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Mirrors > Home > ILE Home > Th. List > dmaddpq | Unicode version |
Description: Domain of addition on positive fractions. (Contributed by NM, 24-Aug-1995.) |
Ref | Expression |
---|---|
dmaddpq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmoprab 5585 | . . 3 | |
2 | df-plqqs 6447 | . . . 4 | |
3 | 2 | dmeqi 4536 | . . 3 |
4 | dmaddpqlem 6475 | . . . . . . . . 9 | |
5 | dmaddpqlem 6475 | . . . . . . . . 9 | |
6 | 4, 5 | anim12i 321 | . . . . . . . 8 |
7 | ee4anv 1809 | . . . . . . . 8 | |
8 | 6, 7 | sylibr 137 | . . . . . . 7 |
9 | enqex 6458 | . . . . . . . . . . . . . 14 | |
10 | ecexg 6110 | . . . . . . . . . . . . . 14 | |
11 | 9, 10 | ax-mp 7 | . . . . . . . . . . . . 13 |
12 | 11 | isseti 2563 | . . . . . . . . . . . 12 |
13 | ax-ia3 101 | . . . . . . . . . . . . 13 | |
14 | 13 | eximdv 1760 | . . . . . . . . . . . 12 |
15 | 12, 14 | mpi 15 | . . . . . . . . . . 11 |
16 | 15 | 2eximi 1492 | . . . . . . . . . 10 |
17 | exrot3 1580 | . . . . . . . . . 10 | |
18 | 16, 17 | sylibr 137 | . . . . . . . . 9 |
19 | 18 | 2eximi 1492 | . . . . . . . 8 |
20 | exrot3 1580 | . . . . . . . 8 | |
21 | 19, 20 | sylibr 137 | . . . . . . 7 |
22 | 8, 21 | syl 14 | . . . . . 6 |
23 | 22 | pm4.71i 371 | . . . . 5 |
24 | 19.42v 1786 | . . . . 5 | |
25 | 23, 24 | bitr4i 176 | . . . 4 |
26 | 25 | opabbii 3824 | . . 3 |
27 | 1, 3, 26 | 3eqtr4i 2070 | . 2 |
28 | df-xp 4351 | . 2 | |
29 | 27, 28 | eqtr4i 2063 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 97 wceq 1243 wex 1381 wcel 1393 cvv 2557 cop 3378 copab 3817 cxp 4343 cdm 4345 (class class class)co 5512 coprab 5513 cec 6104 cplpq 6374 ceq 6377 cnq 6378 cplq 6380 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-br 3765 df-opab 3819 df-iom 4314 df-xp 4351 df-cnv 4353 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-oprab 5516 df-ec 6108 df-qs 6112 df-ni 6402 df-enq 6445 df-nqqs 6446 df-plqqs 6447 |
This theorem is referenced by: (None) |
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