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Mirrors > Home > ILE Home > Th. List > nnmass | Unicode version |
Description: Multiplication of natural numbers is associative. Theorem 4K(4) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.) |
Ref | Expression |
---|---|
nnmass |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 5750 | . . . . . 6 | |
2 | oveq2 5750 | . . . . . . 7 | |
3 | 2 | oveq2d 5758 | . . . . . 6 |
4 | 1, 3 | eqeq12d 2132 | . . . . 5 |
5 | 4 | imbi2d 229 | . . . 4 |
6 | oveq2 5750 | . . . . . 6 | |
7 | oveq2 5750 | . . . . . . 7 | |
8 | 7 | oveq2d 5758 | . . . . . 6 |
9 | 6, 8 | eqeq12d 2132 | . . . . 5 |
10 | oveq2 5750 | . . . . . 6 | |
11 | oveq2 5750 | . . . . . . 7 | |
12 | 11 | oveq2d 5758 | . . . . . 6 |
13 | 10, 12 | eqeq12d 2132 | . . . . 5 |
14 | oveq2 5750 | . . . . . 6 | |
15 | oveq2 5750 | . . . . . . 7 | |
16 | 15 | oveq2d 5758 | . . . . . 6 |
17 | 14, 16 | eqeq12d 2132 | . . . . 5 |
18 | nnmcl 6345 | . . . . . . 7 | |
19 | nnm0 6339 | . . . . . . 7 | |
20 | 18, 19 | syl 14 | . . . . . 6 |
21 | nnm0 6339 | . . . . . . . 8 | |
22 | 21 | oveq2d 5758 | . . . . . . 7 |
23 | nnm0 6339 | . . . . . . 7 | |
24 | 22, 23 | sylan9eqr 2172 | . . . . . 6 |
25 | 20, 24 | eqtr4d 2153 | . . . . 5 |
26 | oveq1 5749 | . . . . . . . . 9 | |
27 | nnmsuc 6341 | . . . . . . . . . . . 12 | |
28 | 18, 27 | sylan 281 | . . . . . . . . . . 11 |
29 | 28 | 3impa 1161 | . . . . . . . . . 10 |
30 | nnmsuc 6341 | . . . . . . . . . . . . 13 | |
31 | 30 | 3adant1 984 | . . . . . . . . . . . 12 |
32 | 31 | oveq2d 5758 | . . . . . . . . . . 11 |
33 | nnmcl 6345 | . . . . . . . . . . . . . . . . 17 | |
34 | nndi 6350 | . . . . . . . . . . . . . . . . 17 | |
35 | 33, 34 | syl3an2 1235 | . . . . . . . . . . . . . . . 16 |
36 | 35 | 3exp 1165 | . . . . . . . . . . . . . . 15 |
37 | 36 | expd 256 | . . . . . . . . . . . . . 14 |
38 | 37 | com34 83 | . . . . . . . . . . . . 13 |
39 | 38 | pm2.43d 50 | . . . . . . . . . . . 12 |
40 | 39 | 3imp 1160 | . . . . . . . . . . 11 |
41 | 32, 40 | eqtrd 2150 | . . . . . . . . . 10 |
42 | 29, 41 | eqeq12d 2132 | . . . . . . . . 9 |
43 | 26, 42 | syl5ibr 155 | . . . . . . . 8 |
44 | 43 | 3exp 1165 | . . . . . . 7 |
45 | 44 | com3r 79 | . . . . . 6 |
46 | 45 | impd 252 | . . . . 5 |
47 | 9, 13, 17, 25, 46 | finds2 4485 | . . . 4 |
48 | 5, 47 | vtoclga 2726 | . . 3 |
49 | 48 | expdcom 1403 | . 2 |
50 | 49 | 3imp 1160 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 947 wceq 1316 wcel 1465 c0 3333 csuc 4257 com 4474 (class class class)co 5742 coa 6278 comu 6279 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-irdg 6235 df-oadd 6285 df-omul 6286 |
This theorem is referenced by: mulasspig 7108 enq0tr 7210 addcmpblnq0 7219 mulcmpblnq0 7220 mulcanenq0ec 7221 distrnq0 7235 addassnq0 7238 |
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