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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 5841 | . 2 | |
2 | df-mpq 7277 | . 2 | |
3 | 1st2nd2 6135 | . . . . . . . . . 10 | |
4 | 3 | eqeq1d 2173 | . . . . . . . . 9 |
5 | 1st2nd2 6135 | . . . . . . . . . 10 | |
6 | 5 | eqeq1d 2173 | . . . . . . . . 9 |
7 | 4, 6 | bi2anan9 596 | . . . . . . . 8 |
8 | 7 | anbi1d 461 | . . . . . . 7 |
9 | 8 | bicomd 140 | . . . . . 6 |
10 | 9 | 4exbidv 1857 | . . . . 5 |
11 | xp1st 6125 | . . . . . . 7 | |
12 | xp2nd 6126 | . . . . . . 7 | |
13 | 11, 12 | jca 304 | . . . . . 6 |
14 | xp1st 6125 | . . . . . . 7 | |
15 | xp2nd 6126 | . . . . . . 7 | |
16 | 14, 15 | jca 304 | . . . . . 6 |
17 | simpll 519 | . . . . . . . . . 10 | |
18 | simprl 521 | . . . . . . . . . 10 | |
19 | 17, 18 | oveq12d 5854 | . . . . . . . . 9 |
20 | simplr 520 | . . . . . . . . . 10 | |
21 | simprr 522 | . . . . . . . . . 10 | |
22 | 20, 21 | oveq12d 5854 | . . . . . . . . 9 |
23 | 19, 22 | opeq12d 3760 | . . . . . . . 8 |
24 | 23 | eqeq2d 2176 | . . . . . . 7 |
25 | 24 | copsex4g 4219 | . . . . . 6 |
26 | 13, 16, 25 | syl2an 287 | . . . . 5 |
27 | 10, 26 | bitr3d 189 | . . . 4 |
28 | 27 | pm5.32i 450 | . . 3 |
29 | 28 | oprabbii 5888 | . 2 |
30 | 1, 2, 29 | 3eqtr4i 2195 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1342 wex 1479 wcel 2135 cop 3573 cxp 4596 cfv 5182 (class class class)co 5836 coprab 5837 cmpo 5838 c1st 6098 c2nd 6099 cnpi 7204 cmi 7206 cmpq 7209 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-iota 5147 df-fun 5184 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-mpq 7277 |
This theorem is referenced by: mulpipqqs 7305 |
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