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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 4589 | . 2 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 = ℝ ∨ 𝑆 = ℂ)) | |
3 | reeff1 15473 | . . . . . 6 ⊢ (exp ↾ ℝ):ℝ–1-1→ℝ+ | |
4 | f1f 6575 | . . . . . 6 ⊢ ((exp ↾ ℝ):ℝ–1-1→ℝ+ → (exp ↾ ℝ):ℝ⟶ℝ+) | |
5 | rpssre 12397 | . . . . . . 7 ⊢ ℝ+ ⊆ ℝ | |
6 | fss 6527 | . . . . . . 7 ⊢ (((exp ↾ ℝ):ℝ⟶ℝ+ ∧ ℝ+ ⊆ ℝ) → (exp ↾ ℝ):ℝ⟶ℝ) | |
7 | 5, 6 | mpan2 689 | . . . . . 6 ⊢ ((exp ↾ ℝ):ℝ⟶ℝ+ → (exp ↾ ℝ):ℝ⟶ℝ) |
8 | 3, 4, 7 | mp2b 10 | . . . . 5 ⊢ (exp ↾ ℝ):ℝ⟶ℝ |
9 | feq23 6498 | . . . . . 6 ⊢ ((𝑆 = ℝ ∧ 𝑆 = ℝ) → ((exp ↾ ℝ):𝑆⟶𝑆 ↔ (exp ↾ ℝ):ℝ⟶ℝ)) | |
10 | 9 | anidms 569 | . . . . 5 ⊢ (𝑆 = ℝ → ((exp ↾ ℝ):𝑆⟶𝑆 ↔ (exp ↾ ℝ):ℝ⟶ℝ)) |
11 | 8, 10 | mpbiri 260 | . . . 4 ⊢ (𝑆 = ℝ → (exp ↾ ℝ):𝑆⟶𝑆) |
12 | reseq2 5848 | . . . . 5 ⊢ (𝑆 = ℝ → (exp ↾ 𝑆) = (exp ↾ ℝ)) | |
13 | 12 | feq1d 6499 | . . . 4 ⊢ (𝑆 = ℝ → ((exp ↾ 𝑆):𝑆⟶𝑆 ↔ (exp ↾ ℝ):𝑆⟶𝑆)) |
14 | 11, 13 | mpbird 259 | . . 3 ⊢ (𝑆 = ℝ → (exp ↾ 𝑆):𝑆⟶𝑆) |
15 | eff 15435 | . . . . . 6 ⊢ exp:ℂ⟶ℂ | |
16 | frel 6519 | . . . . . . . . 9 ⊢ (exp:ℂ⟶ℂ → Rel exp) | |
17 | resdm 5897 | . . . . . . . . 9 ⊢ (Rel exp → (exp ↾ dom exp) = exp) | |
18 | 15, 16, 17 | mp2b 10 | . . . . . . . 8 ⊢ (exp ↾ dom exp) = exp |
19 | 15 | fdmi 6524 | . . . . . . . . 9 ⊢ dom exp = ℂ |
20 | 19 | reseq2i 5850 | . . . . . . . 8 ⊢ (exp ↾ dom exp) = (exp ↾ ℂ) |
21 | 18, 20 | eqtr3i 2846 | . . . . . . 7 ⊢ exp = (exp ↾ ℂ) |
22 | 21 | feq1i 6505 | . . . . . 6 ⊢ (exp:ℂ⟶ℂ ↔ (exp ↾ ℂ):ℂ⟶ℂ) |
23 | 15, 22 | mpbi 232 | . . . . 5 ⊢ (exp ↾ ℂ):ℂ⟶ℂ |
24 | feq23 6498 | . . . . . 6 ⊢ ((𝑆 = ℂ ∧ 𝑆 = ℂ) → ((exp ↾ ℂ):𝑆⟶𝑆 ↔ (exp ↾ ℂ):ℂ⟶ℂ)) | |
25 | 24 | anidms 569 | . . . . 5 ⊢ (𝑆 = ℂ → ((exp ↾ ℂ):𝑆⟶𝑆 ↔ (exp ↾ ℂ):ℂ⟶ℂ)) |
26 | 23, 25 | mpbiri 260 | . . . 4 ⊢ (𝑆 = ℂ → (exp ↾ ℂ):𝑆⟶𝑆) |
27 | reseq2 5848 | . . . . 5 ⊢ (𝑆 = ℂ → (exp ↾ 𝑆) = (exp ↾ ℂ)) | |
28 | 27 | feq1d 6499 | . . . 4 ⊢ (𝑆 = ℂ → ((exp ↾ 𝑆):𝑆⟶𝑆 ↔ (exp ↾ ℂ):𝑆⟶𝑆)) |
29 | 26, 28 | mpbird 259 | . . 3 ⊢ (𝑆 = ℂ → (exp ↾ 𝑆):𝑆⟶𝑆) |
30 | 14, 29 | jaoi 853 | . 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 208 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 {cpr 4569 dom cdm 5555 ↾ cres 5557 Rel wrel 5560 ⟶wf 6351 –1-1→wf1 6352 ℂcc 10535 ℝcr 10536 ℝ+crp 12390 expce 15415 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-ico 12745 df-fz 12894 df-fzo 13035 df-fl 13163 df-seq 13371 df-exp 13431 df-fac 13635 df-bc 13664 df-hash 13692 df-shft 14426 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-sum 15043 df-ef 15421 |
This theorem is referenced by: (None) |
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