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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 6534 | . . 3 ~Q0 ~Q0 | |
2 | elxpi 4604 | . . . . . . 7 | |
3 | 2 | anim1i 338 | . . . . . 6 ~Q0 ~Q0 |
4 | 19.41vv 1883 | . . . . . 6 ~Q0 ~Q0 | |
5 | 3, 4 | sylibr 133 | . . . . 5 ~Q0 ~Q0 |
6 | simplr 520 | . . . . . . 7 ~Q0 | |
7 | simpr 109 | . . . . . . . 8 ~Q0 ~Q0 | |
8 | eceq1 6517 | . . . . . . . . 9 ~Q0 ~Q0 | |
9 | 8 | ad2antrr 480 | . . . . . . . 8 ~Q0 ~Q0 ~Q0 |
10 | 7, 9 | eqtrd 2190 | . . . . . . 7 ~Q0 ~Q0 |
11 | 6, 10 | jca 304 | . . . . . 6 ~Q0 ~Q0 |
12 | 11 | 2eximi 1581 | . . . . 5 ~Q0 ~Q0 |
13 | 5, 12 | syl 14 | . . . 4 ~Q0 ~Q0 |
14 | 13 | rexlimiva 2569 | . . 3 ~Q0 ~Q0 |
15 | 1, 14 | syl 14 | . 2 ~Q0 ~Q0 |
16 | df-nq0 7347 | . 2 Q0 ~Q0 | |
17 | 15, 16 | eleq2s 2252 | 1 Q0 ~Q0 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1335 wex 1472 wcel 2128 wrex 2436 cop 3564 com 4551 cxp 4586 cec 6480 cqs 6481 cnpi 7194 ~Q0 ceq0 7208 Q0cnq0 7209 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-sn 3567 df-pr 3568 df-op 3570 df-br 3968 df-opab 4028 df-xp 4594 df-cnv 4596 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-ec 6484 df-qs 6488 df-nq0 7347 |
This theorem is referenced by: nqpnq0nq 7375 nq0m0r 7378 nq0a0 7379 nq02m 7387 |
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