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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 5520 | . . . . . 6 | |
2 | oveq2 5520 | . . . . . . 7 | |
3 | 2 | oveq2d 5528 | . . . . . 6 |
4 | 1, 3 | eqeq12d 2054 | . . . . 5 |
5 | 4 | imbi2d 219 | . . . 4 |
6 | oveq2 5520 | . . . . . 6 | |
7 | oveq2 5520 | . . . . . . 7 | |
8 | 7 | oveq2d 5528 | . . . . . 6 |
9 | 6, 8 | eqeq12d 2054 | . . . . 5 |
10 | oveq2 5520 | . . . . . 6 | |
11 | oveq2 5520 | . . . . . . 7 | |
12 | 11 | oveq2d 5528 | . . . . . 6 |
13 | 10, 12 | eqeq12d 2054 | . . . . 5 |
14 | oveq2 5520 | . . . . . 6 | |
15 | oveq2 5520 | . . . . . . 7 | |
16 | 15 | oveq2d 5528 | . . . . . 6 |
17 | 14, 16 | eqeq12d 2054 | . . . . 5 |
18 | nnmcl 6060 | . . . . . . 7 | |
19 | nnm0 6054 | . . . . . . 7 | |
20 | 18, 19 | syl 14 | . . . . . 6 |
21 | nnm0 6054 | . . . . . . . 8 | |
22 | 21 | oveq2d 5528 | . . . . . . 7 |
23 | nnm0 6054 | . . . . . . 7 | |
24 | 22, 23 | sylan9eqr 2094 | . . . . . 6 |
25 | 20, 24 | eqtr4d 2075 | . . . . 5 |
26 | oveq1 5519 | . . . . . . . . 9 | |
27 | nnmsuc 6056 | . . . . . . . . . . . 12 | |
28 | 18, 27 | sylan 267 | . . . . . . . . . . 11 |
29 | 28 | 3impa 1099 | . . . . . . . . . 10 |
30 | nnmsuc 6056 | . . . . . . . . . . . . 13 | |
31 | 30 | 3adant1 922 | . . . . . . . . . . . 12 |
32 | 31 | oveq2d 5528 | . . . . . . . . . . 11 |
33 | nnmcl 6060 | . . . . . . . . . . . . . . . . 17 | |
34 | nndi 6065 | . . . . . . . . . . . . . . . . 17 | |
35 | 33, 34 | syl3an2 1169 | . . . . . . . . . . . . . . . 16 |
36 | 35 | 3exp 1103 | . . . . . . . . . . . . . . 15 |
37 | 36 | expd 245 | . . . . . . . . . . . . . 14 |
38 | 37 | com34 77 | . . . . . . . . . . . . 13 |
39 | 38 | pm2.43d 44 | . . . . . . . . . . . 12 |
40 | 39 | 3imp 1098 | . . . . . . . . . . 11 |
41 | 32, 40 | eqtrd 2072 | . . . . . . . . . 10 |
42 | 29, 41 | eqeq12d 2054 | . . . . . . . . 9 |
43 | 26, 42 | syl5ibr 145 | . . . . . . . 8 |
44 | 43 | 3exp 1103 | . . . . . . 7 |
45 | 44 | com3r 73 | . . . . . 6 |
46 | 45 | impd 242 | . . . . 5 |
47 | 9, 13, 17, 25, 46 | finds2 4324 | . . . 4 |
48 | 5, 47 | vtoclga 2619 | . . 3 |
49 | 48 | expdcom 1331 | . 2 |
50 | 49 | 3imp 1098 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 w3a 885 wceq 1243 wcel 1393 c0 3224 csuc 4102 com 4313 (class class class)co 5512 coa 5998 comu 5999 |
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-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 |
This theorem is referenced by: mulasspig 6430 enq0tr 6532 addcmpblnq0 6541 mulcmpblnq0 6542 mulcanenq0ec 6543 distrnq0 6557 addassnq0 6560 |
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