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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-mpo 5779 | . 2 | |
2 | df-mpq 7153 | . 2 | |
3 | 1st2nd2 6073 | . . . . . . . . . 10 | |
4 | 3 | eqeq1d 2148 | . . . . . . . . 9 |
5 | 1st2nd2 6073 | . . . . . . . . . 10 | |
6 | 5 | eqeq1d 2148 | . . . . . . . . 9 |
7 | 4, 6 | bi2anan9 595 | . . . . . . . 8 |
8 | 7 | anbi1d 460 | . . . . . . 7 |
9 | 8 | bicomd 140 | . . . . . 6 |
10 | 9 | 4exbidv 1842 | . . . . 5 |
11 | xp1st 6063 | . . . . . . 7 | |
12 | xp2nd 6064 | . . . . . . 7 | |
13 | 11, 12 | jca 304 | . . . . . 6 |
14 | xp1st 6063 | . . . . . . 7 | |
15 | xp2nd 6064 | . . . . . . 7 | |
16 | 14, 15 | jca 304 | . . . . . 6 |
17 | simpll 518 | . . . . . . . . . 10 | |
18 | simprl 520 | . . . . . . . . . 10 | |
19 | 17, 18 | oveq12d 5792 | . . . . . . . . 9 |
20 | simplr 519 | . . . . . . . . . 10 | |
21 | simprr 521 | . . . . . . . . . 10 | |
22 | 20, 21 | oveq12d 5792 | . . . . . . . . 9 |
23 | 19, 22 | opeq12d 3713 | . . . . . . . 8 |
24 | 23 | eqeq2d 2151 | . . . . . . 7 |
25 | 24 | copsex4g 4169 | . . . . . 6 |
26 | 13, 16, 25 | syl2an 287 | . . . . 5 |
27 | 10, 26 | bitr3d 189 | . . . 4 |
28 | 27 | pm5.32i 449 | . . 3 |
29 | 28 | oprabbii 5826 | . 2 |
30 | 1, 2, 29 | 3eqtr4i 2170 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 cop 3530 cxp 4537 cfv 5123 (class class class)co 5774 coprab 5775 cmpo 5776 c1st 6036 c2nd 6037 cnpi 7080 cmi 7082 cmpq 7085 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-mpq 7153 |
This theorem is referenced by: mulpipqqs 7181 |
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