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Mirrors > Home > ILE Home > Th. List > nnacom | 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 5519 | . . . . 5 | |
2 | oveq2 5520 | . . . . 5 | |
3 | 1, 2 | eqeq12d 2054 | . . . 4 |
4 | 3 | imbi2d 219 | . . 3 |
5 | oveq1 5519 | . . . . 5 | |
6 | oveq2 5520 | . . . . 5 | |
7 | 5, 6 | eqeq12d 2054 | . . . 4 |
8 | oveq1 5519 | . . . . 5 | |
9 | oveq2 5520 | . . . . 5 | |
10 | 8, 9 | eqeq12d 2054 | . . . 4 |
11 | oveq1 5519 | . . . . 5 | |
12 | oveq2 5520 | . . . . 5 | |
13 | 11, 12 | eqeq12d 2054 | . . . 4 |
14 | nna0r 6057 | . . . . 5 | |
15 | nna0 6053 | . . . . 5 | |
16 | 14, 15 | eqtr4d 2075 | . . . 4 |
17 | suceq 4139 | . . . . . 6 | |
18 | oveq2 5520 | . . . . . . . . . . 11 | |
19 | oveq2 5520 | . . . . . . . . . . . 12 | |
20 | suceq 4139 | . . . . . . . . . . . 12 | |
21 | 19, 20 | syl 14 | . . . . . . . . . . 11 |
22 | 18, 21 | eqeq12d 2054 | . . . . . . . . . 10 |
23 | 22 | imbi2d 219 | . . . . . . . . 9 |
24 | oveq2 5520 | . . . . . . . . . . 11 | |
25 | oveq2 5520 | . . . . . . . . . . . 12 | |
26 | suceq 4139 | . . . . . . . . . . . 12 | |
27 | 25, 26 | syl 14 | . . . . . . . . . . 11 |
28 | 24, 27 | eqeq12d 2054 | . . . . . . . . . 10 |
29 | oveq2 5520 | . . . . . . . . . . 11 | |
30 | oveq2 5520 | . . . . . . . . . . . 12 | |
31 | suceq 4139 | . . . . . . . . . . . 12 | |
32 | 30, 31 | syl 14 | . . . . . . . . . . 11 |
33 | 29, 32 | eqeq12d 2054 | . . . . . . . . . 10 |
34 | oveq2 5520 | . . . . . . . . . . 11 | |
35 | oveq2 5520 | . . . . . . . . . . . 12 | |
36 | suceq 4139 | . . . . . . . . . . . 12 | |
37 | 35, 36 | syl 14 | . . . . . . . . . . 11 |
38 | 34, 37 | eqeq12d 2054 | . . . . . . . . . 10 |
39 | peano2 4318 | . . . . . . . . . . . 12 | |
40 | nna0 6053 | . . . . . . . . . . . 12 | |
41 | 39, 40 | syl 14 | . . . . . . . . . . 11 |
42 | nna0 6053 | . . . . . . . . . . . 12 | |
43 | suceq 4139 | . . . . . . . . . . . 12 | |
44 | 42, 43 | syl 14 | . . . . . . . . . . 11 |
45 | 41, 44 | eqtr4d 2075 | . . . . . . . . . 10 |
46 | suceq 4139 | . . . . . . . . . . . 12 | |
47 | nnasuc 6055 | . . . . . . . . . . . . . 14 | |
48 | 39, 47 | sylan 267 | . . . . . . . . . . . . 13 |
49 | nnasuc 6055 | . . . . . . . . . . . . . 14 | |
50 | suceq 4139 | . . . . . . . . . . . . . 14 | |
51 | 49, 50 | syl 14 | . . . . . . . . . . . . 13 |
52 | 48, 51 | eqeq12d 2054 | . . . . . . . . . . . 12 |
53 | 46, 52 | syl5ibr 145 | . . . . . . . . . . 11 |
54 | 53 | expcom 109 | . . . . . . . . . 10 |
55 | 28, 33, 38, 45, 54 | finds2 4324 | . . . . . . . . 9 |
56 | 23, 55 | vtoclga 2619 | . . . . . . . 8 |
57 | 56 | imp 115 | . . . . . . 7 |
58 | nnasuc 6055 | . . . . . . 7 | |
59 | 57, 58 | eqeq12d 2054 | . . . . . 6 |
60 | 17, 59 | syl5ibr 145 | . . . . 5 |
61 | 60 | expcom 109 | . . . 4 |
62 | 7, 10, 13, 16, 61 | finds2 4324 | . . 3 |
63 | 4, 62 | vtoclga 2619 | . 2 |
64 | 63 | imp 115 | 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 |
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 |
This theorem is referenced by: nnmsucr 6067 nnaordi 6081 nnaordr 6083 nnaword 6084 nnaword2 6087 nnawordi 6088 addcompig 6427 nqpnq0nq 6551 prarloclemlt 6591 prarloclemlo 6592 |
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