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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 7155. (Contributed by Jim Kingdon, 15-Sep-2019.) |
Ref | Expression |
---|---|
dmaddpqlem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elqsi 6449 | . . 3 | |
2 | elxpi 4525 | . . . . . . . 8 | |
3 | simpl 108 | . . . . . . . . 9 | |
4 | 3 | 2eximi 1565 | . . . . . . . 8 |
5 | 2, 4 | syl 14 | . . . . . . 7 |
6 | 5 | anim1i 338 | . . . . . 6 |
7 | 19.41vv 1859 | . . . . . 6 | |
8 | 6, 7 | sylibr 133 | . . . . 5 |
9 | simpr 109 | . . . . . . 7 | |
10 | eceq1 6432 | . . . . . . . 8 | |
11 | 10 | adantr 274 | . . . . . . 7 |
12 | 9, 11 | eqtrd 2150 | . . . . . 6 |
13 | 12 | 2eximi 1565 | . . . . 5 |
14 | 8, 13 | syl 14 | . . . 4 |
15 | 14 | rexlimiva 2521 | . . 3 |
16 | 1, 15 | syl 14 | . 2 |
17 | df-nqqs 7124 | . 2 | |
18 | 16, 17 | eleq2s 2212 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wex 1453 wcel 1465 wrex 2394 cop 3500 cxp 4507 cec 6395 cqs 6396 cnpi 7048 ceq 7055 cnq 7056 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-xp 4515 df-cnv 4517 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-ec 6399 df-qs 6403 df-nqqs 7124 |
This theorem is referenced by: dmaddpq 7155 dmmulpq 7156 |
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