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Mirrors > Home > ILE Home > Th. List > nqpi | Unicode version |
Description: Decomposition of a positive fraction into numerator and denominator. Similar to dmaddpqlem 6567 but also shows that the numerator and denominator are positive integers. (Contributed by Jim Kingdon, 20-Sep-2019.) |
Ref | Expression |
---|---|
nqpi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elqsi 6181 | . . 3 | |
2 | elxpi 4379 | . . . . . . 7 | |
3 | 2 | anim1i 333 | . . . . . 6 |
4 | 19.41vv 1824 | . . . . . 6 | |
5 | 3, 4 | sylibr 132 | . . . . 5 |
6 | simplr 496 | . . . . . . 7 | |
7 | simpr 108 | . . . . . . . 8 | |
8 | eceq1 6164 | . . . . . . . . 9 | |
9 | 8 | ad2antrr 471 | . . . . . . . 8 |
10 | 7, 9 | eqtrd 2113 | . . . . . . 7 |
11 | 6, 10 | jca 300 | . . . . . 6 |
12 | 11 | 2eximi 1532 | . . . . 5 |
13 | 5, 12 | syl 14 | . . . 4 |
14 | 13 | rexlimiva 2472 | . . 3 |
15 | 1, 14 | syl 14 | . 2 |
16 | df-nqqs 6538 | . 2 | |
17 | 15, 16 | eleq2s 2173 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wex 1421 wcel 1433 wrex 2349 cop 3401 cxp 4361 cec 6127 cqs 6128 cnpi 6462 ceq 6469 cnq 6470 |
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-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-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-xp 4369 df-cnv 4371 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-ec 6131 df-qs 6135 df-nqqs 6538 |
This theorem is referenced by: ltdcnq 6587 archnqq 6607 nqpnq0nq 6643 nqnq0a 6644 nqnq0m 6645 |
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