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Mirrors > Home > MPE Home > Th. List > divdenle | Structured version Visualization version GIF version |
Description: Reducing a quotient never increases the denominator. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
Ref | Expression |
---|---|
divdenle | β’ ((π΄ β β€ β§ π΅ β β) β (denomβ(π΄ / π΅)) β€ π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divnumden 16693 | . . 3 β’ ((π΄ β β€ β§ π΅ β β) β ((numerβ(π΄ / π΅)) = (π΄ / (π΄ gcd π΅)) β§ (denomβ(π΄ / π΅)) = (π΅ / (π΄ gcd π΅)))) | |
2 | 1 | simprd 495 | . 2 β’ ((π΄ β β€ β§ π΅ β β) β (denomβ(π΄ / π΅)) = (π΅ / (π΄ gcd π΅))) |
3 | simpl 482 | . . . . . 6 β’ ((π΄ β β€ β§ π΅ β β) β π΄ β β€) | |
4 | nnz 12583 | . . . . . . 7 β’ (π΅ β β β π΅ β β€) | |
5 | 4 | adantl 481 | . . . . . 6 β’ ((π΄ β β€ β§ π΅ β β) β π΅ β β€) |
6 | nnne0 12250 | . . . . . . . . 9 β’ (π΅ β β β π΅ β 0) | |
7 | 6 | neneqd 2939 | . . . . . . . 8 β’ (π΅ β β β Β¬ π΅ = 0) |
8 | 7 | adantl 481 | . . . . . . 7 β’ ((π΄ β β€ β§ π΅ β β) β Β¬ π΅ = 0) |
9 | 8 | intnand 488 | . . . . . 6 β’ ((π΄ β β€ β§ π΅ β β) β Β¬ (π΄ = 0 β§ π΅ = 0)) |
10 | gcdn0cl 16450 | . . . . . 6 β’ (((π΄ β β€ β§ π΅ β β€) β§ Β¬ (π΄ = 0 β§ π΅ = 0)) β (π΄ gcd π΅) β β) | |
11 | 3, 5, 9, 10 | syl21anc 835 | . . . . 5 β’ ((π΄ β β€ β§ π΅ β β) β (π΄ gcd π΅) β β) |
12 | 11 | nnge1d 12264 | . . . 4 β’ ((π΄ β β€ β§ π΅ β β) β 1 β€ (π΄ gcd π΅)) |
13 | 1red 11219 | . . . . 5 β’ ((π΄ β β€ β§ π΅ β β) β 1 β β) | |
14 | 0lt1 11740 | . . . . . 6 β’ 0 < 1 | |
15 | 14 | a1i 11 | . . . . 5 β’ ((π΄ β β€ β§ π΅ β β) β 0 < 1) |
16 | 11 | nnred 12231 | . . . . 5 β’ ((π΄ β β€ β§ π΅ β β) β (π΄ gcd π΅) β β) |
17 | 11 | nngt0d 12265 | . . . . 5 β’ ((π΄ β β€ β§ π΅ β β) β 0 < (π΄ gcd π΅)) |
18 | nnre 12223 | . . . . . 6 β’ (π΅ β β β π΅ β β) | |
19 | 18 | adantl 481 | . . . . 5 β’ ((π΄ β β€ β§ π΅ β β) β π΅ β β) |
20 | nngt0 12247 | . . . . . 6 β’ (π΅ β β β 0 < π΅) | |
21 | 20 | adantl 481 | . . . . 5 β’ ((π΄ β β€ β§ π΅ β β) β 0 < π΅) |
22 | lediv2 12108 | . . . . 5 β’ (((1 β β β§ 0 < 1) β§ ((π΄ gcd π΅) β β β§ 0 < (π΄ gcd π΅)) β§ (π΅ β β β§ 0 < π΅)) β (1 β€ (π΄ gcd π΅) β (π΅ / (π΄ gcd π΅)) β€ (π΅ / 1))) | |
23 | 13, 15, 16, 17, 19, 21, 22 | syl222anc 1383 | . . . 4 β’ ((π΄ β β€ β§ π΅ β β) β (1 β€ (π΄ gcd π΅) β (π΅ / (π΄ gcd π΅)) β€ (π΅ / 1))) |
24 | 12, 23 | mpbid 231 | . . 3 β’ ((π΄ β β€ β§ π΅ β β) β (π΅ / (π΄ gcd π΅)) β€ (π΅ / 1)) |
25 | nncn 12224 | . . . . 5 β’ (π΅ β β β π΅ β β) | |
26 | 25 | adantl 481 | . . . 4 β’ ((π΄ β β€ β§ π΅ β β) β π΅ β β) |
27 | 26 | div1d 11986 | . . 3 β’ ((π΄ β β€ β§ π΅ β β) β (π΅ / 1) = π΅) |
28 | 24, 27 | breqtrd 5167 | . 2 β’ ((π΄ β β€ β§ π΅ β β) β (π΅ / (π΄ gcd π΅)) β€ π΅) |
29 | 2, 28 | eqbrtrd 5163 | 1 β’ ((π΄ β β€ β§ π΅ β β) β (denomβ(π΄ / π΅)) β€ π΅) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 395 = wceq 1533 β wcel 2098 class class class wbr 5141 βcfv 6537 (class class class)co 7405 βcc 11110 βcr 11111 0cc0 11112 1c1 11113 < clt 11252 β€ cle 11253 / cdiv 11875 βcn 12216 β€cz 12562 gcd cgcd 16442 numercnumer 16678 denomcdenom 16679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-q 12937 df-rp 12981 df-fl 13763 df-mod 13841 df-seq 13973 df-exp 14033 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-dvds 16205 df-gcd 16443 df-numer 16680 df-denom 16681 |
This theorem is referenced by: qden1elz 16702 irrapxlem5 42142 |
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