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Mirrors > Home > ILE Home > Th. List > addassnqg | Unicode version |
Description: Addition of positive fractions is associative. (Contributed by Jim Kingdon, 16-Sep-2019.) |
Ref | Expression |
---|---|
addassnqg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nqqs 6446 | . 2 | |
2 | addpipqqs 6468 | . 2 | |
3 | addpipqqs 6468 | . 2 | |
4 | addpipqqs 6468 | . 2 | |
5 | addpipqqs 6468 | . 2 | |
6 | mulclpi 6426 | . . . . 5 | |
7 | 6 | ad2ant2rl 480 | . . . 4 |
8 | mulclpi 6426 | . . . . 5 | |
9 | 8 | ad2ant2lr 479 | . . . 4 |
10 | addclpi 6425 | . . . 4 | |
11 | 7, 9, 10 | syl2anc 391 | . . 3 |
12 | mulclpi 6426 | . . . 4 | |
13 | 12 | ad2ant2l 477 | . . 3 |
14 | 11, 13 | jca 290 | . 2 |
15 | mulclpi 6426 | . . . . 5 | |
16 | 15 | ad2ant2rl 480 | . . . 4 |
17 | mulclpi 6426 | . . . . 5 | |
18 | 17 | ad2ant2lr 479 | . . . 4 |
19 | addclpi 6425 | . . . 4 | |
20 | 16, 18, 19 | syl2anc 391 | . . 3 |
21 | mulclpi 6426 | . . . 4 | |
22 | 21 | ad2ant2l 477 | . . 3 |
23 | 20, 22 | jca 290 | . 2 |
24 | simp1l 928 | . . . . 5 | |
25 | simp2r 931 | . . . . . 6 | |
26 | simp3r 933 | . . . . . 6 | |
27 | 25, 26, 21 | syl2anc 391 | . . . . 5 |
28 | mulclpi 6426 | . . . . 5 | |
29 | 24, 27, 28 | syl2anc 391 | . . . 4 |
30 | simp1r 929 | . . . . 5 | |
31 | simp2l 930 | . . . . . 6 | |
32 | 31, 26, 15 | syl2anc 391 | . . . . 5 |
33 | mulclpi 6426 | . . . . 5 | |
34 | 30, 32, 33 | syl2anc 391 | . . . 4 |
35 | simp3l 932 | . . . . . 6 | |
36 | 25, 35, 17 | syl2anc 391 | . . . . 5 |
37 | mulclpi 6426 | . . . . 5 | |
38 | 30, 36, 37 | syl2anc 391 | . . . 4 |
39 | addasspig 6428 | . . . 4 | |
40 | 29, 34, 38, 39 | syl3anc 1135 | . . 3 |
41 | mulcompig 6429 | . . . . . 6 | |
42 | 41 | adantl 262 | . . . . 5 |
43 | distrpig 6431 | . . . . . . . 8 | |
44 | 43 | 3coml 1111 | . . . . . . 7 |
45 | addclpi 6425 | . . . . . . . . . 10 | |
46 | mulcompig 6429 | . . . . . . . . . 10 | |
47 | 45, 46 | sylan2 270 | . . . . . . . . 9 |
48 | 47 | ancoms 255 | . . . . . . . 8 |
49 | 48 | 3impa 1099 | . . . . . . 7 |
50 | mulcompig 6429 | . . . . . . . . . 10 | |
51 | 50 | ancoms 255 | . . . . . . . . 9 |
52 | 51 | 3adant2 923 | . . . . . . . 8 |
53 | mulcompig 6429 | . . . . . . . . . 10 | |
54 | 53 | ancoms 255 | . . . . . . . . 9 |
55 | 54 | 3adant1 922 | . . . . . . . 8 |
56 | 52, 55 | oveq12d 5530 | . . . . . . 7 |
57 | 44, 49, 56 | 3eqtr3d 2080 | . . . . . 6 |
58 | 57 | adantl 262 | . . . . 5 |
59 | mulasspig 6430 | . . . . . 6 | |
60 | 59 | adantl 262 | . . . . 5 |
61 | mulclpi 6426 | . . . . . 6 | |
62 | 61 | adantl 262 | . . . . 5 |
63 | 42, 58, 60, 62, 24, 30, 25, 31, 26 | caovdilemd 5692 | . . . 4 |
64 | mulasspig 6430 | . . . . . . 7 | |
65 | 64 | 3adant1l 1127 | . . . . . 6 |
66 | 65 | 3adant2l 1129 | . . . . 5 |
67 | 66 | 3adant3r 1132 | . . . 4 |
68 | 63, 67 | oveq12d 5530 | . . 3 |
69 | distrpig 6431 | . . . . 5 | |
70 | 30, 32, 36, 69 | syl3anc 1135 | . . . 4 |
71 | 70 | oveq2d 5528 | . . 3 |
72 | 40, 68, 71 | 3eqtr4d 2082 | . 2 |
73 | mulasspig 6430 | . . . . 5 | |
74 | 73 | 3adant1l 1127 | . . . 4 |
75 | 74 | 3adant2l 1129 | . . 3 |
76 | 75 | 3adant3l 1131 | . 2 |
77 | 1, 2, 3, 4, 5, 14, 23, 72, 76 | ecoviass 6216 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 w3a 885 wceq 1243 wcel 1393 (class class class)co 5512 cnpi 6370 cpli 6371 cmi 6372 ceq 6377 cnq 6378 cplq 6380 |
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-in1 544 ax-in2 545 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-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 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-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-plpq 6442 df-enq 6445 df-nqqs 6446 df-plqqs 6447 |
This theorem is referenced by: ltaddnq 6505 addlocprlemeqgt 6630 addassprg 6677 ltexprlemloc 6705 ltexprlemrl 6708 ltexprlemru 6710 addcanprleml 6712 addcanprlemu 6713 cauappcvgprlemdisj 6749 cauappcvgprlemloc 6750 cauappcvgprlemladdfl 6753 cauappcvgprlemladdru 6754 cauappcvgprlemladdrl 6755 cauappcvgprlem1 6757 caucvgprlemloc 6773 caucvgprlemladdrl 6776 caucvgprprlemloccalc 6782 |
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