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Mirrors > Home > ILE Home > Th. List > mulpipqqs | Unicode version |
Description: Multiplication of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.) |
Ref | Expression |
---|---|
mulpipqqs |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulclpi 7136 | . . . 4 | |
2 | mulclpi 7136 | . . . 4 | |
3 | opelxpi 4571 | . . . 4 | |
4 | 1, 2, 3 | syl2an 287 | . . 3 |
5 | 4 | an4s 577 | . 2 |
6 | mulclpi 7136 | . . . 4 | |
7 | mulclpi 7136 | . . . 4 | |
8 | opelxpi 4571 | . . . 4 | |
9 | 6, 7, 8 | syl2an 287 | . . 3 |
10 | 9 | an4s 577 | . 2 |
11 | mulclpi 7136 | . . . 4 | |
12 | mulclpi 7136 | . . . 4 | |
13 | opelxpi 4571 | . . . 4 | |
14 | 11, 12, 13 | syl2an 287 | . . 3 |
15 | 14 | an4s 577 | . 2 |
16 | enqex 7168 | . 2 | |
17 | enqer 7166 | . 2 | |
18 | df-enq 7155 | . 2 | |
19 | simpll 518 | . . . 4 | |
20 | simprr 521 | . . . 4 | |
21 | 19, 20 | oveq12d 5792 | . . 3 |
22 | simplr 519 | . . . 4 | |
23 | simprl 520 | . . . 4 | |
24 | 22, 23 | oveq12d 5792 | . . 3 |
25 | 21, 24 | eqeq12d 2154 | . 2 |
26 | simpll 518 | . . . 4 | |
27 | simprr 521 | . . . 4 | |
28 | 26, 27 | oveq12d 5792 | . . 3 |
29 | simplr 519 | . . . 4 | |
30 | simprl 520 | . . . 4 | |
31 | 29, 30 | oveq12d 5792 | . . 3 |
32 | 28, 31 | eqeq12d 2154 | . 2 |
33 | dfmpq2 7163 | . 2 | |
34 | simpll 518 | . . . 4 | |
35 | simprl 520 | . . . 4 | |
36 | 34, 35 | oveq12d 5792 | . . 3 |
37 | simplr 519 | . . . 4 | |
38 | simprr 521 | . . . 4 | |
39 | 37, 38 | oveq12d 5792 | . . 3 |
40 | 36, 39 | opeq12d 3713 | . 2 |
41 | simpll 518 | . . . 4 | |
42 | simprl 520 | . . . 4 | |
43 | 41, 42 | oveq12d 5792 | . . 3 |
44 | simplr 519 | . . . 4 | |
45 | simprr 521 | . . . 4 | |
46 | 44, 45 | oveq12d 5792 | . . 3 |
47 | 43, 46 | opeq12d 3713 | . 2 |
48 | simpll 518 | . . . 4 | |
49 | simprl 520 | . . . 4 | |
50 | 48, 49 | oveq12d 5792 | . . 3 |
51 | simplr 519 | . . . 4 | |
52 | simprr 521 | . . . 4 | |
53 | 51, 52 | oveq12d 5792 | . . 3 |
54 | 50, 53 | opeq12d 3713 | . 2 |
55 | df-mqqs 7158 | . 2 | |
56 | df-nqqs 7156 | . 2 | |
57 | mulcmpblnq 7176 | . 2 | |
58 | 5, 10, 15, 16, 17, 18, 25, 32, 33, 40, 47, 54, 55, 56, 57 | oviec 6535 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cop 3530 cxp 4537 (class class class)co 5774 cec 6427 cnpi 7080 cmi 7082 cmpq 7085 ceq 7087 cnq 7088 cmq 7091 |
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 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
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 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-oadd 6317 df-omul 6318 df-er 6429 df-ec 6431 df-qs 6435 df-ni 7112 df-mi 7114 df-mpq 7153 df-enq 7155 df-nqqs 7156 df-mqqs 7158 |
This theorem is referenced by: mulclnq 7184 mulcomnqg 7191 mulassnqg 7192 distrnqg 7195 mulidnq 7197 recexnq 7198 ltmnqg 7209 nqnq0m 7263 |
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