Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 9993 | . . 3 ⊢ ℝ ⊆ ℂ | |
2 | 1re 10039 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
3 | resqcl 12931 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → (𝑦↑2) ∈ ℝ) | |
4 | readdcl 10019 | . . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ (𝑦↑2) ∈ ℝ) → (1 + (𝑦↑2)) ∈ ℝ) | |
5 | 2, 3, 4 | sylancr 695 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → (1 + (𝑦↑2)) ∈ ℝ) |
6 | 5 | recnd 10068 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → (1 + (𝑦↑2)) ∈ ℂ) |
7 | 0re 10040 | . . . . . . . . . 10 ⊢ 0 ∈ ℝ | |
8 | 7 | a1i 11 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → 0 ∈ ℝ) |
9 | 2 | a1i 11 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → 1 ∈ ℝ) |
10 | 0lt1 10550 | . . . . . . . . . 10 ⊢ 0 < 1 | |
11 | 10 | a1i 11 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → 0 < 1) |
12 | sqge0 12940 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ → 0 ≤ (𝑦↑2)) | |
13 | addge01 10538 | . . . . . . . . . . 11 ⊢ ((1 ∈ ℝ ∧ (𝑦↑2) ∈ ℝ) → (0 ≤ (𝑦↑2) ↔ 1 ≤ (1 + (𝑦↑2)))) | |
14 | 2, 3, 13 | sylancr 695 | . . . . . . . . . 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 10197 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → 0 < (1 + (𝑦↑2))) |
17 | ltnle 10117 | . . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ (1 + (𝑦↑2)) ∈ ℝ) → (0 < (1 + (𝑦↑2)) ↔ ¬ (1 + (𝑦↑2)) ≤ 0)) | |
18 | 7, 5, 17 | sylancr 695 | . . . . . . . 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 10096 | . . . . . . . . 9 ⊢ -∞ ∈ ℝ* | |
21 | elioc2 12236 | . . . . . . . . 9 ⊢ ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ) → ((1 + (𝑦↑2)) ∈ (-∞(,]0) ↔ ((1 + (𝑦↑2)) ∈ ℝ ∧ -∞ < (1 + (𝑦↑2)) ∧ (1 + (𝑦↑2)) ≤ 0))) | |
22 | 20, 7, 21 | mp2an 708 | . . . . . . . 8 ⊢ ((1 + (𝑦↑2)) ∈ (-∞(,]0) ↔ ((1 + (𝑦↑2)) ∈ ℝ ∧ -∞ < (1 + (𝑦↑2)) ∧ (1 + (𝑦↑2)) ≤ 0)) |
23 | 22 | simp3bi 1078 | . . . . . . 7 ⊢ ((1 + (𝑦↑2)) ∈ (-∞(,]0) → (1 + (𝑦↑2)) ≤ 0) |
24 | 19, 23 | nsyl 135 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → ¬ (1 + (𝑦↑2)) ∈ (-∞(,]0)) |
25 | 6, 24 | eldifd 3585 | . . . . 5 ⊢ (𝑦 ∈ ℝ → (1 + (𝑦↑2)) ∈ (ℂ ∖ (-∞(,]0))) |
26 | atansopn.d | . . . . 5 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) | |
27 | 25, 26 | syl6eleqr 2712 | . . . 4 ⊢ (𝑦 ∈ ℝ → (1 + (𝑦↑2)) ∈ 𝐷) |
28 | 27 | rgen 2922 | . . 3 ⊢ ∀𝑦 ∈ ℝ (1 + (𝑦↑2)) ∈ 𝐷 |
29 | ssrab 3680 | . . 3 ⊢ (ℝ ⊆ {𝑦 ∈ ℂ ∣ (1 + (𝑦↑2)) ∈ 𝐷} ↔ (ℝ ⊆ ℂ ∧ ∀𝑦 ∈ ℝ (1 + (𝑦↑2)) ∈ 𝐷)) | |
30 | 1, 28, 29 | mpbir2an 955 | . 2 ⊢ ℝ ⊆ {𝑦 ∈ ℂ ∣ (1 + (𝑦↑2)) ∈ 𝐷} |
31 | atansopn.s | . 2 ⊢ 𝑆 = {𝑦 ∈ ℂ ∣ (1 + (𝑦↑2)) ∈ 𝐷} | |
32 | 30, 31 | sseqtr4i 3638 | 1 ⊢ ℝ ⊆ 𝑆 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 {crab 2916 ∖ cdif 3571 ⊆ wss 3574 class class class wbr 4653 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 -∞cmnf 10072 ℝ*cxr 10073 < clt 10074 ≤ cle 10075 2c2 11070 (,]cioc 12176 ↑cexp 12860 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-ioc 12180 df-seq 12802 df-exp 12861 |
This theorem is referenced by: leibpi 24669 |
Copyright terms: Public domain | W3C validator |