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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 12255 | . 2 ⊢ (𝐴 ∈ ℝ+ → 0 < 𝐴) | |
2 | nnz 11858 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ) | |
3 | 2 | adantl 482 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ) |
4 | rpxr 12252 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ*) | |
5 | 4 | adantr 481 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ*) |
6 | xrsmulgzz 30335 | . . . . . 6 ⊢ ((𝑛 ∈ ℤ ∧ 𝐴 ∈ ℝ*) → (𝑛(.g‘ℝ*𝑠)𝐴) = (𝑛 ·e 𝐴)) | |
7 | 3, 5, 6 | syl2anc 584 | . . . . 5 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑛 ∈ ℕ) → (𝑛(.g‘ℝ*𝑠)𝐴) = (𝑛 ·e 𝐴)) |
8 | 3 | zred 11941 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ) |
9 | rpre 12251 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) | |
10 | 9 | adantr 481 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ) |
11 | rexmul 12518 | . . . . . . 7 ⊢ ((𝑛 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑛 ·e 𝐴) = (𝑛 · 𝐴)) | |
12 | remulcl 10475 | . . . . . . 7 ⊢ ((𝑛 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑛 · 𝐴) ∈ ℝ) | |
13 | 11, 12 | eqeltrd 2885 | . . . . . 6 ⊢ ((𝑛 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑛 ·e 𝐴) ∈ ℝ) |
14 | 8, 10, 13 | syl2anc 584 | . . . . 5 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑛 ∈ ℕ) → (𝑛 ·e 𝐴) ∈ ℝ) |
15 | 7, 14 | eqeltrd 2885 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑛 ∈ ℕ) → (𝑛(.g‘ℝ*𝑠)𝐴) ∈ ℝ) |
16 | ltpnf 12369 | . . . 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 3151 | . 2 ⊢ (𝐴 ∈ ℝ+ → ∀𝑛 ∈ ℕ (𝑛(.g‘ℝ*𝑠)𝐴) < +∞) |
19 | xrsex 20246 | . . . 4 ⊢ ℝ*𝑠 ∈ V | |
20 | pnfxr 10548 | . . . 4 ⊢ +∞ ∈ ℝ* | |
21 | xrsbas 20247 | . . . . 5 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
22 | xrs0 30332 | . . . . 5 ⊢ 0 = (0g‘ℝ*𝑠) | |
23 | eqid 2797 | . . . . 5 ⊢ (.g‘ℝ*𝑠) = (.g‘ℝ*𝑠) | |
24 | xrslt 30333 | . . . . 5 ⊢ < = (lt‘ℝ*𝑠) | |
25 | 21, 22, 23, 24 | isinftm 30444 | . . . 4 ⊢ ((ℝ*𝑠 ∈ V ∧ 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝐴(⋘‘ℝ*𝑠)+∞ ↔ (0 < 𝐴 ∧ ∀𝑛 ∈ ℕ (𝑛(.g‘ℝ*𝑠)𝐴) < +∞))) |
26 | 19, 20, 25 | mp3an13 1444 | . . 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 709 | 1 ⊢ (𝐴 ∈ ℝ+ → 𝐴(⋘‘ℝ*𝑠)+∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1525 ∈ wcel 2083 ∀wral 3107 Vcvv 3440 class class class wbr 4968 ‘cfv 6232 (class class class)co 7023 ℝcr 10389 0cc0 10390 · cmul 10395 +∞cpnf 10525 ℝ*cxr 10527 < clt 10528 ℕcn 11492 ℤcz 11835 ℝ+crp 12243 ·e cxmu 12360 ℝ*𝑠cxrs 16606 .gcmg 17985 ⋘cinftm 30439 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-1st 7552 df-2nd 7553 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-oadd 7964 df-er 8146 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-nn 11493 df-2 11554 df-3 11555 df-4 11556 df-5 11557 df-6 11558 df-7 11559 df-8 11560 df-9 11561 df-n0 11752 df-z 11836 df-dec 11953 df-uz 12098 df-rp 12244 df-xneg 12361 df-xadd 12362 df-xmul 12363 df-fz 12747 df-seq 13224 df-struct 16318 df-ndx 16319 df-slot 16320 df-base 16322 df-plusg 16411 df-mulr 16412 df-tset 16417 df-ple 16418 df-ds 16420 df-0g 16548 df-xrs 16608 df-plt 17401 df-minusg 17869 df-mulg 17986 df-inftm 30441 |
This theorem is referenced by: xrnarchi 30447 |
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