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Mirrors > Home > ILE Home > Th. List > ledivp1 | GIF version |
Description: Less-than-or-equal-to and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.) (Contributed by NM, 28-Sep-2005.) |
Ref | Expression |
---|---|
ledivp1 | ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴 / (𝐵 + 1)) · 𝐵) ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprl 497 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → 𝐵 ∈ ℝ) | |
2 | peano2re 7244 | . . . 4 ⊢ (𝐵 ∈ ℝ → (𝐵 + 1) ∈ ℝ) | |
3 | 1, 2 | syl 14 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → (𝐵 + 1) ∈ ℝ) |
4 | simpll 495 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → 𝐴 ∈ ℝ) | |
5 | 0red 7120 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → 0 ∈ ℝ) | |
6 | simprr 498 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → 0 ≤ 𝐵) | |
7 | ltp1 7922 | . . . . . . . 8 ⊢ (𝐵 ∈ ℝ → 𝐵 < (𝐵 + 1)) | |
8 | 1, 7 | syl 14 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → 𝐵 < (𝐵 + 1)) |
9 | 5, 1, 3, 6, 8 | lelttrd 7234 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → 0 < (𝐵 + 1)) |
10 | 3, 9 | gt0ap0d 7728 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → (𝐵 + 1) # 0) |
11 | 4, 3, 10 | redivclapd 7920 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → (𝐴 / (𝐵 + 1)) ∈ ℝ) |
12 | simpl 107 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) | |
13 | divge0 7951 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ ((𝐵 + 1) ∈ ℝ ∧ 0 < (𝐵 + 1))) → 0 ≤ (𝐴 / (𝐵 + 1))) | |
14 | 12, 3, 9, 13 | syl12anc 1167 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → 0 ≤ (𝐴 / (𝐵 + 1))) |
15 | 11, 14 | jca 300 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴 / (𝐵 + 1)) ∈ ℝ ∧ 0 ≤ (𝐴 / (𝐵 + 1)))) |
16 | 1, 3, 8 | ltled 7228 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → 𝐵 ≤ (𝐵 + 1)) |
17 | lemul2a 7937 | . . 3 ⊢ (((𝐵 ∈ ℝ ∧ (𝐵 + 1) ∈ ℝ ∧ ((𝐴 / (𝐵 + 1)) ∈ ℝ ∧ 0 ≤ (𝐴 / (𝐵 + 1)))) ∧ 𝐵 ≤ (𝐵 + 1)) → ((𝐴 / (𝐵 + 1)) · 𝐵) ≤ ((𝐴 / (𝐵 + 1)) · (𝐵 + 1))) | |
18 | 1, 3, 15, 16, 17 | syl31anc 1172 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴 / (𝐵 + 1)) · 𝐵) ≤ ((𝐴 / (𝐵 + 1)) · (𝐵 + 1))) |
19 | 4 | recnd 7147 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → 𝐴 ∈ ℂ) |
20 | 3 | recnd 7147 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → (𝐵 + 1) ∈ ℂ) |
21 | 19, 20, 10 | divcanap1d 7878 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴 / (𝐵 + 1)) · (𝐵 + 1)) = 𝐴) |
22 | 18, 21 | breqtrd 3809 | 1 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴 / (𝐵 + 1)) · 𝐵) ≤ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∈ wcel 1433 class class class wbr 3785 (class class class)co 5532 ℝcr 6980 0cc0 6981 1c1 6982 + caddc 6984 · cmul 6986 < clt 7153 ≤ cle 7154 / cdiv 7760 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-po 4051 df-iso 4052 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 |
This theorem is referenced by: (None) |
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