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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xdivpnfrp | Structured version Visualization version GIF version |
Description: Plus infinity divided by a positive real number is plus infinity. (Contributed by Thierry Arnoux, 18-Dec-2016.) |
Ref | Expression |
---|---|
xdivpnfrp | ⊢ (𝐴 ∈ ℝ+ → (+∞ /𝑒 𝐴) = +∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rprene0 12034 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 𝐴 ≠ 0)) | |
2 | pnfxr 10276 | . . . . 5 ⊢ +∞ ∈ ℝ* | |
3 | 1, 2 | jctil 561 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → (+∞ ∈ ℝ* ∧ (𝐴 ∈ ℝ ∧ 𝐴 ≠ 0))) |
4 | 3anass 1081 | . . . 4 ⊢ ((+∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) ↔ (+∞ ∈ ℝ* ∧ (𝐴 ∈ ℝ ∧ 𝐴 ≠ 0))) | |
5 | 3, 4 | sylibr 224 | . . 3 ⊢ (𝐴 ∈ ℝ+ → (+∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ ∧ 𝐴 ≠ 0)) |
6 | xdivval 29928 | . . 3 ⊢ ((+∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (+∞ /𝑒 𝐴) = (℩𝑥 ∈ ℝ* (𝐴 ·e 𝑥) = +∞)) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (𝐴 ∈ ℝ+ → (+∞ /𝑒 𝐴) = (℩𝑥 ∈ ℝ* (𝐴 ·e 𝑥) = +∞)) |
8 | 2 | a1i 11 | . . 3 ⊢ (𝐴 ∈ ℝ+ → +∞ ∈ ℝ*) |
9 | xlemul2 12306 | . . . . . . 7 ⊢ ((+∞ ∈ ℝ* ∧ 𝑥 ∈ ℝ* ∧ 𝐴 ∈ ℝ+) → (+∞ ≤ 𝑥 ↔ (𝐴 ·e +∞) ≤ (𝐴 ·e 𝑥))) | |
10 | 2, 9 | mp3an1 1552 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ* ∧ 𝐴 ∈ ℝ+) → (+∞ ≤ 𝑥 ↔ (𝐴 ·e +∞) ≤ (𝐴 ·e 𝑥))) |
11 | 10 | ancoms 468 | . . . . 5 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑥 ∈ ℝ*) → (+∞ ≤ 𝑥 ↔ (𝐴 ·e +∞) ≤ (𝐴 ·e 𝑥))) |
12 | rpxr 12025 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ*) | |
13 | rpgt0 12029 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ+ → 0 < 𝐴) | |
14 | xmulpnf1 12289 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (𝐴 ·e +∞) = +∞) | |
15 | 12, 13, 14 | syl2anc 696 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ+ → (𝐴 ·e +∞) = +∞) |
16 | 15 | adantr 472 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑥 ∈ ℝ*) → (𝐴 ·e +∞) = +∞) |
17 | 16 | breq1d 4806 | . . . . 5 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑥 ∈ ℝ*) → ((𝐴 ·e +∞) ≤ (𝐴 ·e 𝑥) ↔ +∞ ≤ (𝐴 ·e 𝑥))) |
18 | 11, 17 | bitr2d 269 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑥 ∈ ℝ*) → (+∞ ≤ (𝐴 ·e 𝑥) ↔ +∞ ≤ 𝑥)) |
19 | xmulcl 12288 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝑥 ∈ ℝ*) → (𝐴 ·e 𝑥) ∈ ℝ*) | |
20 | 12, 19 | sylan 489 | . . . . 5 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑥 ∈ ℝ*) → (𝐴 ·e 𝑥) ∈ ℝ*) |
21 | xgepnf 12181 | . . . . 5 ⊢ ((𝐴 ·e 𝑥) ∈ ℝ* → (+∞ ≤ (𝐴 ·e 𝑥) ↔ (𝐴 ·e 𝑥) = +∞)) | |
22 | 20, 21 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑥 ∈ ℝ*) → (+∞ ≤ (𝐴 ·e 𝑥) ↔ (𝐴 ·e 𝑥) = +∞)) |
23 | xgepnf 12181 | . . . . 5 ⊢ (𝑥 ∈ ℝ* → (+∞ ≤ 𝑥 ↔ 𝑥 = +∞)) | |
24 | 23 | adantl 473 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑥 ∈ ℝ*) → (+∞ ≤ 𝑥 ↔ 𝑥 = +∞)) |
25 | 18, 22, 24 | 3bitr3d 298 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑥 ∈ ℝ*) → ((𝐴 ·e 𝑥) = +∞ ↔ 𝑥 = +∞)) |
26 | 8, 25 | riota5 6792 | . 2 ⊢ (𝐴 ∈ ℝ+ → (℩𝑥 ∈ ℝ* (𝐴 ·e 𝑥) = +∞) = +∞) |
27 | 7, 26 | eqtrd 2786 | 1 ⊢ (𝐴 ∈ ℝ+ → (+∞ /𝑒 𝐴) = +∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 ∧ w3a 1072 = wceq 1624 ∈ wcel 2131 ≠ wne 2924 class class class wbr 4796 ℩crio 6765 (class class class)co 6805 ℝcr 10119 0cc0 10120 +∞cpnf 10255 ℝ*cxr 10257 < clt 10258 ≤ cle 10259 ℝ+crp 12017 ·e cxmu 12130 /𝑒 cxdiv 29926 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-8 2133 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-sep 4925 ax-nul 4933 ax-pow 4984 ax-pr 5047 ax-un 7106 ax-cnex 10176 ax-resscn 10177 ax-1cn 10178 ax-icn 10179 ax-addcl 10180 ax-addrcl 10181 ax-mulcl 10182 ax-mulrcl 10183 ax-mulcom 10184 ax-addass 10185 ax-mulass 10186 ax-distr 10187 ax-i2m1 10188 ax-1ne0 10189 ax-1rid 10190 ax-rnegex 10191 ax-rrecex 10192 ax-cnre 10193 ax-pre-lttri 10194 ax-pre-lttrn 10195 ax-pre-ltadd 10196 ax-pre-mulgt0 10197 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ne 2925 df-nel 3028 df-ral 3047 df-rex 3048 df-reu 3049 df-rmo 3050 df-rab 3051 df-v 3334 df-sbc 3569 df-csb 3667 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-nul 4051 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-op 4320 df-uni 4581 df-iun 4666 df-br 4797 df-opab 4857 df-mpt 4874 df-id 5166 df-po 5179 df-so 5180 df-xp 5264 df-rel 5265 df-cnv 5266 df-co 5267 df-dm 5268 df-rn 5269 df-res 5270 df-ima 5271 df-iota 6004 df-fun 6043 df-fn 6044 df-f 6045 df-f1 6046 df-fo 6047 df-f1o 6048 df-fv 6049 df-riota 6766 df-ov 6808 df-oprab 6809 df-mpt2 6810 df-1st 7325 df-2nd 7326 df-er 7903 df-en 8114 df-dom 8115 df-sdom 8116 df-pnf 10260 df-mnf 10261 df-xr 10262 df-ltxr 10263 df-le 10264 df-sub 10452 df-neg 10453 df-div 10869 df-rp 12018 df-xneg 12131 df-xmul 12133 df-xdiv 29927 |
This theorem is referenced by: xrpxdivcld 29944 |
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