|   | Mathbox for Thierry Arnoux | < Previous  
      Next > Nearby theorems | |
| 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 13048 | . 2 ⊢ (𝐴 ∈ ℝ+ → 0 < 𝐴) | |
| 2 | nnz 12636 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ) | |
| 3 | 2 | adantl 481 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ) | 
| 4 | rpxr 13045 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ*) | |
| 5 | 4 | adantr 480 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ*) | 
| 6 | xrsmulgzz 33012 | . . . . . 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 12724 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ) | 
| 9 | rpre 13044 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) | |
| 10 | 9 | adantr 480 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ) | 
| 11 | rexmul 13314 | . . . . . . 7 ⊢ ((𝑛 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑛 ·e 𝐴) = (𝑛 · 𝐴)) | |
| 12 | remulcl 11241 | . . . . . . 7 ⊢ ((𝑛 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑛 · 𝐴) ∈ ℝ) | |
| 13 | 11, 12 | eqeltrd 2840 | . . . . . 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 2840 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑛 ∈ ℕ) → (𝑛(.g‘ℝ*𝑠)𝐴) ∈ ℝ) | 
| 16 | ltpnf 13163 | . . . 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 3145 | . 2 ⊢ (𝐴 ∈ ℝ+ → ∀𝑛 ∈ ℕ (𝑛(.g‘ℝ*𝑠)𝐴) < +∞) | 
| 19 | xrsex 21396 | . . . 4 ⊢ ℝ*𝑠 ∈ V | |
| 20 | pnfxr 11316 | . . . 4 ⊢ +∞ ∈ ℝ* | |
| 21 | xrsbas 21397 | . . . . 5 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
| 22 | xrs0 33009 | . . . . 5 ⊢ 0 = (0g‘ℝ*𝑠) | |
| 23 | eqid 2736 | . . . . 5 ⊢ (.g‘ℝ*𝑠) = (.g‘ℝ*𝑠) | |
| 24 | xrslt 33010 | . . . . 5 ⊢ < = (lt‘ℝ*𝑠) | |
| 25 | 21, 22, 23, 24 | isinftm 33189 | . . . 4 ⊢ ((ℝ*𝑠 ∈ V ∧ 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝐴(⋘‘ℝ*𝑠)+∞ ↔ (0 < 𝐴 ∧ ∀𝑛 ∈ ℕ (𝑛(.g‘ℝ*𝑠)𝐴) < +∞))) | 
| 26 | 19, 20, 25 | mp3an13 1453 | . . 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 713 | 1 ⊢ (𝐴 ∈ ℝ+ → 𝐴(⋘‘ℝ*𝑠)+∞) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∀wral 3060 Vcvv 3479 class class class wbr 5142 ‘cfv 6560 (class class class)co 7432 ℝcr 11155 0cc0 11156 · cmul 11161 +∞cpnf 11293 ℝ*cxr 11295 < clt 11296 ℕcn 12267 ℤcz 12615 ℝ+crp 13035 ·e cxmu 13154 ℝ*𝑠cxrs 17546 .gcmg 19086 ⋘cinftm 33184 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-fz 13549 df-seq 14044 df-struct 17185 df-slot 17220 df-ndx 17232 df-base 17249 df-plusg 17311 df-mulr 17312 df-tset 17317 df-ple 17318 df-ds 17320 df-0g 17487 df-xrs 17548 df-plt 18376 df-minusg 18956 df-mulg 19087 df-inftm 33186 | 
| This theorem is referenced by: xrnarchi 33192 | 
| Copyright terms: Public domain | W3C validator |