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Mirrors > Home > MPE Home > Th. List > ressatans | Structured version Visualization version GIF version |
Description: The real number line is a subset of the domain of continuity of the arctangent. (Contributed by Mario Carneiro, 7-Apr-2015.) |
Ref | Expression |
---|---|
atansopn.d | ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) |
atansopn.s | ⊢ 𝑆 = {𝑦 ∈ ℂ ∣ (1 + (𝑦↑2)) ∈ 𝐷} |
Ref | Expression |
---|---|
ressatans | ⊢ ℝ ⊆ 𝑆 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-resscn 10199 | . . 3 ⊢ ℝ ⊆ ℂ | |
2 | 1re 10245 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
3 | resqcl 13138 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → (𝑦↑2) ∈ ℝ) | |
4 | readdcl 10225 | . . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ (𝑦↑2) ∈ ℝ) → (1 + (𝑦↑2)) ∈ ℝ) | |
5 | 2, 3, 4 | sylancr 575 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → (1 + (𝑦↑2)) ∈ ℝ) |
6 | 5 | recnd 10274 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → (1 + (𝑦↑2)) ∈ ℂ) |
7 | 0re 10246 | . . . . . . . . . 10 ⊢ 0 ∈ ℝ | |
8 | 7 | a1i 11 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → 0 ∈ ℝ) |
9 | 2 | a1i 11 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → 1 ∈ ℝ) |
10 | 0lt1 10756 | . . . . . . . . . 10 ⊢ 0 < 1 | |
11 | 10 | a1i 11 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → 0 < 1) |
12 | sqge0 13147 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ → 0 ≤ (𝑦↑2)) | |
13 | addge01 10744 | . . . . . . . . . . 11 ⊢ ((1 ∈ ℝ ∧ (𝑦↑2) ∈ ℝ) → (0 ≤ (𝑦↑2) ↔ 1 ≤ (1 + (𝑦↑2)))) | |
14 | 2, 3, 13 | sylancr 575 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ → (0 ≤ (𝑦↑2) ↔ 1 ≤ (1 + (𝑦↑2)))) |
15 | 12, 14 | mpbid 222 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → 1 ≤ (1 + (𝑦↑2))) |
16 | 8, 9, 5, 11, 15 | ltletrd 10403 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → 0 < (1 + (𝑦↑2))) |
17 | ltnle 10323 | . . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ (1 + (𝑦↑2)) ∈ ℝ) → (0 < (1 + (𝑦↑2)) ↔ ¬ (1 + (𝑦↑2)) ≤ 0)) | |
18 | 7, 5, 17 | sylancr 575 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → (0 < (1 + (𝑦↑2)) ↔ ¬ (1 + (𝑦↑2)) ≤ 0)) |
19 | 16, 18 | mpbid 222 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → ¬ (1 + (𝑦↑2)) ≤ 0) |
20 | mnfxr 10302 | . . . . . . . . 9 ⊢ -∞ ∈ ℝ* | |
21 | elioc2 12441 | . . . . . . . . 9 ⊢ ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ) → ((1 + (𝑦↑2)) ∈ (-∞(,]0) ↔ ((1 + (𝑦↑2)) ∈ ℝ ∧ -∞ < (1 + (𝑦↑2)) ∧ (1 + (𝑦↑2)) ≤ 0))) | |
22 | 20, 7, 21 | mp2an 672 | . . . . . . . 8 ⊢ ((1 + (𝑦↑2)) ∈ (-∞(,]0) ↔ ((1 + (𝑦↑2)) ∈ ℝ ∧ -∞ < (1 + (𝑦↑2)) ∧ (1 + (𝑦↑2)) ≤ 0)) |
23 | 22 | simp3bi 1141 | . . . . . . 7 ⊢ ((1 + (𝑦↑2)) ∈ (-∞(,]0) → (1 + (𝑦↑2)) ≤ 0) |
24 | 19, 23 | nsyl 137 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → ¬ (1 + (𝑦↑2)) ∈ (-∞(,]0)) |
25 | 6, 24 | eldifd 3734 | . . . . 5 ⊢ (𝑦 ∈ ℝ → (1 + (𝑦↑2)) ∈ (ℂ ∖ (-∞(,]0))) |
26 | atansopn.d | . . . . 5 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) | |
27 | 25, 26 | syl6eleqr 2861 | . . . 4 ⊢ (𝑦 ∈ ℝ → (1 + (𝑦↑2)) ∈ 𝐷) |
28 | 27 | rgen 3071 | . . 3 ⊢ ∀𝑦 ∈ ℝ (1 + (𝑦↑2)) ∈ 𝐷 |
29 | ssrab 3829 | . . 3 ⊢ (ℝ ⊆ {𝑦 ∈ ℂ ∣ (1 + (𝑦↑2)) ∈ 𝐷} ↔ (ℝ ⊆ ℂ ∧ ∀𝑦 ∈ ℝ (1 + (𝑦↑2)) ∈ 𝐷)) | |
30 | 1, 28, 29 | mpbir2an 690 | . 2 ⊢ ℝ ⊆ {𝑦 ∈ ℂ ∣ (1 + (𝑦↑2)) ∈ 𝐷} |
31 | atansopn.s | . 2 ⊢ 𝑆 = {𝑦 ∈ ℂ ∣ (1 + (𝑦↑2)) ∈ 𝐷} | |
32 | 30, 31 | sseqtr4i 3787 | 1 ⊢ ℝ ⊆ 𝑆 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 ∀wral 3061 {crab 3065 ∖ cdif 3720 ⊆ wss 3723 class class class wbr 4787 (class class class)co 6796 ℂcc 10140 ℝcr 10141 0cc0 10142 1c1 10143 + caddc 10145 -∞cmnf 10278 ℝ*cxr 10279 < clt 10280 ≤ cle 10281 2c2 11276 (,]cioc 12381 ↑cexp 13067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-cnex 10198 ax-resscn 10199 ax-1cn 10200 ax-icn 10201 ax-addcl 10202 ax-addrcl 10203 ax-mulcl 10204 ax-mulrcl 10205 ax-mulcom 10206 ax-addass 10207 ax-mulass 10208 ax-distr 10209 ax-i2m1 10210 ax-1ne0 10211 ax-1rid 10212 ax-rnegex 10213 ax-rrecex 10214 ax-cnre 10215 ax-pre-lttri 10216 ax-pre-lttrn 10217 ax-pre-ltadd 10218 ax-pre-mulgt0 10219 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-riota 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-om 7217 df-2nd 7320 df-wrecs 7563 df-recs 7625 df-rdg 7663 df-er 7900 df-en 8114 df-dom 8115 df-sdom 8116 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 df-sub 10474 df-neg 10475 df-nn 11227 df-2 11285 df-n0 11500 df-z 11585 df-uz 11894 df-ioc 12385 df-seq 13009 df-exp 13068 |
This theorem is referenced by: leibpi 24890 |
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