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Mirrors > Home > ILE Home > Th. List > nnmsucr | Unicode version |
Description: Multiplication with successor. Exercise 16 of [Enderton] p. 82. (Contributed by NM, 21-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Ref | Expression |
---|---|
nnmsucr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 5520 | . . . . 5 | |
2 | oveq2 5520 | . . . . . 6 | |
3 | id 19 | . . . . . 6 | |
4 | 2, 3 | oveq12d 5530 | . . . . 5 |
5 | 1, 4 | eqeq12d 2054 | . . . 4 |
6 | 5 | imbi2d 219 | . . 3 |
7 | oveq2 5520 | . . . . 5 | |
8 | oveq2 5520 | . . . . . 6 | |
9 | id 19 | . . . . . 6 | |
10 | 8, 9 | oveq12d 5530 | . . . . 5 |
11 | 7, 10 | eqeq12d 2054 | . . . 4 |
12 | oveq2 5520 | . . . . 5 | |
13 | oveq2 5520 | . . . . . 6 | |
14 | id 19 | . . . . . 6 | |
15 | 13, 14 | oveq12d 5530 | . . . . 5 |
16 | 12, 15 | eqeq12d 2054 | . . . 4 |
17 | oveq2 5520 | . . . . 5 | |
18 | oveq2 5520 | . . . . . 6 | |
19 | id 19 | . . . . . 6 | |
20 | 18, 19 | oveq12d 5530 | . . . . 5 |
21 | 17, 20 | eqeq12d 2054 | . . . 4 |
22 | peano2 4318 | . . . . . . 7 | |
23 | nnm0 6054 | . . . . . . 7 | |
24 | 22, 23 | syl 14 | . . . . . 6 |
25 | nnm0 6054 | . . . . . 6 | |
26 | 24, 25 | eqtr4d 2075 | . . . . 5 |
27 | peano1 4317 | . . . . . . 7 | |
28 | nnmcl 6060 | . . . . . . 7 | |
29 | 27, 28 | mpan2 401 | . . . . . 6 |
30 | nna0 6053 | . . . . . 6 | |
31 | 29, 30 | syl 14 | . . . . 5 |
32 | 26, 31 | eqtr4d 2075 | . . . 4 |
33 | oveq1 5519 | . . . . . 6 | |
34 | peano2b 4337 | . . . . . . . 8 | |
35 | nnmsuc 6056 | . . . . . . . 8 | |
36 | 34, 35 | sylanb 268 | . . . . . . 7 |
37 | nnmcl 6060 | . . . . . . . . . . 11 | |
38 | peano2b 4337 | . . . . . . . . . . . 12 | |
39 | nnaass 6064 | . . . . . . . . . . . 12 | |
40 | 38, 39 | syl3an3b 1173 | . . . . . . . . . . 11 |
41 | 37, 40 | syl3an1 1168 | . . . . . . . . . 10 |
42 | 41 | 3expb 1105 | . . . . . . . . 9 |
43 | 42 | anidms 377 | . . . . . . . 8 |
44 | nnmsuc 6056 | . . . . . . . . 9 | |
45 | 44 | oveq1d 5527 | . . . . . . . 8 |
46 | nnaass 6064 | . . . . . . . . . . . . . 14 | |
47 | 34, 46 | syl3an3b 1173 | . . . . . . . . . . . . 13 |
48 | 37, 47 | syl3an1 1168 | . . . . . . . . . . . 12 |
49 | 48 | 3expb 1105 | . . . . . . . . . . 11 |
50 | 49 | an42s 523 | . . . . . . . . . 10 |
51 | 50 | anidms 377 | . . . . . . . . 9 |
52 | nnacom 6063 | . . . . . . . . . . . 12 | |
53 | suceq 4139 | . . . . . . . . . . . 12 | |
54 | 52, 53 | syl 14 | . . . . . . . . . . 11 |
55 | nnasuc 6055 | . . . . . . . . . . 11 | |
56 | nnasuc 6055 | . . . . . . . . . . . 12 | |
57 | 56 | ancoms 255 | . . . . . . . . . . 11 |
58 | 54, 55, 57 | 3eqtr4d 2082 | . . . . . . . . . 10 |
59 | 58 | oveq2d 5528 | . . . . . . . . 9 |
60 | 51, 59 | eqtr4d 2075 | . . . . . . . 8 |
61 | 43, 45, 60 | 3eqtr4d 2082 | . . . . . . 7 |
62 | 36, 61 | eqeq12d 2054 | . . . . . 6 |
63 | 33, 62 | syl5ibr 145 | . . . . 5 |
64 | 63 | expcom 109 | . . . 4 |
65 | 11, 16, 21, 32, 64 | finds2 4324 | . . 3 |
66 | 6, 65 | vtoclga 2619 | . 2 |
67 | 66 | impcom 116 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 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: nnmcom 6068 |
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