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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pnfinf | Structured version Visualization version GIF version |
Description: Plus infinity is an infinite for the completed real line, as any real number is infinitesimal compared to it. (Contributed by Thierry Arnoux, 1-Feb-2018.) |
Ref | Expression |
---|---|
pnfinf | ⊢ (𝐴 ∈ ℝ+ → 𝐴(⋘‘ℝ*𝑠)+∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpgt0 13026 | . 2 ⊢ (𝐴 ∈ ℝ+ → 0 < 𝐴) | |
2 | nnz 12617 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ) | |
3 | 2 | adantl 480 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ) |
4 | rpxr 13023 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ*) | |
5 | 4 | adantr 479 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ*) |
6 | xrsmulgzz 32846 | . . . . . 6 ⊢ ((𝑛 ∈ ℤ ∧ 𝐴 ∈ ℝ*) → (𝑛(.g‘ℝ*𝑠)𝐴) = (𝑛 ·e 𝐴)) | |
7 | 3, 5, 6 | syl2anc 582 | . . . . 5 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑛 ∈ ℕ) → (𝑛(.g‘ℝ*𝑠)𝐴) = (𝑛 ·e 𝐴)) |
8 | 3 | zred 12704 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ) |
9 | rpre 13022 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) | |
10 | 9 | adantr 479 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ) |
11 | rexmul 13290 | . . . . . . 7 ⊢ ((𝑛 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑛 ·e 𝐴) = (𝑛 · 𝐴)) | |
12 | remulcl 11230 | . . . . . . 7 ⊢ ((𝑛 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑛 · 𝐴) ∈ ℝ) | |
13 | 11, 12 | eqeltrd 2825 | . . . . . 6 ⊢ ((𝑛 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑛 ·e 𝐴) ∈ ℝ) |
14 | 8, 10, 13 | syl2anc 582 | . . . . 5 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑛 ∈ ℕ) → (𝑛 ·e 𝐴) ∈ ℝ) |
15 | 7, 14 | eqeltrd 2825 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑛 ∈ ℕ) → (𝑛(.g‘ℝ*𝑠)𝐴) ∈ ℝ) |
16 | ltpnf 13140 | . . . 4 ⊢ ((𝑛(.g‘ℝ*𝑠)𝐴) ∈ ℝ → (𝑛(.g‘ℝ*𝑠)𝐴) < +∞) | |
17 | 15, 16 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑛 ∈ ℕ) → (𝑛(.g‘ℝ*𝑠)𝐴) < +∞) |
18 | 17 | ralrimiva 3135 | . 2 ⊢ (𝐴 ∈ ℝ+ → ∀𝑛 ∈ ℕ (𝑛(.g‘ℝ*𝑠)𝐴) < +∞) |
19 | xrsex 21344 | . . . 4 ⊢ ℝ*𝑠 ∈ V | |
20 | pnfxr 11305 | . . . 4 ⊢ +∞ ∈ ℝ* | |
21 | xrsbas 21345 | . . . . 5 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
22 | xrs0 32843 | . . . . 5 ⊢ 0 = (0g‘ℝ*𝑠) | |
23 | eqid 2725 | . . . . 5 ⊢ (.g‘ℝ*𝑠) = (.g‘ℝ*𝑠) | |
24 | xrslt 32844 | . . . . 5 ⊢ < = (lt‘ℝ*𝑠) | |
25 | 21, 22, 23, 24 | isinftm 33002 | . . . 4 ⊢ ((ℝ*𝑠 ∈ V ∧ 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝐴(⋘‘ℝ*𝑠)+∞ ↔ (0 < 𝐴 ∧ ∀𝑛 ∈ ℕ (𝑛(.g‘ℝ*𝑠)𝐴) < +∞))) |
26 | 19, 20, 25 | mp3an13 1448 | . . 3 ⊢ (𝐴 ∈ ℝ* → (𝐴(⋘‘ℝ*𝑠)+∞ ↔ (0 < 𝐴 ∧ ∀𝑛 ∈ ℕ (𝑛(.g‘ℝ*𝑠)𝐴) < +∞))) |
27 | 4, 26 | syl 17 | . 2 ⊢ (𝐴 ∈ ℝ+ → (𝐴(⋘‘ℝ*𝑠)+∞ ↔ (0 < 𝐴 ∧ ∀𝑛 ∈ ℕ (𝑛(.g‘ℝ*𝑠)𝐴) < +∞))) |
28 | 1, 18, 27 | mpbir2and 711 | 1 ⊢ (𝐴 ∈ ℝ+ → 𝐴(⋘‘ℝ*𝑠)+∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3050 Vcvv 3461 class class class wbr 5149 ‘cfv 6549 (class class class)co 7419 ℝcr 11144 0cc0 11145 · cmul 11150 +∞cpnf 11282 ℝ*cxr 11284 < clt 11285 ℕcn 12250 ℤcz 12596 ℝ+crp 13014 ·e cxmu 13131 ℝ*𝑠cxrs 17501 .gcmg 19047 ⋘cinftm 32997 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-fz 13525 df-seq 14008 df-struct 17135 df-slot 17170 df-ndx 17182 df-base 17200 df-plusg 17265 df-mulr 17266 df-tset 17271 df-ple 17272 df-ds 17274 df-0g 17442 df-xrs 17503 df-plt 18341 df-minusg 18918 df-mulg 19048 df-inftm 32999 |
This theorem is referenced by: xrnarchi 33005 |
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