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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 5934 | . . 3 | |
2 | df-mqqs 7312 | . . . 4 | |
3 | 2 | dmeqi 4812 | . . 3 |
4 | dmaddpqlem 7339 | . . . . . . . . 9 | |
5 | dmaddpqlem 7339 | . . . . . . . . 9 | |
6 | 4, 5 | anim12i 336 | . . . . . . . 8 |
7 | ee4anv 1927 | . . . . . . . 8 | |
8 | 6, 7 | sylibr 133 | . . . . . . 7 |
9 | enqex 7322 | . . . . . . . . . . . . . 14 | |
10 | ecexg 6517 | . . . . . . . . . . . . . 14 | |
11 | 9, 10 | ax-mp 5 | . . . . . . . . . . . . 13 |
12 | 11 | isseti 2738 | . . . . . . . . . . . 12 |
13 | ax-ia3 107 | . . . . . . . . . . . . 13 | |
14 | 13 | eximdv 1873 | . . . . . . . . . . . 12 |
15 | 12, 14 | mpi 15 | . . . . . . . . . . 11 |
16 | 15 | 2eximi 1594 | . . . . . . . . . 10 |
17 | exrot3 1683 | . . . . . . . . . 10 | |
18 | 16, 17 | sylibr 133 | . . . . . . . . 9 |
19 | 18 | 2eximi 1594 | . . . . . . . 8 |
20 | exrot3 1683 | . . . . . . . 8 | |
21 | 19, 20 | sylibr 133 | . . . . . . 7 |
22 | 8, 21 | syl 14 | . . . . . 6 |
23 | 22 | pm4.71i 389 | . . . . 5 |
24 | 19.42v 1899 | . . . . 5 | |
25 | 23, 24 | bitr4i 186 | . . . 4 |
26 | 25 | opabbii 4056 | . . 3 |
27 | 1, 3, 26 | 3eqtr4i 2201 | . 2 |
28 | df-xp 4617 | . 2 | |
29 | 27, 28 | eqtr4i 2194 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1348 wex 1485 wcel 2141 cvv 2730 cop 3586 copab 4049 cxp 4609 cdm 4611 (class class class)co 5853 coprab 5854 cec 6511 cmpq 7239 ceq 7241 cnq 7242 cmq 7245 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-iom 4575 df-xp 4617 df-cnv 4619 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-oprab 5857 df-ec 6515 df-qs 6519 df-ni 7266 df-enq 7309 df-nqqs 7310 df-mqqs 7312 |
This theorem is referenced by: (None) |
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