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Mirrors > Home > ILE Home > Th. List > dfmpq2 | Unicode version |
Description: Alternative definition of pre-multiplication on positive fractions. (Contributed by Jim Kingdon, 13-Sep-2019.) |
Ref | Expression |
---|---|
dfmpq2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mpt2 5517 | . 2 | |
2 | df-mpq 6443 | . 2 | |
3 | 1st2nd2 5801 | . . . . . . . . . 10 | |
4 | 3 | eqeq1d 2048 | . . . . . . . . 9 |
5 | 1st2nd2 5801 | . . . . . . . . . 10 | |
6 | 5 | eqeq1d 2048 | . . . . . . . . 9 |
7 | 4, 6 | bi2anan9 538 | . . . . . . . 8 |
8 | 7 | anbi1d 438 | . . . . . . 7 |
9 | 8 | bicomd 129 | . . . . . 6 |
10 | 9 | 4exbidv 1750 | . . . . 5 |
11 | xp1st 5792 | . . . . . . 7 | |
12 | xp2nd 5793 | . . . . . . 7 | |
13 | 11, 12 | jca 290 | . . . . . 6 |
14 | xp1st 5792 | . . . . . . 7 | |
15 | xp2nd 5793 | . . . . . . 7 | |
16 | 14, 15 | jca 290 | . . . . . 6 |
17 | simpll 481 | . . . . . . . . . 10 | |
18 | simprl 483 | . . . . . . . . . 10 | |
19 | 17, 18 | oveq12d 5530 | . . . . . . . . 9 |
20 | simplr 482 | . . . . . . . . . 10 | |
21 | simprr 484 | . . . . . . . . . 10 | |
22 | 20, 21 | oveq12d 5530 | . . . . . . . . 9 |
23 | 19, 22 | opeq12d 3557 | . . . . . . . 8 |
24 | 23 | eqeq2d 2051 | . . . . . . 7 |
25 | 24 | copsex4g 3984 | . . . . . 6 |
26 | 13, 16, 25 | syl2an 273 | . . . . 5 |
27 | 10, 26 | bitr3d 179 | . . . 4 |
28 | 27 | pm5.32i 427 | . . 3 |
29 | 28 | oprabbii 5560 | . 2 |
30 | 1, 2, 29 | 3eqtr4i 2070 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 97 wb 98 wceq 1243 wex 1381 wcel 1393 cop 3378 cxp 4343 cfv 4902 (class class class)co 5512 coprab 5513 cmpt2 5514 c1st 5765 c2nd 5766 cnpi 6370 cmi 6372 cmpq 6375 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-iota 4867 df-fun 4904 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-mpq 6443 |
This theorem is referenced by: mulpipqqs 6471 |
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