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Mirrors > Home > ILE Home > Th. List > appdivnq | Unicode version |
Description: Approximate division for positive rationals. Proposition 12.7 of [BauerTaylor], p. 55 (a special case where and are positive, as well as ). Our proof is simpler than the one in BauerTaylor because we have reciprocals. (Contributed by Jim Kingdon, 8-Dec-2019.) |
Ref | Expression |
---|---|
appdivnq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 | . . . 4 | |
2 | ltrelnq 7166 | . . . . . . . 8 | |
3 | 2 | brel 4586 | . . . . . . 7 |
4 | 3 | adantr 274 | . . . . . 6 |
5 | 4 | simpld 111 | . . . . 5 |
6 | 4 | simprd 113 | . . . . 5 |
7 | recclnq 7193 | . . . . . 6 | |
8 | 7 | adantl 275 | . . . . 5 |
9 | ltmnqg 7202 | . . . . 5 | |
10 | 5, 6, 8, 9 | syl3anc 1216 | . . . 4 |
11 | 1, 10 | mpbid 146 | . . 3 |
12 | ltbtwnnqq 7216 | . . 3 | |
13 | 11, 12 | sylib 121 | . 2 |
14 | 8 | adantr 274 | . . . . . . . . 9 |
15 | 5 | adantr 274 | . . . . . . . . 9 |
16 | mulclnq 7177 | . . . . . . . . 9 | |
17 | 14, 15, 16 | syl2anc 408 | . . . . . . . 8 |
18 | simpr 109 | . . . . . . . 8 | |
19 | simplr 519 | . . . . . . . 8 | |
20 | ltmnqg 7202 | . . . . . . . 8 | |
21 | 17, 18, 19, 20 | syl3anc 1216 | . . . . . . 7 |
22 | recidnq 7194 | . . . . . . . . . . 11 | |
23 | 22 | oveq1d 5782 | . . . . . . . . . 10 |
24 | 23 | ad2antlr 480 | . . . . . . . . 9 |
25 | mulassnqg 7185 | . . . . . . . . . 10 | |
26 | 19, 14, 15, 25 | syl3anc 1216 | . . . . . . . . 9 |
27 | 1nq 7167 | . . . . . . . . . . . 12 | |
28 | mulcomnqg 7184 | . . . . . . . . . . . 12 | |
29 | 27, 28 | mpan 420 | . . . . . . . . . . 11 |
30 | mulidnq 7190 | . . . . . . . . . . 11 | |
31 | 29, 30 | eqtrd 2170 | . . . . . . . . . 10 |
32 | 15, 31 | syl 14 | . . . . . . . . 9 |
33 | 24, 26, 32 | 3eqtr3d 2178 | . . . . . . . 8 |
34 | 33 | breq1d 3934 | . . . . . . 7 |
35 | 21, 34 | bitrd 187 | . . . . . 6 |
36 | 6 | adantr 274 | . . . . . . . . 9 |
37 | mulclnq 7177 | . . . . . . . . 9 | |
38 | 14, 36, 37 | syl2anc 408 | . . . . . . . 8 |
39 | ltmnqg 7202 | . . . . . . . 8 | |
40 | 18, 38, 19, 39 | syl3anc 1216 | . . . . . . 7 |
41 | 22 | oveq1d 5782 | . . . . . . . . . 10 |
42 | 41 | ad2antlr 480 | . . . . . . . . 9 |
43 | mulassnqg 7185 | . . . . . . . . . 10 | |
44 | 19, 14, 36, 43 | syl3anc 1216 | . . . . . . . . 9 |
45 | mulcomnqg 7184 | . . . . . . . . . . . 12 | |
46 | 27, 45 | mpan 420 | . . . . . . . . . . 11 |
47 | mulidnq 7190 | . . . . . . . . . . 11 | |
48 | 46, 47 | eqtrd 2170 | . . . . . . . . . 10 |
49 | 36, 48 | syl 14 | . . . . . . . . 9 |
50 | 42, 44, 49 | 3eqtr3d 2178 | . . . . . . . 8 |
51 | 50 | breq2d 3936 | . . . . . . 7 |
52 | 40, 51 | bitrd 187 | . . . . . 6 |
53 | 35, 52 | anbi12d 464 | . . . . 5 |
54 | mulcomnqg 7184 | . . . . . . . 8 | |
55 | 19, 18, 54 | syl2anc 408 | . . . . . . 7 |
56 | 55 | breq2d 3936 | . . . . . 6 |
57 | 55 | breq1d 3934 | . . . . . 6 |
58 | 56, 57 | anbi12d 464 | . . . . 5 |
59 | 53, 58 | bitrd 187 | . . . 4 |
60 | 59 | biimpd 143 | . . 3 |
61 | 60 | reximdva 2532 | . 2 |
62 | 13, 61 | mpd 13 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wrex 2415 class class class wbr 3924 cfv 5118 (class class class)co 5767 cnq 7081 c1q 7082 cmq 7084 crq 7085 cltq 7086 |
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 |
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-ral 2419 df-rex 2420 df-reu 2421 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-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-eprel 4206 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 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-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-1o 6306 df-oadd 6310 df-omul 6311 df-er 6422 df-ec 6424 df-qs 6428 df-ni 7105 df-pli 7106 df-mi 7107 df-lti 7108 df-plpq 7145 df-mpq 7146 df-enq 7148 df-nqqs 7149 df-plqqs 7150 df-mqqs 7151 df-1nqqs 7152 df-rq 7153 df-ltnqqs 7154 |
This theorem is referenced by: appdiv0nq 7365 mullocpr 7372 |
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