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Mirrors > Home > MPE Home > Th. List > nnacom | Structured version Visualization version Unicode version |
Description: Addition of natural numbers is commutative. Theorem 4K(2) of [Enderton] p. 81. (Contributed by NM, 6-May-1995.) (Revised by Mario Carneiro, 15-Nov-2014.) |
Ref | Expression |
---|---|
nnacom |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 6657 | . . . . 5 | |
2 | oveq2 6658 | . . . . 5 | |
3 | 1, 2 | eqeq12d 2637 | . . . 4 |
4 | 3 | imbi2d 330 | . . 3 |
5 | oveq1 6657 | . . . . 5 | |
6 | oveq2 6658 | . . . . 5 | |
7 | 5, 6 | eqeq12d 2637 | . . . 4 |
8 | oveq1 6657 | . . . . 5 | |
9 | oveq2 6658 | . . . . 5 | |
10 | 8, 9 | eqeq12d 2637 | . . . 4 |
11 | oveq1 6657 | . . . . 5 | |
12 | oveq2 6658 | . . . . 5 | |
13 | 11, 12 | eqeq12d 2637 | . . . 4 |
14 | nna0r 7689 | . . . . 5 | |
15 | nna0 7684 | . . . . 5 | |
16 | 14, 15 | eqtr4d 2659 | . . . 4 |
17 | suceq 5790 | . . . . . 6 | |
18 | oveq2 6658 | . . . . . . . . . . 11 | |
19 | oveq2 6658 | . . . . . . . . . . . 12 | |
20 | suceq 5790 | . . . . . . . . . . . 12 | |
21 | 19, 20 | syl 17 | . . . . . . . . . . 11 |
22 | 18, 21 | eqeq12d 2637 | . . . . . . . . . 10 |
23 | 22 | imbi2d 330 | . . . . . . . . 9 |
24 | oveq2 6658 | . . . . . . . . . . 11 | |
25 | oveq2 6658 | . . . . . . . . . . . 12 | |
26 | suceq 5790 | . . . . . . . . . . . 12 | |
27 | 25, 26 | syl 17 | . . . . . . . . . . 11 |
28 | 24, 27 | eqeq12d 2637 | . . . . . . . . . 10 |
29 | oveq2 6658 | . . . . . . . . . . 11 | |
30 | oveq2 6658 | . . . . . . . . . . . 12 | |
31 | suceq 5790 | . . . . . . . . . . . 12 | |
32 | 30, 31 | syl 17 | . . . . . . . . . . 11 |
33 | 29, 32 | eqeq12d 2637 | . . . . . . . . . 10 |
34 | oveq2 6658 | . . . . . . . . . . 11 | |
35 | oveq2 6658 | . . . . . . . . . . . 12 | |
36 | suceq 5790 | . . . . . . . . . . . 12 | |
37 | 35, 36 | syl 17 | . . . . . . . . . . 11 |
38 | 34, 37 | eqeq12d 2637 | . . . . . . . . . 10 |
39 | peano2 7086 | . . . . . . . . . . . 12 | |
40 | nna0 7684 | . . . . . . . . . . . 12 | |
41 | 39, 40 | syl 17 | . . . . . . . . . . 11 |
42 | nna0 7684 | . . . . . . . . . . . 12 | |
43 | suceq 5790 | . . . . . . . . . . . 12 | |
44 | 42, 43 | syl 17 | . . . . . . . . . . 11 |
45 | 41, 44 | eqtr4d 2659 | . . . . . . . . . 10 |
46 | suceq 5790 | . . . . . . . . . . . 12 | |
47 | nnasuc 7686 | . . . . . . . . . . . . . 14 | |
48 | 39, 47 | sylan 488 | . . . . . . . . . . . . 13 |
49 | nnasuc 7686 | . . . . . . . . . . . . . 14 | |
50 | suceq 5790 | . . . . . . . . . . . . . 14 | |
51 | 49, 50 | syl 17 | . . . . . . . . . . . . 13 |
52 | 48, 51 | eqeq12d 2637 | . . . . . . . . . . . 12 |
53 | 46, 52 | syl5ibr 236 | . . . . . . . . . . 11 |
54 | 53 | expcom 451 | . . . . . . . . . 10 |
55 | 28, 33, 38, 45, 54 | finds2 7094 | . . . . . . . . 9 |
56 | 23, 55 | vtoclga 3272 | . . . . . . . 8 |
57 | 56 | imp 445 | . . . . . . 7 |
58 | nnasuc 7686 | . . . . . . 7 | |
59 | 57, 58 | eqeq12d 2637 | . . . . . 6 |
60 | 17, 59 | syl5ibr 236 | . . . . 5 |
61 | 60 | expcom 451 | . . . 4 |
62 | 7, 10, 13, 16, 61 | finds2 7094 | . . 3 |
63 | 4, 62 | vtoclga 3272 | . 2 |
64 | 63 | imp 445 | 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 |
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 |
This theorem is referenced by: nnaordr 7700 nnmsucr 7705 nnaword2 7710 omopthlem2 7736 omopthi 7737 addcompi 9716 finxpreclem4 33231 |
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