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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 5605 | . . 3 | |
2 | df-mqqs 6540 | . . . 4 | |
3 | 2 | dmeqi 4554 | . . 3 |
4 | dmaddpqlem 6567 | . . . . . . . . 9 | |
5 | dmaddpqlem 6567 | . . . . . . . . 9 | |
6 | 4, 5 | anim12i 331 | . . . . . . . 8 |
7 | ee4anv 1850 | . . . . . . . 8 | |
8 | 6, 7 | sylibr 132 | . . . . . . 7 |
9 | enqex 6550 | . . . . . . . . . . . . . 14 | |
10 | ecexg 6133 | . . . . . . . . . . . . . 14 | |
11 | 9, 10 | ax-mp 7 | . . . . . . . . . . . . 13 |
12 | 11 | isseti 2607 | . . . . . . . . . . . 12 |
13 | ax-ia3 106 | . . . . . . . . . . . . 13 | |
14 | 13 | eximdv 1801 | . . . . . . . . . . . 12 |
15 | 12, 14 | mpi 15 | . . . . . . . . . . 11 |
16 | 15 | 2eximi 1532 | . . . . . . . . . 10 |
17 | exrot3 1620 | . . . . . . . . . 10 | |
18 | 16, 17 | sylibr 132 | . . . . . . . . 9 |
19 | 18 | 2eximi 1532 | . . . . . . . 8 |
20 | exrot3 1620 | . . . . . . . 8 | |
21 | 19, 20 | sylibr 132 | . . . . . . 7 |
22 | 8, 21 | syl 14 | . . . . . 6 |
23 | 22 | pm4.71i 383 | . . . . 5 |
24 | 19.42v 1827 | . . . . 5 | |
25 | 23, 24 | bitr4i 185 | . . . 4 |
26 | 25 | opabbii 3845 | . . 3 |
27 | 1, 3, 26 | 3eqtr4i 2111 | . 2 |
28 | df-xp 4369 | . 2 | |
29 | 27, 28 | eqtr4i 2104 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 102 wceq 1284 wex 1421 wcel 1433 cvv 2601 cop 3401 copab 3838 cxp 4361 cdm 4363 (class class class)co 5532 coprab 5533 cec 6127 cmpq 6467 ceq 6469 cnq 6470 cmq 6473 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-iom 4332 df-xp 4369 df-cnv 4371 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-oprab 5536 df-ec 6131 df-qs 6135 df-ni 6494 df-enq 6537 df-nqqs 6538 df-mqqs 6540 |
This theorem is referenced by: (None) |
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