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Mirrors > Home > ILE Home > Th. List > dmaddpqlem | Unicode version |
Description: Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 7187. (Contributed by Jim Kingdon, 15-Sep-2019.) |
Ref | Expression |
---|---|
dmaddpqlem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elqsi 6481 | . . 3 | |
2 | elxpi 4555 | . . . . . . . 8 | |
3 | simpl 108 | . . . . . . . . 9 | |
4 | 3 | 2eximi 1580 | . . . . . . . 8 |
5 | 2, 4 | syl 14 | . . . . . . 7 |
6 | 5 | anim1i 338 | . . . . . 6 |
7 | 19.41vv 1875 | . . . . . 6 | |
8 | 6, 7 | sylibr 133 | . . . . 5 |
9 | simpr 109 | . . . . . . 7 | |
10 | eceq1 6464 | . . . . . . . 8 | |
11 | 10 | adantr 274 | . . . . . . 7 |
12 | 9, 11 | eqtrd 2172 | . . . . . 6 |
13 | 12 | 2eximi 1580 | . . . . 5 |
14 | 8, 13 | syl 14 | . . . 4 |
15 | 14 | rexlimiva 2544 | . . 3 |
16 | 1, 15 | syl 14 | . 2 |
17 | df-nqqs 7156 | . 2 | |
18 | 16, 17 | eleq2s 2234 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wex 1468 wcel 1480 wrex 2417 cop 3530 cxp 4537 cec 6427 cqs 6428 cnpi 7080 ceq 7087 cnq 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 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-xp 4545 df-cnv 4547 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-ec 6431 df-qs 6435 df-nqqs 7156 |
This theorem is referenced by: dmaddpq 7187 dmmulpq 7188 |
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