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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 5846 | . . . . 5 | |
2 | oveq2 5847 | . . . . 5 | |
3 | 1, 2 | eqeq12d 2179 | . . . 4 |
4 | 3 | imbi2d 229 | . . 3 |
5 | oveq1 5846 | . . . . 5 | |
6 | oveq2 5847 | . . . . 5 | |
7 | 5, 6 | eqeq12d 2179 | . . . 4 |
8 | oveq1 5846 | . . . . 5 | |
9 | oveq2 5847 | . . . . 5 | |
10 | 8, 9 | eqeq12d 2179 | . . . 4 |
11 | oveq1 5846 | . . . . 5 | |
12 | oveq2 5847 | . . . . 5 | |
13 | 11, 12 | eqeq12d 2179 | . . . 4 |
14 | nna0r 6440 | . . . . 5 | |
15 | nna0 6436 | . . . . 5 | |
16 | 14, 15 | eqtr4d 2200 | . . . 4 |
17 | suceq 4377 | . . . . . 6 | |
18 | oveq2 5847 | . . . . . . . . . . 11 | |
19 | oveq2 5847 | . . . . . . . . . . . 12 | |
20 | suceq 4377 | . . . . . . . . . . . 12 | |
21 | 19, 20 | syl 14 | . . . . . . . . . . 11 |
22 | 18, 21 | eqeq12d 2179 | . . . . . . . . . 10 |
23 | 22 | imbi2d 229 | . . . . . . . . 9 |
24 | oveq2 5847 | . . . . . . . . . . 11 | |
25 | oveq2 5847 | . . . . . . . . . . . 12 | |
26 | suceq 4377 | . . . . . . . . . . . 12 | |
27 | 25, 26 | syl 14 | . . . . . . . . . . 11 |
28 | 24, 27 | eqeq12d 2179 | . . . . . . . . . 10 |
29 | oveq2 5847 | . . . . . . . . . . 11 | |
30 | oveq2 5847 | . . . . . . . . . . . 12 | |
31 | suceq 4377 | . . . . . . . . . . . 12 | |
32 | 30, 31 | syl 14 | . . . . . . . . . . 11 |
33 | 29, 32 | eqeq12d 2179 | . . . . . . . . . 10 |
34 | oveq2 5847 | . . . . . . . . . . 11 | |
35 | oveq2 5847 | . . . . . . . . . . . 12 | |
36 | suceq 4377 | . . . . . . . . . . . 12 | |
37 | 35, 36 | syl 14 | . . . . . . . . . . 11 |
38 | 34, 37 | eqeq12d 2179 | . . . . . . . . . 10 |
39 | peano2 4569 | . . . . . . . . . . . 12 | |
40 | nna0 6436 | . . . . . . . . . . . 12 | |
41 | 39, 40 | syl 14 | . . . . . . . . . . 11 |
42 | nna0 6436 | . . . . . . . . . . . 12 | |
43 | suceq 4377 | . . . . . . . . . . . 12 | |
44 | 42, 43 | syl 14 | . . . . . . . . . . 11 |
45 | 41, 44 | eqtr4d 2200 | . . . . . . . . . 10 |
46 | suceq 4377 | . . . . . . . . . . . 12 | |
47 | nnasuc 6438 | . . . . . . . . . . . . . 14 | |
48 | 39, 47 | sylan 281 | . . . . . . . . . . . . 13 |
49 | nnasuc 6438 | . . . . . . . . . . . . . 14 | |
50 | suceq 4377 | . . . . . . . . . . . . . 14 | |
51 | 49, 50 | syl 14 | . . . . . . . . . . . . 13 |
52 | 48, 51 | eqeq12d 2179 | . . . . . . . . . . . 12 |
53 | 46, 52 | syl5ibr 155 | . . . . . . . . . . 11 |
54 | 53 | expcom 115 | . . . . . . . . . 10 |
55 | 28, 33, 38, 45, 54 | finds2 4575 | . . . . . . . . 9 |
56 | 23, 55 | vtoclga 2790 | . . . . . . . 8 |
57 | 56 | imp 123 | . . . . . . 7 |
58 | nnasuc 6438 | . . . . . . 7 | |
59 | 57, 58 | eqeq12d 2179 | . . . . . 6 |
60 | 17, 59 | syl5ibr 155 | . . . . 5 |
61 | 60 | expcom 115 | . . . 4 |
62 | 7, 10, 13, 16, 61 | finds2 4575 | . . 3 |
63 | 4, 62 | vtoclga 2790 | . 2 |
64 | 63 | imp 123 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1342 wcel 2135 c0 3407 csuc 4340 com 4564 (class class class)co 5839 coa 6375 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4094 ax-sep 4097 ax-nul 4105 ax-pow 4150 ax-pr 4184 ax-un 4408 ax-setind 4511 ax-iinf 4562 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2726 df-sbc 2950 df-csb 3044 df-dif 3116 df-un 3118 df-in 3120 df-ss 3127 df-nul 3408 df-pw 3558 df-sn 3579 df-pr 3580 df-op 3582 df-uni 3787 df-int 3822 df-iun 3865 df-br 3980 df-opab 4041 df-mpt 4042 df-tr 4078 df-id 4268 df-iord 4341 df-on 4343 df-suc 4346 df-iom 4565 df-xp 4607 df-rel 4608 df-cnv 4609 df-co 4610 df-dm 4611 df-rn 4612 df-res 4613 df-ima 4614 df-iota 5150 df-fun 5187 df-fn 5188 df-f 5189 df-f1 5190 df-fo 5191 df-f1o 5192 df-fv 5193 df-ov 5842 df-oprab 5843 df-mpo 5844 df-1st 6103 df-2nd 6104 df-recs 6267 df-irdg 6332 df-oadd 6382 |
This theorem is referenced by: nnmsucr 6450 nnaordi 6470 nnaordr 6472 nnaword 6473 nnaword2 6476 nnawordi 6477 addcompig 7264 nqpnq0nq 7388 prarloclemlt 7428 prarloclemlo 7429 |
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