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Mirrors > Home > ILE Home > Th. List > dmmulpq | Unicode version |
Description: Domain of multiplication on positive fractions. (Contributed by NM, 24-Aug-1995.) |
Ref | Expression |
---|---|
dmmulpq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmoprab 5852 | . . 3 | |
2 | df-mqqs 7158 | . . . 4 | |
3 | 2 | dmeqi 4740 | . . 3 |
4 | dmaddpqlem 7185 | . . . . . . . . 9 | |
5 | dmaddpqlem 7185 | . . . . . . . . 9 | |
6 | 4, 5 | anim12i 336 | . . . . . . . 8 |
7 | ee4anv 1906 | . . . . . . . 8 | |
8 | 6, 7 | sylibr 133 | . . . . . . 7 |
9 | enqex 7168 | . . . . . . . . . . . . . 14 | |
10 | ecexg 6433 | . . . . . . . . . . . . . 14 | |
11 | 9, 10 | ax-mp 5 | . . . . . . . . . . . . 13 |
12 | 11 | isseti 2694 | . . . . . . . . . . . 12 |
13 | ax-ia3 107 | . . . . . . . . . . . . 13 | |
14 | 13 | eximdv 1852 | . . . . . . . . . . . 12 |
15 | 12, 14 | mpi 15 | . . . . . . . . . . 11 |
16 | 15 | 2eximi 1580 | . . . . . . . . . 10 |
17 | exrot3 1668 | . . . . . . . . . 10 | |
18 | 16, 17 | sylibr 133 | . . . . . . . . 9 |
19 | 18 | 2eximi 1580 | . . . . . . . 8 |
20 | exrot3 1668 | . . . . . . . 8 | |
21 | 19, 20 | sylibr 133 | . . . . . . 7 |
22 | 8, 21 | syl 14 | . . . . . 6 |
23 | 22 | pm4.71i 388 | . . . . 5 |
24 | 19.42v 1878 | . . . . 5 | |
25 | 23, 24 | bitr4i 186 | . . . 4 |
26 | 25 | opabbii 3995 | . . 3 |
27 | 1, 3, 26 | 3eqtr4i 2170 | . 2 |
28 | df-xp 4545 | . 2 | |
29 | 27, 28 | eqtr4i 2163 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1331 wex 1468 wcel 1480 cvv 2686 cop 3530 copab 3988 cxp 4537 cdm 4539 (class class class)co 5774 coprab 5775 cec 6427 cmpq 7085 ceq 7087 cnq 7088 cmq 7091 |
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-in1 603 ax-in2 604 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-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-iom 4505 df-xp 4545 df-cnv 4547 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-oprab 5778 df-ec 6431 df-qs 6435 df-ni 7112 df-enq 7155 df-nqqs 7156 df-mqqs 7158 |
This theorem is referenced by: (None) |
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