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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 7312 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 6547 | . . 3 | |
2 | elxpi 4617 | . . . . . . 7 | |
3 | 2 | anim1i 338 | . . . . . 6 |
4 | 19.41vv 1890 | . . . . . 6 | |
5 | 3, 4 | sylibr 133 | . . . . 5 |
6 | simplr 520 | . . . . . . 7 | |
7 | simpr 109 | . . . . . . . 8 | |
8 | eceq1 6530 | . . . . . . . . 9 | |
9 | 8 | ad2antrr 480 | . . . . . . . 8 |
10 | 7, 9 | eqtrd 2197 | . . . . . . 7 |
11 | 6, 10 | jca 304 | . . . . . 6 |
12 | 11 | 2eximi 1588 | . . . . 5 |
13 | 5, 12 | syl 14 | . . . 4 |
14 | 13 | rexlimiva 2576 | . . 3 |
15 | 1, 14 | syl 14 | . 2 |
16 | df-nqqs 7283 | . 2 | |
17 | 15, 16 | eleq2s 2259 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1342 wex 1479 wcel 2135 wrex 2443 cop 3576 cxp 4599 cec 6493 cqs 6494 cnpi 7207 ceq 7214 cnq 7215 |
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-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2726 df-un 3118 df-in 3120 df-ss 3127 df-sn 3579 df-pr 3580 df-op 3582 df-br 3980 df-opab 4041 df-xp 4607 df-cnv 4609 df-dm 4611 df-rn 4612 df-res 4613 df-ima 4614 df-ec 6497 df-qs 6501 df-nqqs 7283 |
This theorem is referenced by: ltdcnq 7332 archnqq 7352 nqpnq0nq 7388 nqnq0a 7389 nqnq0m 7390 |
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