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| Mirrors > Home > MPE Home > Th. List > recp1lt1 | Structured version Visualization version GIF version | ||
| Description: Construct a number less than 1 from any nonnegative number. (Contributed by NM, 30-Dec-2005.) | 
| Ref | Expression | 
|---|---|
| recp1lt1 | ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 / (1 + 𝐴)) < 1) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ltp1 12107 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 < (𝐴 + 1)) | |
| 2 | recn 11245 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
| 3 | ax-1cn 11213 | . . . . . 6 ⊢ 1 ∈ ℂ | |
| 4 | addcom 11447 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 + 1) = (1 + 𝐴)) | |
| 5 | 2, 3, 4 | sylancl 586 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 + 1) = (1 + 𝐴)) | 
| 6 | 1, 5 | breqtrd 5169 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 < (1 + 𝐴)) | 
| 7 | 6 | adantr 480 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 < (1 + 𝐴)) | 
| 8 | 2 | adantr 480 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ ℂ) | 
| 9 | 1re 11261 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
| 10 | readdcl 11238 | . . . . . . 7 ⊢ ((1 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (1 + 𝐴) ∈ ℝ) | |
| 11 | 9, 10 | mpan 690 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (1 + 𝐴) ∈ ℝ) | 
| 12 | 11 | adantr 480 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (1 + 𝐴) ∈ ℝ) | 
| 13 | 12 | recnd 11289 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (1 + 𝐴) ∈ ℂ) | 
| 14 | 0lt1 11785 | . . . . . . 7 ⊢ 0 < 1 | |
| 15 | addgtge0 11751 | . . . . . . 7 ⊢ (((1 ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ (0 < 1 ∧ 0 ≤ 𝐴)) → 0 < (1 + 𝐴)) | |
| 16 | 14, 15 | mpanr1 703 | . . . . . 6 ⊢ (((1 ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ 0 ≤ 𝐴) → 0 < (1 + 𝐴)) | 
| 17 | 9, 16 | mpanl1 700 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 0 < (1 + 𝐴)) | 
| 18 | 17 | gt0ne0d 11827 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (1 + 𝐴) ≠ 0) | 
| 19 | 8, 13, 18 | divcan1d 12044 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((𝐴 / (1 + 𝐴)) · (1 + 𝐴)) = 𝐴) | 
| 20 | 11 | recnd 11289 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (1 + 𝐴) ∈ ℂ) | 
| 21 | 20 | mullidd 11279 | . . . 4 ⊢ (𝐴 ∈ ℝ → (1 · (1 + 𝐴)) = (1 + 𝐴)) | 
| 22 | 21 | adantr 480 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (1 · (1 + 𝐴)) = (1 + 𝐴)) | 
| 23 | 7, 19, 22 | 3brtr4d 5175 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((𝐴 / (1 + 𝐴)) · (1 + 𝐴)) < (1 · (1 + 𝐴))) | 
| 24 | simpl 482 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ ℝ) | |
| 25 | 24, 12, 18 | redivcld 12095 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 / (1 + 𝐴)) ∈ ℝ) | 
| 26 | ltmul1 12117 | . . . 4 ⊢ (((𝐴 / (1 + 𝐴)) ∈ ℝ ∧ 1 ∈ ℝ ∧ ((1 + 𝐴) ∈ ℝ ∧ 0 < (1 + 𝐴))) → ((𝐴 / (1 + 𝐴)) < 1 ↔ ((𝐴 / (1 + 𝐴)) · (1 + 𝐴)) < (1 · (1 + 𝐴)))) | |
| 27 | 9, 26 | mp3an2 1451 | . . 3 ⊢ (((𝐴 / (1 + 𝐴)) ∈ ℝ ∧ ((1 + 𝐴) ∈ ℝ ∧ 0 < (1 + 𝐴))) → ((𝐴 / (1 + 𝐴)) < 1 ↔ ((𝐴 / (1 + 𝐴)) · (1 + 𝐴)) < (1 · (1 + 𝐴)))) | 
| 28 | 25, 12, 17, 27 | syl12anc 837 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((𝐴 / (1 + 𝐴)) < 1 ↔ ((𝐴 / (1 + 𝐴)) · (1 + 𝐴)) < (1 · (1 + 𝐴)))) | 
| 29 | 23, 28 | mpbird 257 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 / (1 + 𝐴)) < 1) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 < clt 11295 ≤ cle 11296 / cdiv 11920 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 | 
| This theorem is referenced by: (None) | 
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