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Mirrors > Home > ILE Home > Th. List > nq0nn | Unicode version |
Description: Decomposition of a nonnegative fraction into numerator and denominator. (Contributed by Jim Kingdon, 24-Nov-2019.) |
Ref | Expression |
---|---|
nq0nn | Q0 ~Q0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elqsi 6474 | . . 3 ~Q0 ~Q0 | |
2 | elxpi 4550 | . . . . . . 7 | |
3 | 2 | anim1i 338 | . . . . . 6 ~Q0 ~Q0 |
4 | 19.41vv 1875 | . . . . . 6 ~Q0 ~Q0 | |
5 | 3, 4 | sylibr 133 | . . . . 5 ~Q0 ~Q0 |
6 | simplr 519 | . . . . . . 7 ~Q0 | |
7 | simpr 109 | . . . . . . . 8 ~Q0 ~Q0 | |
8 | eceq1 6457 | . . . . . . . . 9 ~Q0 ~Q0 | |
9 | 8 | ad2antrr 479 | . . . . . . . 8 ~Q0 ~Q0 ~Q0 |
10 | 7, 9 | eqtrd 2170 | . . . . . . 7 ~Q0 ~Q0 |
11 | 6, 10 | jca 304 | . . . . . 6 ~Q0 ~Q0 |
12 | 11 | 2eximi 1580 | . . . . 5 ~Q0 ~Q0 |
13 | 5, 12 | syl 14 | . . . 4 ~Q0 ~Q0 |
14 | 13 | rexlimiva 2542 | . . 3 ~Q0 ~Q0 |
15 | 1, 14 | syl 14 | . 2 ~Q0 ~Q0 |
16 | df-nq0 7226 | . 2 Q0 ~Q0 | |
17 | 15, 16 | eleq2s 2232 | 1 Q0 ~Q0 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wex 1468 wcel 1480 wrex 2415 cop 3525 com 4499 cxp 4532 cec 6420 cqs 6421 cnpi 7073 ~Q0 ceq0 7087 Q0cnq0 7088 |
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 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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-xp 4540 df-cnv 4542 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-ec 6424 df-qs 6428 df-nq0 7226 |
This theorem is referenced by: nqpnq0nq 7254 nq0m0r 7257 nq0a0 7258 nq02m 7266 |
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