Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > nnmsucr | Structured version Visualization version 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 6658 | . . . . 5 | |
2 | oveq2 6658 | . . . . . 6 | |
3 | id 22 | . . . . . 6 | |
4 | 2, 3 | oveq12d 6668 | . . . . 5 |
5 | 1, 4 | eqeq12d 2637 | . . . 4 |
6 | 5 | imbi2d 330 | . . 3 |
7 | oveq2 6658 | . . . . 5 | |
8 | oveq2 6658 | . . . . . 6 | |
9 | id 22 | . . . . . 6 | |
10 | 8, 9 | oveq12d 6668 | . . . . 5 |
11 | 7, 10 | eqeq12d 2637 | . . . 4 |
12 | oveq2 6658 | . . . . 5 | |
13 | oveq2 6658 | . . . . . 6 | |
14 | id 22 | . . . . . 6 | |
15 | 13, 14 | oveq12d 6668 | . . . . 5 |
16 | 12, 15 | eqeq12d 2637 | . . . 4 |
17 | oveq2 6658 | . . . . 5 | |
18 | oveq2 6658 | . . . . . 6 | |
19 | id 22 | . . . . . 6 | |
20 | 18, 19 | oveq12d 6668 | . . . . 5 |
21 | 17, 20 | eqeq12d 2637 | . . . 4 |
22 | peano2 7086 | . . . . . . 7 | |
23 | nnm0 7685 | . . . . . . 7 | |
24 | 22, 23 | syl 17 | . . . . . 6 |
25 | nnm0 7685 | . . . . . 6 | |
26 | 24, 25 | eqtr4d 2659 | . . . . 5 |
27 | peano1 7085 | . . . . . . 7 | |
28 | nnmcl 7692 | . . . . . . 7 | |
29 | 27, 28 | mpan2 707 | . . . . . 6 |
30 | nna0 7684 | . . . . . 6 | |
31 | 29, 30 | syl 17 | . . . . 5 |
32 | 26, 31 | eqtr4d 2659 | . . . 4 |
33 | oveq1 6657 | . . . . . 6 | |
34 | peano2b 7081 | . . . . . . . 8 | |
35 | nnmsuc 7687 | . . . . . . . 8 | |
36 | 34, 35 | sylanb 489 | . . . . . . 7 |
37 | nnmcl 7692 | . . . . . . . . . . 11 | |
38 | peano2b 7081 | . . . . . . . . . . . 12 | |
39 | nnaass 7702 | . . . . . . . . . . . 12 | |
40 | 38, 39 | syl3an3b 1364 | . . . . . . . . . . 11 |
41 | 37, 40 | syl3an1 1359 | . . . . . . . . . 10 |
42 | 41 | 3expb 1266 | . . . . . . . . 9 |
43 | 42 | anidms 677 | . . . . . . . 8 |
44 | nnmsuc 7687 | . . . . . . . . 9 | |
45 | 44 | oveq1d 6665 | . . . . . . . 8 |
46 | nnaass 7702 | . . . . . . . . . . . . . 14 | |
47 | 34, 46 | syl3an3b 1364 | . . . . . . . . . . . . 13 |
48 | 37, 47 | syl3an1 1359 | . . . . . . . . . . . 12 |
49 | 48 | 3expb 1266 | . . . . . . . . . . 11 |
50 | 49 | an42s 870 | . . . . . . . . . 10 |
51 | 50 | anidms 677 | . . . . . . . . 9 |
52 | nnacom 7697 | . . . . . . . . . . . 12 | |
53 | suceq 5790 | . . . . . . . . . . . 12 | |
54 | 52, 53 | syl 17 | . . . . . . . . . . 11 |
55 | nnasuc 7686 | . . . . . . . . . . 11 | |
56 | nnasuc 7686 | . . . . . . . . . . . 12 | |
57 | 56 | ancoms 469 | . . . . . . . . . . 11 |
58 | 54, 55, 57 | 3eqtr4d 2666 | . . . . . . . . . 10 |
59 | 58 | oveq2d 6666 | . . . . . . . . 9 |
60 | 51, 59 | eqtr4d 2659 | . . . . . . . 8 |
61 | 43, 45, 60 | 3eqtr4d 2666 | . . . . . . 7 |
62 | 36, 61 | eqeq12d 2637 | . . . . . 6 |
63 | 33, 62 | syl5ibr 236 | . . . . 5 |
64 | 63 | expcom 451 | . . . 4 |
65 | 11, 16, 21, 32, 64 | finds2 7094 | . . 3 |
66 | 6, 65 | vtoclga 3272 | . 2 |
67 | 66 | impcom 446 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 c0 3915 csuc 5725 (class class class)co 6650 com 7065 coa 7557 comu 7558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-oadd 7564 df-omul 7565 |
This theorem is referenced by: nnmcom 7706 |
Copyright terms: Public domain | W3C validator |