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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 5845 | . . 3 | |
2 | df-mqqs 7151 | . . . 4 | |
3 | 2 | dmeqi 4735 | . . 3 |
4 | dmaddpqlem 7178 | . . . . . . . . 9 | |
5 | dmaddpqlem 7178 | . . . . . . . . 9 | |
6 | 4, 5 | anim12i 336 | . . . . . . . 8 |
7 | ee4anv 1904 | . . . . . . . 8 | |
8 | 6, 7 | sylibr 133 | . . . . . . 7 |
9 | enqex 7161 | . . . . . . . . . . . . . 14 | |
10 | ecexg 6426 | . . . . . . . . . . . . . 14 | |
11 | 9, 10 | ax-mp 5 | . . . . . . . . . . . . 13 |
12 | 11 | isseti 2689 | . . . . . . . . . . . 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 3990 | . . 3 |
27 | 1, 3, 26 | 3eqtr4i 2168 | . 2 |
28 | df-xp 4540 | . 2 | |
29 | 27, 28 | eqtr4i 2161 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1331 wex 1468 wcel 1480 cvv 2681 cop 3525 copab 3983 cxp 4532 cdm 4534 (class class class)co 5767 coprab 5768 cec 6420 cmpq 7078 ceq 7080 cnq 7081 cmq 7084 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-iinf 4497 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-iom 4500 df-xp 4540 df-cnv 4542 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-oprab 5771 df-ec 6424 df-qs 6428 df-ni 7105 df-enq 7148 df-nqqs 7149 df-mqqs 7151 |
This theorem is referenced by: (None) |
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