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 12395 | . 2 ⊢ (𝐴 ∈ ℝ+ → 0 < 𝐴) | |
2 | nnz 11998 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ) | |
3 | 2 | adantl 484 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ) |
4 | rpxr 12392 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ*) | |
5 | 4 | adantr 483 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ*) |
6 | xrsmulgzz 30660 | . . . . . 6 ⊢ ((𝑛 ∈ ℤ ∧ 𝐴 ∈ ℝ*) → (𝑛(.g‘ℝ*𝑠)𝐴) = (𝑛 ·e 𝐴)) | |
7 | 3, 5, 6 | syl2anc 586 | . . . . 5 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑛 ∈ ℕ) → (𝑛(.g‘ℝ*𝑠)𝐴) = (𝑛 ·e 𝐴)) |
8 | 3 | zred 12081 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ) |
9 | rpre 12391 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) | |
10 | 9 | adantr 483 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ) |
11 | rexmul 12658 | . . . . . . 7 ⊢ ((𝑛 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑛 ·e 𝐴) = (𝑛 · 𝐴)) | |
12 | remulcl 10616 | . . . . . . 7 ⊢ ((𝑛 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑛 · 𝐴) ∈ ℝ) | |
13 | 11, 12 | eqeltrd 2913 | . . . . . 6 ⊢ ((𝑛 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑛 ·e 𝐴) ∈ ℝ) |
14 | 8, 10, 13 | syl2anc 586 | . . . . 5 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑛 ∈ ℕ) → (𝑛 ·e 𝐴) ∈ ℝ) |
15 | 7, 14 | eqeltrd 2913 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑛 ∈ ℕ) → (𝑛(.g‘ℝ*𝑠)𝐴) ∈ ℝ) |
16 | ltpnf 12509 | . . . 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 3182 | . 2 ⊢ (𝐴 ∈ ℝ+ → ∀𝑛 ∈ ℕ (𝑛(.g‘ℝ*𝑠)𝐴) < +∞) |
19 | xrsex 20554 | . . . 4 ⊢ ℝ*𝑠 ∈ V | |
20 | pnfxr 10689 | . . . 4 ⊢ +∞ ∈ ℝ* | |
21 | xrsbas 20555 | . . . . 5 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
22 | xrs0 30657 | . . . . 5 ⊢ 0 = (0g‘ℝ*𝑠) | |
23 | eqid 2821 | . . . . 5 ⊢ (.g‘ℝ*𝑠) = (.g‘ℝ*𝑠) | |
24 | xrslt 30658 | . . . . 5 ⊢ < = (lt‘ℝ*𝑠) | |
25 | 21, 22, 23, 24 | isinftm 30805 | . . . 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 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 Vcvv 3495 class class class wbr 5059 ‘cfv 6350 (class class class)co 7150 ℝcr 10530 0cc0 10531 · cmul 10536 +∞cpnf 10666 ℝ*cxr 10668 < clt 10669 ℕcn 11632 ℤcz 11975 ℝ+crp 12383 ·e cxmu 12500 ℝ*𝑠cxrs 16767 .gcmg 18218 ⋘cinftm 30800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-fz 12887 df-seq 13364 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-plusg 16572 df-mulr 16573 df-tset 16578 df-ple 16579 df-ds 16581 df-0g 16709 df-xrs 16769 df-plt 17562 df-minusg 18101 df-mulg 18219 df-inftm 30802 |
This theorem is referenced by: xrnarchi 30808 |
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