . . . 7 ⊢ exp = (exp ↾ ℂ) |
22 | 21 | feq1i 5949 | . . . . . 6 ⊢ (exp:ℂ⟶ℂ ↔ (exp ↾ ℂ):ℂ⟶ℂ) |
23 | 15, 22 | mpbi 219 | . . . . 5 ⊢ (exp ↾ ℂ):ℂ⟶ℂ |
24 | feq23 5942 | . . . . . 6 ⊢ ((𝑆 = ℂ ∧ 𝑆 = ℂ) → ((exp ↾ ℂ):𝑆⟶𝑆 ↔ (exp ↾ ℂ):ℂ⟶ℂ)) | |
25 | 24 | anidms 675 | . . . . 5 ⊢ (𝑆 = ℂ → ((exp ↾ ℂ):𝑆⟶𝑆 ↔ (exp ↾ ℂ):ℂ⟶ℂ)) |
26 | 23, 25 | mpbiri 247 | . . . 4 ⊢ (𝑆 = ℂ → (exp ↾ ℂ):𝑆⟶𝑆) |
27 | reseq2 5312 | . . . . 5 ⊢ (𝑆 = ℂ → (exp ↾ 𝑆) = (exp ↾ ℂ)) | |
28 | 27 | feq1d 5943 | . . . 4 ⊢ (𝑆 = ℂ → ((exp ↾ 𝑆):𝑆⟶𝑆 ↔ (exp ↾ ℂ):𝑆⟶𝑆)) |
29 | 26, 28 | mpbird 246 | . . 3 ⊢ (𝑆 = ℂ → (exp ↾ 𝑆):𝑆⟶𝑆) |
30 | 14, 29 | jaoi 393 | . 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 195 ∨ wo 382 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 {cpr 4127 dom cdm 5038 ↾ cres 5040 Rel wrel 5043 ⟶wf 5800 –1-1→wf1 5801 ℂcc 9813 ℝcr 9814 ℝ+crp 11708 expce 14631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-ico 12052 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 |
This theorem is referenced by: (None) |
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