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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elqsi 6601 |
. . 3
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2 | elxpi 4654 |
. . . . . . 7
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3 | 2 | anim1i 340 |
. . . . . 6
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4 | 19.41vv 1913 |
. . . . . 6
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5 | 3, 4 | sylibr 134 |
. . . . 5
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6 | simplr 528 |
. . . . . . 7
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7 | simpr 110 |
. . . . . . . 8
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8 | eceq1 6584 |
. . . . . . . . 9
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9 | 8 | ad2antrr 488 |
. . . . . . . 8
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10 | 7, 9 | eqtrd 2220 |
. . . . . . 7
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11 | 6, 10 | jca 306 |
. . . . . 6
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12 | 11 | 2eximi 1611 |
. . . . 5
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13 | 5, 12 | syl 14 |
. . . 4
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14 | 13 | rexlimiva 2599 |
. . 3
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15 | 1, 14 | syl 14 |
. 2
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16 | df-nq0 7438 |
. 2
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17 | 15, 16 | eleq2s 2282 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-un 3145 df-in 3147 df-ss 3154 df-sn 3610 df-pr 3611 df-op 3613 df-br 4016 df-opab 4077 df-xp 4644 df-cnv 4646 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-ec 6551 df-qs 6555 df-nq0 7438 |
This theorem is referenced by: nqpnq0nq 7466 nq0m0r 7469 nq0a0 7470 nq02m 7478 |
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