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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 11067 | . . 3 ⊢ ℝ ⊆ ℂ | |
2 | 1re 11114 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
3 | resqcl 13984 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → (𝑦↑2) ∈ ℝ) | |
4 | readdcl 11093 | . . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ (𝑦↑2) ∈ ℝ) → (1 + (𝑦↑2)) ∈ ℝ) | |
5 | 2, 3, 4 | sylancr 588 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → (1 + (𝑦↑2)) ∈ ℝ) |
6 | 5 | recnd 11142 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → (1 + (𝑦↑2)) ∈ ℂ) |
7 | 2 | a1i 11 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → 1 ∈ ℝ) |
8 | 0lt1 11636 | . . . . . . . . . 10 ⊢ 0 < 1 | |
9 | 8 | a1i 11 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → 0 < 1) |
10 | sqge0 13995 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → 0 ≤ (𝑦↑2)) | |
11 | 7, 3, 9, 10 | addgtge0d 11688 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → 0 < (1 + (𝑦↑2))) |
12 | 0re 11116 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
13 | ltnle 11193 | . . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ (1 + (𝑦↑2)) ∈ ℝ) → (0 < (1 + (𝑦↑2)) ↔ ¬ (1 + (𝑦↑2)) ≤ 0)) | |
14 | 12, 5, 13 | sylancr 588 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → (0 < (1 + (𝑦↑2)) ↔ ¬ (1 + (𝑦↑2)) ≤ 0)) |
15 | 11, 14 | mpbid 231 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → ¬ (1 + (𝑦↑2)) ≤ 0) |
16 | mnfxr 11171 | . . . . . . . . 9 ⊢ -∞ ∈ ℝ* | |
17 | elioc2 13282 | . . . . . . . . 9 ⊢ ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ) → ((1 + (𝑦↑2)) ∈ (-∞(,]0) ↔ ((1 + (𝑦↑2)) ∈ ℝ ∧ -∞ < (1 + (𝑦↑2)) ∧ (1 + (𝑦↑2)) ≤ 0))) | |
18 | 16, 12, 17 | mp2an 691 | . . . . . . . 8 ⊢ ((1 + (𝑦↑2)) ∈ (-∞(,]0) ↔ ((1 + (𝑦↑2)) ∈ ℝ ∧ -∞ < (1 + (𝑦↑2)) ∧ (1 + (𝑦↑2)) ≤ 0)) |
19 | 18 | simp3bi 1148 | . . . . . . 7 ⊢ ((1 + (𝑦↑2)) ∈ (-∞(,]0) → (1 + (𝑦↑2)) ≤ 0) |
20 | 15, 19 | nsyl 140 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → ¬ (1 + (𝑦↑2)) ∈ (-∞(,]0)) |
21 | 6, 20 | eldifd 3920 | . . . . 5 ⊢ (𝑦 ∈ ℝ → (1 + (𝑦↑2)) ∈ (ℂ ∖ (-∞(,]0))) |
22 | atansopn.d | . . . . 5 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) | |
23 | 21, 22 | eleqtrrdi 2850 | . . . 4 ⊢ (𝑦 ∈ ℝ → (1 + (𝑦↑2)) ∈ 𝐷) |
24 | 23 | rgen 3065 | . . 3 ⊢ ∀𝑦 ∈ ℝ (1 + (𝑦↑2)) ∈ 𝐷 |
25 | ssrab 4029 | . . 3 ⊢ (ℝ ⊆ {𝑦 ∈ ℂ ∣ (1 + (𝑦↑2)) ∈ 𝐷} ↔ (ℝ ⊆ ℂ ∧ ∀𝑦 ∈ ℝ (1 + (𝑦↑2)) ∈ 𝐷)) | |
26 | 1, 24, 25 | mpbir2an 710 | . 2 ⊢ ℝ ⊆ {𝑦 ∈ ℂ ∣ (1 + (𝑦↑2)) ∈ 𝐷} |
27 | atansopn.s | . 2 ⊢ 𝑆 = {𝑦 ∈ ℂ ∣ (1 + (𝑦↑2)) ∈ 𝐷} | |
28 | 26, 27 | sseqtrri 3980 | 1 ⊢ ℝ ⊆ 𝑆 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3063 {crab 3406 ∖ cdif 3906 ⊆ wss 3909 class class class wbr 5104 (class class class)co 7352 ℂcc 11008 ℝcr 11009 0cc0 11010 1c1 11011 + caddc 11013 -∞cmnf 11146 ℝ*cxr 11147 < clt 11148 ≤ cle 11149 2c2 12167 (,]cioc 13220 ↑cexp 13922 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 ax-cnex 11066 ax-resscn 11067 ax-1cn 11068 ax-icn 11069 ax-addcl 11070 ax-addrcl 11071 ax-mulcl 11072 ax-mulrcl 11073 ax-mulcom 11074 ax-addass 11075 ax-mulass 11076 ax-distr 11077 ax-i2m1 11078 ax-1ne0 11079 ax-1rid 11080 ax-rnegex 11081 ax-rrecex 11082 ax-cnre 11083 ax-pre-lttri 11084 ax-pre-lttrn 11085 ax-pre-ltadd 11086 ax-pre-mulgt0 11087 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7308 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7796 df-2nd 7915 df-frecs 8205 df-wrecs 8236 df-recs 8310 df-rdg 8349 df-er 8607 df-en 8843 df-dom 8844 df-sdom 8845 df-pnf 11150 df-mnf 11151 df-xr 11152 df-ltxr 11153 df-le 11154 df-sub 11346 df-neg 11347 df-nn 12113 df-2 12175 df-n0 12373 df-z 12459 df-uz 12723 df-ioc 13224 df-seq 13862 df-exp 13923 |
This theorem is referenced by: leibpi 26244 |
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