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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 7035. (Contributed by Jim Kingdon, 15-Sep-2019.) |
Ref | Expression |
---|---|
dmaddpqlem |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elqsi 6384 |
. . 3
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2 | elxpi 4483 |
. . . . . . . 8
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3 | simpl 108 |
. . . . . . . . 9
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4 | 3 | 2eximi 1544 |
. . . . . . . 8
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5 | 2, 4 | syl 14 |
. . . . . . 7
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6 | 5 | anim1i 334 |
. . . . . 6
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7 | 19.41vv 1838 |
. . . . . 6
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8 | 6, 7 | sylibr 133 |
. . . . 5
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9 | simpr 109 |
. . . . . . 7
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10 | eceq1 6367 |
. . . . . . . 8
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11 | 10 | adantr 271 |
. . . . . . 7
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12 | 9, 11 | eqtrd 2127 |
. . . . . 6
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13 | 12 | 2eximi 1544 |
. . . . 5
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14 | 8, 13 | syl 14 |
. . . 4
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15 | 14 | rexlimiva 2497 |
. . 3
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16 | 1, 15 | syl 14 |
. 2
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17 | df-nqqs 7004 |
. 2
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18 | 16, 17 | eleq2s 2189 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-v 2635 df-un 3017 df-in 3019 df-ss 3026 df-sn 3472 df-pr 3473 df-op 3475 df-br 3868 df-opab 3922 df-xp 4473 df-cnv 4475 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-ec 6334 df-qs 6338 df-nqqs 7004 |
This theorem is referenced by: dmaddpq 7035 dmmulpq 7036 |
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