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Mirrors > Home > ILE Home > Th. List > dfmpq2 | Unicode version |
Description: Alternate definition of pre-multiplication on positive fractions. (Contributed by Jim Kingdon, 13-Sep-2019.) |
Ref | Expression |
---|---|
dfmpq2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mpt2 5537 | . 2 | |
2 | df-mpq 6535 | . 2 | |
3 | 1st2nd2 5821 | . . . . . . . . . 10 | |
4 | 3 | eqeq1d 2089 | . . . . . . . . 9 |
5 | 1st2nd2 5821 | . . . . . . . . . 10 | |
6 | 5 | eqeq1d 2089 | . . . . . . . . 9 |
7 | 4, 6 | bi2anan9 570 | . . . . . . . 8 |
8 | 7 | anbi1d 452 | . . . . . . 7 |
9 | 8 | bicomd 139 | . . . . . 6 |
10 | 9 | 4exbidv 1791 | . . . . 5 |
11 | xp1st 5812 | . . . . . . 7 | |
12 | xp2nd 5813 | . . . . . . 7 | |
13 | 11, 12 | jca 300 | . . . . . 6 |
14 | xp1st 5812 | . . . . . . 7 | |
15 | xp2nd 5813 | . . . . . . 7 | |
16 | 14, 15 | jca 300 | . . . . . 6 |
17 | simpll 495 | . . . . . . . . . 10 | |
18 | simprl 497 | . . . . . . . . . 10 | |
19 | 17, 18 | oveq12d 5550 | . . . . . . . . 9 |
20 | simplr 496 | . . . . . . . . . 10 | |
21 | simprr 498 | . . . . . . . . . 10 | |
22 | 20, 21 | oveq12d 5550 | . . . . . . . . 9 |
23 | 19, 22 | opeq12d 3578 | . . . . . . . 8 |
24 | 23 | eqeq2d 2092 | . . . . . . 7 |
25 | 24 | copsex4g 4002 | . . . . . 6 |
26 | 13, 16, 25 | syl2an 283 | . . . . 5 |
27 | 10, 26 | bitr3d 188 | . . . 4 |
28 | 27 | pm5.32i 441 | . . 3 |
29 | 28 | oprabbii 5580 | . 2 |
30 | 1, 2, 29 | 3eqtr4i 2111 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 102 wb 103 wceq 1284 wex 1421 wcel 1433 cop 3401 cxp 4361 cfv 4922 (class class class)co 5532 coprab 5533 cmpt2 5534 c1st 5785 c2nd 5786 cnpi 6462 cmi 6464 cmpq 6467 |
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-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 |
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-sbc 2816 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-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-iota 4887 df-fun 4924 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-mpq 6535 |
This theorem is referenced by: mulpipqqs 6563 |
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