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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 10392 | . . 3 ⊢ ℝ ⊆ ℂ | |
2 | 1re 10439 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
3 | resqcl 13305 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → (𝑦↑2) ∈ ℝ) | |
4 | readdcl 10418 | . . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ (𝑦↑2) ∈ ℝ) → (1 + (𝑦↑2)) ∈ ℝ) | |
5 | 2, 3, 4 | sylancr 578 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → (1 + (𝑦↑2)) ∈ ℝ) |
6 | 5 | recnd 10468 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → (1 + (𝑦↑2)) ∈ ℂ) |
7 | 2 | a1i 11 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → 1 ∈ ℝ) |
8 | 0lt1 10963 | . . . . . . . . . 10 ⊢ 0 < 1 | |
9 | 8 | a1i 11 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → 0 < 1) |
10 | sqge0 13316 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → 0 ≤ (𝑦↑2)) | |
11 | 7, 3, 9, 10 | addgtge0d 11015 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → 0 < (1 + (𝑦↑2))) |
12 | 0re 10441 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
13 | ltnle 10520 | . . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ (1 + (𝑦↑2)) ∈ ℝ) → (0 < (1 + (𝑦↑2)) ↔ ¬ (1 + (𝑦↑2)) ≤ 0)) | |
14 | 12, 5, 13 | sylancr 578 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → (0 < (1 + (𝑦↑2)) ↔ ¬ (1 + (𝑦↑2)) ≤ 0)) |
15 | 11, 14 | mpbid 224 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → ¬ (1 + (𝑦↑2)) ≤ 0) |
16 | mnfxr 10498 | . . . . . . . . 9 ⊢ -∞ ∈ ℝ* | |
17 | elioc2 12615 | . . . . . . . . 9 ⊢ ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ) → ((1 + (𝑦↑2)) ∈ (-∞(,]0) ↔ ((1 + (𝑦↑2)) ∈ ℝ ∧ -∞ < (1 + (𝑦↑2)) ∧ (1 + (𝑦↑2)) ≤ 0))) | |
18 | 16, 12, 17 | mp2an 679 | . . . . . . . 8 ⊢ ((1 + (𝑦↑2)) ∈ (-∞(,]0) ↔ ((1 + (𝑦↑2)) ∈ ℝ ∧ -∞ < (1 + (𝑦↑2)) ∧ (1 + (𝑦↑2)) ≤ 0)) |
19 | 18 | simp3bi 1127 | . . . . . . 7 ⊢ ((1 + (𝑦↑2)) ∈ (-∞(,]0) → (1 + (𝑦↑2)) ≤ 0) |
20 | 15, 19 | nsyl 138 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → ¬ (1 + (𝑦↑2)) ∈ (-∞(,]0)) |
21 | 6, 20 | eldifd 3840 | . . . . 5 ⊢ (𝑦 ∈ ℝ → (1 + (𝑦↑2)) ∈ (ℂ ∖ (-∞(,]0))) |
22 | atansopn.d | . . . . 5 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) | |
23 | 21, 22 | syl6eleqr 2877 | . . . 4 ⊢ (𝑦 ∈ ℝ → (1 + (𝑦↑2)) ∈ 𝐷) |
24 | 23 | rgen 3098 | . . 3 ⊢ ∀𝑦 ∈ ℝ (1 + (𝑦↑2)) ∈ 𝐷 |
25 | ssrab 3939 | . . 3 ⊢ (ℝ ⊆ {𝑦 ∈ ℂ ∣ (1 + (𝑦↑2)) ∈ 𝐷} ↔ (ℝ ⊆ ℂ ∧ ∀𝑦 ∈ ℝ (1 + (𝑦↑2)) ∈ 𝐷)) | |
26 | 1, 24, 25 | mpbir2an 698 | . 2 ⊢ ℝ ⊆ {𝑦 ∈ ℂ ∣ (1 + (𝑦↑2)) ∈ 𝐷} |
27 | atansopn.s | . 2 ⊢ 𝑆 = {𝑦 ∈ ℂ ∣ (1 + (𝑦↑2)) ∈ 𝐷} | |
28 | 26, 27 | sseqtr4i 3894 | 1 ⊢ ℝ ⊆ 𝑆 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 198 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 ∀wral 3088 {crab 3092 ∖ cdif 3826 ⊆ wss 3829 class class class wbr 4929 (class class class)co 6976 ℂcc 10333 ℝcr 10334 0cc0 10335 1c1 10336 + caddc 10338 -∞cmnf 10472 ℝ*cxr 10473 < clt 10474 ≤ cle 10475 2c2 11495 (,]cioc 12555 ↑cexp 13244 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-nn 11440 df-2 11503 df-n0 11708 df-z 11794 df-uz 12059 df-ioc 12559 df-seq 13185 df-exp 13245 |
This theorem is referenced by: leibpi 25222 |
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