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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 5899 | . . 3 | |
2 | df-mqqs 7265 | . . . 4 | |
3 | 2 | dmeqi 4786 | . . 3 |
4 | dmaddpqlem 7292 | . . . . . . . . 9 | |
5 | dmaddpqlem 7292 | . . . . . . . . 9 | |
6 | 4, 5 | anim12i 336 | . . . . . . . 8 |
7 | ee4anv 1914 | . . . . . . . 8 | |
8 | 6, 7 | sylibr 133 | . . . . . . 7 |
9 | enqex 7275 | . . . . . . . . . . . . . 14 | |
10 | ecexg 6481 | . . . . . . . . . . . . . 14 | |
11 | 9, 10 | ax-mp 5 | . . . . . . . . . . . . 13 |
12 | 11 | isseti 2720 | . . . . . . . . . . . 12 |
13 | ax-ia3 107 | . . . . . . . . . . . . 13 | |
14 | 13 | eximdv 1860 | . . . . . . . . . . . 12 |
15 | 12, 14 | mpi 15 | . . . . . . . . . . 11 |
16 | 15 | 2eximi 1581 | . . . . . . . . . 10 |
17 | exrot3 1670 | . . . . . . . . . 10 | |
18 | 16, 17 | sylibr 133 | . . . . . . . . 9 |
19 | 18 | 2eximi 1581 | . . . . . . . 8 |
20 | exrot3 1670 | . . . . . . . 8 | |
21 | 19, 20 | sylibr 133 | . . . . . . 7 |
22 | 8, 21 | syl 14 | . . . . . 6 |
23 | 22 | pm4.71i 389 | . . . . 5 |
24 | 19.42v 1886 | . . . . 5 | |
25 | 23, 24 | bitr4i 186 | . . . 4 |
26 | 25 | opabbii 4031 | . . 3 |
27 | 1, 3, 26 | 3eqtr4i 2188 | . 2 |
28 | df-xp 4591 | . 2 | |
29 | 27, 28 | eqtr4i 2181 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1335 wex 1472 wcel 2128 cvv 2712 cop 3563 copab 4024 cxp 4583 cdm 4585 (class class class)co 5821 coprab 5822 cec 6475 cmpq 7192 ceq 7194 cnq 7195 cmq 7198 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-iinf 4546 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-br 3966 df-opab 4026 df-iom 4549 df-xp 4591 df-cnv 4593 df-dm 4595 df-rn 4596 df-res 4597 df-ima 4598 df-oprab 5825 df-ec 6479 df-qs 6483 df-ni 7219 df-enq 7262 df-nqqs 7263 df-mqqs 7265 |
This theorem is referenced by: (None) |
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