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Mirrors > Home > ILE Home > Th. List > divdenle | GIF version |
Description: Reducing a quotient never increases the denominator. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
Ref | Expression |
---|---|
divdenle | ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (denom‘(𝐴 / 𝐵)) ≤ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divnumden 11863 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → ((numer‘(𝐴 / 𝐵)) = (𝐴 / (𝐴 gcd 𝐵)) ∧ (denom‘(𝐴 / 𝐵)) = (𝐵 / (𝐴 gcd 𝐵)))) | |
2 | 1 | simprd 113 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (denom‘(𝐴 / 𝐵)) = (𝐵 / (𝐴 gcd 𝐵))) |
3 | simpl 108 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝐴 ∈ ℤ) | |
4 | nnz 9066 | . . . . . . 7 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ) | |
5 | 4 | adantl 275 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℤ) |
6 | nnne0 8741 | . . . . . . . . 9 ⊢ (𝐵 ∈ ℕ → 𝐵 ≠ 0) | |
7 | 6 | neneqd 2327 | . . . . . . . 8 ⊢ (𝐵 ∈ ℕ → ¬ 𝐵 = 0) |
8 | 7 | adantl 275 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → ¬ 𝐵 = 0) |
9 | 8 | intnand 916 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → ¬ (𝐴 = 0 ∧ 𝐵 = 0)) |
10 | gcdn0cl 11640 | . . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (𝐴 gcd 𝐵) ∈ ℕ) | |
11 | 3, 5, 9, 10 | syl21anc 1215 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 gcd 𝐵) ∈ ℕ) |
12 | 11 | nnge1d 8756 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 1 ≤ (𝐴 gcd 𝐵)) |
13 | 1red 7774 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 1 ∈ ℝ) | |
14 | 0lt1 7882 | . . . . . 6 ⊢ 0 < 1 | |
15 | 14 | a1i 9 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 0 < 1) |
16 | 11 | nnred 8726 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 gcd 𝐵) ∈ ℝ) |
17 | 11 | nngt0d 8757 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 0 < (𝐴 gcd 𝐵)) |
18 | nnre 8720 | . . . . . 6 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ) | |
19 | 18 | adantl 275 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℝ) |
20 | nngt0 8738 | . . . . . 6 ⊢ (𝐵 ∈ ℕ → 0 < 𝐵) | |
21 | 20 | adantl 275 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 0 < 𝐵) |
22 | lediv2 8642 | . . . . 5 ⊢ (((1 ∈ ℝ ∧ 0 < 1) ∧ ((𝐴 gcd 𝐵) ∈ ℝ ∧ 0 < (𝐴 gcd 𝐵)) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → (1 ≤ (𝐴 gcd 𝐵) ↔ (𝐵 / (𝐴 gcd 𝐵)) ≤ (𝐵 / 1))) | |
23 | 13, 15, 16, 17, 19, 21, 22 | syl222anc 1232 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (1 ≤ (𝐴 gcd 𝐵) ↔ (𝐵 / (𝐴 gcd 𝐵)) ≤ (𝐵 / 1))) |
24 | 12, 23 | mpbid 146 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐵 / (𝐴 gcd 𝐵)) ≤ (𝐵 / 1)) |
25 | nncn 8721 | . . . . 5 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℂ) | |
26 | 25 | adantl 275 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℂ) |
27 | 26 | div1d 8533 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐵 / 1) = 𝐵) |
28 | 24, 27 | breqtrd 3949 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐵 / (𝐴 gcd 𝐵)) ≤ 𝐵) |
29 | 2, 28 | eqbrtrd 3945 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (denom‘(𝐴 / 𝐵)) ≤ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 class class class wbr 3924 ‘cfv 5118 (class class class)co 5767 ℂcc 7611 ℝcr 7612 0cc0 7613 1c1 7614 < clt 7793 ≤ cle 7794 / cdiv 8425 ℕcn 8713 ℤcz 9047 gcd cgcd 11624 numercnumer 11848 denomcdenom 11849 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 ax-arch 7732 ax-caucvg 7733 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-ilim 4286 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-frec 6281 df-sup 6864 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 df-inn 8714 df-2 8772 df-3 8773 df-4 8774 df-n0 8971 df-z 9048 df-uz 9320 df-q 9405 df-rp 9435 df-fz 9784 df-fzo 9913 df-fl 10036 df-mod 10089 df-seqfrec 10212 df-exp 10286 df-cj 10607 df-re 10608 df-im 10609 df-rsqrt 10763 df-abs 10764 df-dvds 11483 df-gcd 11625 df-numer 11850 df-denom 11851 |
This theorem is referenced by: qden1elz 11872 |
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