Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 5540 | . . . . . 6 | |
2 | oveq2 5540 | . . . . . . 7 | |
3 | 2 | oveq2d 5548 | . . . . . 6 |
4 | 1, 3 | eqeq12d 2095 | . . . . 5 |
5 | 4 | imbi2d 228 | . . . 4 |
6 | oveq2 5540 | . . . . . 6 | |
7 | oveq2 5540 | . . . . . . 7 | |
8 | 7 | oveq2d 5548 | . . . . . 6 |
9 | 6, 8 | eqeq12d 2095 | . . . . 5 |
10 | oveq2 5540 | . . . . . 6 | |
11 | oveq2 5540 | . . . . . . 7 | |
12 | 11 | oveq2d 5548 | . . . . . 6 |
13 | 10, 12 | eqeq12d 2095 | . . . . 5 |
14 | oveq2 5540 | . . . . . 6 | |
15 | oveq2 5540 | . . . . . . 7 | |
16 | 15 | oveq2d 5548 | . . . . . 6 |
17 | 14, 16 | eqeq12d 2095 | . . . . 5 |
18 | nnmcl 6083 | . . . . . . 7 | |
19 | nnm0 6077 | . . . . . . 7 | |
20 | 18, 19 | syl 14 | . . . . . 6 |
21 | nnm0 6077 | . . . . . . . 8 | |
22 | 21 | oveq2d 5548 | . . . . . . 7 |
23 | nnm0 6077 | . . . . . . 7 | |
24 | 22, 23 | sylan9eqr 2135 | . . . . . 6 |
25 | 20, 24 | eqtr4d 2116 | . . . . 5 |
26 | oveq1 5539 | . . . . . . . . 9 | |
27 | nnmsuc 6079 | . . . . . . . . . . . 12 | |
28 | 18, 27 | sylan 277 | . . . . . . . . . . 11 |
29 | 28 | 3impa 1133 | . . . . . . . . . 10 |
30 | nnmsuc 6079 | . . . . . . . . . . . . 13 | |
31 | 30 | 3adant1 956 | . . . . . . . . . . . 12 |
32 | 31 | oveq2d 5548 | . . . . . . . . . . 11 |
33 | nnmcl 6083 | . . . . . . . . . . . . . . . . 17 | |
34 | nndi 6088 | . . . . . . . . . . . . . . . . 17 | |
35 | 33, 34 | syl3an2 1203 | . . . . . . . . . . . . . . . 16 |
36 | 35 | 3exp 1137 | . . . . . . . . . . . . . . 15 |
37 | 36 | expd 254 | . . . . . . . . . . . . . 14 |
38 | 37 | com34 82 | . . . . . . . . . . . . 13 |
39 | 38 | pm2.43d 49 | . . . . . . . . . . . 12 |
40 | 39 | 3imp 1132 | . . . . . . . . . . 11 |
41 | 32, 40 | eqtrd 2113 | . . . . . . . . . 10 |
42 | 29, 41 | eqeq12d 2095 | . . . . . . . . 9 |
43 | 26, 42 | syl5ibr 154 | . . . . . . . 8 |
44 | 43 | 3exp 1137 | . . . . . . 7 |
45 | 44 | com3r 78 | . . . . . 6 |
46 | 45 | impd 251 | . . . . 5 |
47 | 9, 13, 17, 25, 46 | finds2 4342 | . . . 4 |
48 | 5, 47 | vtoclga 2664 | . . 3 |
49 | 48 | expdcom 1371 | . 2 |
50 | 49 | 3imp 1132 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 w3a 919 wceq 1284 wcel 1433 c0 3251 csuc 4120 com 4331 (class class class)co 5532 coa 6021 comu 6022 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-oadd 6028 df-omul 6029 |
This theorem is referenced by: mulasspig 6522 enq0tr 6624 addcmpblnq0 6633 mulcmpblnq0 6634 mulcanenq0ec 6635 distrnq0 6649 addassnq0 6652 |
Copyright terms: Public domain | W3C validator |