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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 6565 | . . 3 ~Q0 ~Q0 | |
2 | elxpi 4627 | . . . . . . 7 | |
3 | 2 | anim1i 338 | . . . . . 6 ~Q0 ~Q0 |
4 | 19.41vv 1896 | . . . . . 6 ~Q0 ~Q0 | |
5 | 3, 4 | sylibr 133 | . . . . 5 ~Q0 ~Q0 |
6 | simplr 525 | . . . . . . 7 ~Q0 | |
7 | simpr 109 | . . . . . . . 8 ~Q0 ~Q0 | |
8 | eceq1 6548 | . . . . . . . . 9 ~Q0 ~Q0 | |
9 | 8 | ad2antrr 485 | . . . . . . . 8 ~Q0 ~Q0 ~Q0 |
10 | 7, 9 | eqtrd 2203 | . . . . . . 7 ~Q0 ~Q0 |
11 | 6, 10 | jca 304 | . . . . . 6 ~Q0 ~Q0 |
12 | 11 | 2eximi 1594 | . . . . 5 ~Q0 ~Q0 |
13 | 5, 12 | syl 14 | . . . 4 ~Q0 ~Q0 |
14 | 13 | rexlimiva 2582 | . . 3 ~Q0 ~Q0 |
15 | 1, 14 | syl 14 | . 2 ~Q0 ~Q0 |
16 | df-nq0 7387 | . 2 Q0 ~Q0 | |
17 | 15, 16 | eleq2s 2265 | 1 Q0 ~Q0 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wex 1485 wcel 2141 wrex 2449 cop 3586 com 4574 cxp 4609 cec 6511 cqs 6512 cnpi 7234 ~Q0 ceq0 7248 Q0cnq0 7249 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-sn 3589 df-pr 3590 df-op 3592 df-br 3990 df-opab 4051 df-xp 4617 df-cnv 4619 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-ec 6515 df-qs 6519 df-nq0 7387 |
This theorem is referenced by: nqpnq0nq 7415 nq0m0r 7418 nq0a0 7419 nq02m 7427 |
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