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Mirrors > Home > ILE Home > Th. List > mulassnqg | Unicode version |
Description: Multiplication of positive fractions is associative. (Contributed by Jim Kingdon, 17-Sep-2019.) |
Ref | Expression |
---|---|
mulassnqg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nqqs 7280 | . 2 | |
2 | mulpipqqs 7305 | . 2 | |
3 | mulpipqqs 7305 | . 2 | |
4 | mulpipqqs 7305 | . 2 | |
5 | mulpipqqs 7305 | . 2 | |
6 | mulclpi 7260 | . . . 4 | |
7 | 6 | ad2ant2r 501 | . . 3 |
8 | mulclpi 7260 | . . . 4 | |
9 | 8 | ad2ant2l 500 | . . 3 |
10 | 7, 9 | jca 304 | . 2 |
11 | mulclpi 7260 | . . . 4 | |
12 | 11 | ad2ant2r 501 | . . 3 |
13 | mulclpi 7260 | . . . 4 | |
14 | 13 | ad2ant2l 500 | . . 3 |
15 | 12, 14 | jca 304 | . 2 |
16 | mulasspig 7264 | . . . . 5 | |
17 | 16 | 3adant1r 1220 | . . . 4 |
18 | 17 | 3adant2r 1222 | . . 3 |
19 | 18 | 3adant3r 1224 | . 2 |
20 | mulasspig 7264 | . . . . 5 | |
21 | 20 | 3adant1l 1219 | . . . 4 |
22 | 21 | 3adant2l 1221 | . . 3 |
23 | 22 | 3adant3l 1223 | . 2 |
24 | 1, 2, 3, 4, 5, 10, 15, 19, 23 | ecoviass 6602 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 967 wceq 1342 wcel 2135 (class class class)co 5836 cnpi 7204 cmi 7206 ceq 7211 cnq 7212 cmq 7215 |
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-in1 604 ax-in2 605 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-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 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-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-oadd 6379 df-omul 6380 df-er 6492 df-ec 6494 df-qs 6498 df-ni 7236 df-mi 7238 df-mpq 7277 df-enq 7279 df-nqqs 7280 df-mqqs 7282 |
This theorem is referenced by: recmulnqg 7323 halfnqq 7342 prarloclemarch 7350 ltrnqg 7352 addnqprl 7461 addnqpru 7462 appdivnq 7495 mulnqprl 7500 mulnqpru 7501 mullocprlem 7502 mulassprg 7513 1idprl 7522 1idpru 7523 recexprlem1ssl 7565 recexprlem1ssu 7566 |
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