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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 11212 | . . 3 ⊢ ℝ ⊆ ℂ | |
| 2 | 1re 11261 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
| 3 | resqcl 14164 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → (𝑦↑2) ∈ ℝ) | |
| 4 | readdcl 11238 | . . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ (𝑦↑2) ∈ ℝ) → (1 + (𝑦↑2)) ∈ ℝ) | |
| 5 | 2, 3, 4 | sylancr 587 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → (1 + (𝑦↑2)) ∈ ℝ) |
| 6 | 5 | recnd 11289 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → (1 + (𝑦↑2)) ∈ ℂ) |
| 7 | 2 | a1i 11 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → 1 ∈ ℝ) |
| 8 | 0lt1 11785 | . . . . . . . . . 10 ⊢ 0 < 1 | |
| 9 | 8 | a1i 11 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → 0 < 1) |
| 10 | sqge0 14176 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → 0 ≤ (𝑦↑2)) | |
| 11 | 7, 3, 9, 10 | addgtge0d 11837 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → 0 < (1 + (𝑦↑2))) |
| 12 | 0re 11263 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
| 13 | ltnle 11340 | . . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ (1 + (𝑦↑2)) ∈ ℝ) → (0 < (1 + (𝑦↑2)) ↔ ¬ (1 + (𝑦↑2)) ≤ 0)) | |
| 14 | 12, 5, 13 | sylancr 587 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → (0 < (1 + (𝑦↑2)) ↔ ¬ (1 + (𝑦↑2)) ≤ 0)) |
| 15 | 11, 14 | mpbid 232 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → ¬ (1 + (𝑦↑2)) ≤ 0) |
| 16 | mnfxr 11318 | . . . . . . . . 9 ⊢ -∞ ∈ ℝ* | |
| 17 | elioc2 13450 | . . . . . . . . 9 ⊢ ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ) → ((1 + (𝑦↑2)) ∈ (-∞(,]0) ↔ ((1 + (𝑦↑2)) ∈ ℝ ∧ -∞ < (1 + (𝑦↑2)) ∧ (1 + (𝑦↑2)) ≤ 0))) | |
| 18 | 16, 12, 17 | mp2an 692 | . . . . . . . 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 3962 | . . . . 5 ⊢ (𝑦 ∈ ℝ → (1 + (𝑦↑2)) ∈ (ℂ ∖ (-∞(,]0))) |
| 22 | atansopn.d | . . . . 5 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) | |
| 23 | 21, 22 | eleqtrrdi 2852 | . . . 4 ⊢ (𝑦 ∈ ℝ → (1 + (𝑦↑2)) ∈ 𝐷) |
| 24 | 23 | rgen 3063 | . . 3 ⊢ ∀𝑦 ∈ ℝ (1 + (𝑦↑2)) ∈ 𝐷 |
| 25 | ssrab 4073 | . . 3 ⊢ (ℝ ⊆ {𝑦 ∈ ℂ ∣ (1 + (𝑦↑2)) ∈ 𝐷} ↔ (ℝ ⊆ ℂ ∧ ∀𝑦 ∈ ℝ (1 + (𝑦↑2)) ∈ 𝐷)) | |
| 26 | 1, 24, 25 | mpbir2an 711 | . 2 ⊢ ℝ ⊆ {𝑦 ∈ ℂ ∣ (1 + (𝑦↑2)) ∈ 𝐷} |
| 27 | atansopn.s | . 2 ⊢ 𝑆 = {𝑦 ∈ ℂ ∣ (1 + (𝑦↑2)) ∈ 𝐷} | |
| 28 | 26, 27 | sseqtrri 4033 | 1 ⊢ ℝ ⊆ 𝑆 |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 206 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 {crab 3436 ∖ cdif 3948 ⊆ wss 3951 class class class wbr 5143 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 -∞cmnf 11293 ℝ*cxr 11294 < clt 11295 ≤ cle 11296 2c2 12321 (,]cioc 13388 ↑cexp 14102 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-ioc 13392 df-seq 14043 df-exp 14103 |
| This theorem is referenced by: leibpi 26985 |
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