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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 7156 | . 2 | |
2 | addpipqqs 7178 | . 2 | |
3 | addpipqqs 7178 | . 2 | |
4 | addpipqqs 7178 | . 2 | |
5 | addpipqqs 7178 | . 2 | |
6 | mulclpi 7136 | . . . . 5 | |
7 | 6 | ad2ant2rl 502 | . . . 4 |
8 | mulclpi 7136 | . . . . 5 | |
9 | 8 | ad2ant2lr 501 | . . . 4 |
10 | addclpi 7135 | . . . 4 | |
11 | 7, 9, 10 | syl2anc 408 | . . 3 |
12 | mulclpi 7136 | . . . 4 | |
13 | 12 | ad2ant2l 499 | . . 3 |
14 | 11, 13 | jca 304 | . 2 |
15 | mulclpi 7136 | . . . . 5 | |
16 | 15 | ad2ant2rl 502 | . . . 4 |
17 | mulclpi 7136 | . . . . 5 | |
18 | 17 | ad2ant2lr 501 | . . . 4 |
19 | addclpi 7135 | . . . 4 | |
20 | 16, 18, 19 | syl2anc 408 | . . 3 |
21 | mulclpi 7136 | . . . 4 | |
22 | 21 | ad2ant2l 499 | . . 3 |
23 | 20, 22 | jca 304 | . 2 |
24 | simp1l 1005 | . . . . 5 | |
25 | simp2r 1008 | . . . . . 6 | |
26 | simp3r 1010 | . . . . . 6 | |
27 | 25, 26, 21 | syl2anc 408 | . . . . 5 |
28 | mulclpi 7136 | . . . . 5 | |
29 | 24, 27, 28 | syl2anc 408 | . . . 4 |
30 | simp1r 1006 | . . . . 5 | |
31 | simp2l 1007 | . . . . . 6 | |
32 | 31, 26, 15 | syl2anc 408 | . . . . 5 |
33 | mulclpi 7136 | . . . . 5 | |
34 | 30, 32, 33 | syl2anc 408 | . . . 4 |
35 | simp3l 1009 | . . . . . 6 | |
36 | 25, 35, 17 | syl2anc 408 | . . . . 5 |
37 | mulclpi 7136 | . . . . 5 | |
38 | 30, 36, 37 | syl2anc 408 | . . . 4 |
39 | addasspig 7138 | . . . 4 | |
40 | 29, 34, 38, 39 | syl3anc 1216 | . . 3 |
41 | mulcompig 7139 | . . . . . 6 | |
42 | 41 | adantl 275 | . . . . 5 |
43 | distrpig 7141 | . . . . . . . 8 | |
44 | 43 | 3coml 1188 | . . . . . . 7 |
45 | addclpi 7135 | . . . . . . . . . 10 | |
46 | mulcompig 7139 | . . . . . . . . . 10 | |
47 | 45, 46 | sylan2 284 | . . . . . . . . 9 |
48 | 47 | ancoms 266 | . . . . . . . 8 |
49 | 48 | 3impa 1176 | . . . . . . 7 |
50 | mulcompig 7139 | . . . . . . . . . 10 | |
51 | 50 | ancoms 266 | . . . . . . . . 9 |
52 | 51 | 3adant2 1000 | . . . . . . . 8 |
53 | mulcompig 7139 | . . . . . . . . . 10 | |
54 | 53 | ancoms 266 | . . . . . . . . 9 |
55 | 54 | 3adant1 999 | . . . . . . . 8 |
56 | 52, 55 | oveq12d 5792 | . . . . . . 7 |
57 | 44, 49, 56 | 3eqtr3d 2180 | . . . . . 6 |
58 | 57 | adantl 275 | . . . . 5 |
59 | mulasspig 7140 | . . . . . 6 | |
60 | 59 | adantl 275 | . . . . 5 |
61 | mulclpi 7136 | . . . . . 6 | |
62 | 61 | adantl 275 | . . . . 5 |
63 | 42, 58, 60, 62, 24, 30, 25, 31, 26 | caovdilemd 5962 | . . . 4 |
64 | mulasspig 7140 | . . . . . . 7 | |
65 | 64 | 3adant1l 1208 | . . . . . 6 |
66 | 65 | 3adant2l 1210 | . . . . 5 |
67 | 66 | 3adant3r 1213 | . . . 4 |
68 | 63, 67 | oveq12d 5792 | . . 3 |
69 | distrpig 7141 | . . . . 5 | |
70 | 30, 32, 36, 69 | syl3anc 1216 | . . . 4 |
71 | 70 | oveq2d 5790 | . . 3 |
72 | 40, 68, 71 | 3eqtr4d 2182 | . 2 |
73 | mulasspig 7140 | . . . . 5 | |
74 | 73 | 3adant1l 1208 | . . . 4 |
75 | 74 | 3adant2l 1210 | . . 3 |
76 | 75 | 3adant3l 1212 | . 2 |
77 | 1, 2, 3, 4, 5, 14, 23, 72, 76 | ecoviass 6539 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wceq 1331 wcel 1480 (class class class)co 5774 cnpi 7080 cpli 7081 cmi 7082 ceq 7087 cnq 7088 cplq 7090 |
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 603 ax-in2 604 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-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 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-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-oadd 6317 df-omul 6318 df-er 6429 df-ec 6431 df-qs 6435 df-ni 7112 df-pli 7113 df-mi 7114 df-plpq 7152 df-enq 7155 df-nqqs 7156 df-plqqs 7157 |
This theorem is referenced by: ltaddnq 7215 addlocprlemeqgt 7340 addassprg 7387 ltexprlemloc 7415 ltexprlemrl 7418 ltexprlemru 7420 addcanprleml 7422 addcanprlemu 7423 cauappcvgprlemdisj 7459 cauappcvgprlemloc 7460 cauappcvgprlemladdfl 7463 cauappcvgprlemladdru 7464 cauappcvgprlemladdrl 7465 cauappcvgprlem1 7467 caucvgprlemloc 7483 caucvgprlemladdrl 7486 caucvgprprlemloccalc 7492 |
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