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Mirrors > Home > ILE Home > Th. List > nnaass | Unicode version |
Description: Addition of natural numbers is associative. Theorem 4K(1) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.) |
Ref | Expression |
---|---|
nnaass |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 5790 | . . . . . 6 | |
2 | oveq2 5790 | . . . . . . 7 | |
3 | 2 | oveq2d 5798 | . . . . . 6 |
4 | 1, 3 | eqeq12d 2155 | . . . . 5 |
5 | 4 | imbi2d 229 | . . . 4 |
6 | oveq2 5790 | . . . . . 6 | |
7 | oveq2 5790 | . . . . . . 7 | |
8 | 7 | oveq2d 5798 | . . . . . 6 |
9 | 6, 8 | eqeq12d 2155 | . . . . 5 |
10 | oveq2 5790 | . . . . . 6 | |
11 | oveq2 5790 | . . . . . . 7 | |
12 | 11 | oveq2d 5798 | . . . . . 6 |
13 | 10, 12 | eqeq12d 2155 | . . . . 5 |
14 | oveq2 5790 | . . . . . 6 | |
15 | oveq2 5790 | . . . . . . 7 | |
16 | 15 | oveq2d 5798 | . . . . . 6 |
17 | 14, 16 | eqeq12d 2155 | . . . . 5 |
18 | nnacl 6384 | . . . . . . 7 | |
19 | nna0 6378 | . . . . . . 7 | |
20 | 18, 19 | syl 14 | . . . . . 6 |
21 | nna0 6378 | . . . . . . . 8 | |
22 | 21 | oveq2d 5798 | . . . . . . 7 |
23 | 22 | adantl 275 | . . . . . 6 |
24 | 20, 23 | eqtr4d 2176 | . . . . 5 |
25 | suceq 4332 | . . . . . . 7 | |
26 | nnasuc 6380 | . . . . . . . . 9 | |
27 | 18, 26 | sylan 281 | . . . . . . . 8 |
28 | nnasuc 6380 | . . . . . . . . . . . 12 | |
29 | 28 | oveq2d 5798 | . . . . . . . . . . 11 |
30 | 29 | adantl 275 | . . . . . . . . . 10 |
31 | nnacl 6384 | . . . . . . . . . . 11 | |
32 | nnasuc 6380 | . . . . . . . . . . 11 | |
33 | 31, 32 | sylan2 284 | . . . . . . . . . 10 |
34 | 30, 33 | eqtrd 2173 | . . . . . . . . 9 |
35 | 34 | anassrs 398 | . . . . . . . 8 |
36 | 27, 35 | eqeq12d 2155 | . . . . . . 7 |
37 | 25, 36 | syl5ibr 155 | . . . . . 6 |
38 | 37 | expcom 115 | . . . . 5 |
39 | 9, 13, 17, 24, 38 | finds2 4523 | . . . 4 |
40 | 5, 39 | vtoclga 2755 | . . 3 |
41 | 40 | com12 30 | . 2 |
42 | 41 | 3impia 1179 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 963 wceq 1332 wcel 1481 c0 3368 csuc 4295 com 4512 (class class class)co 5782 coa 6318 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-oadd 6325 |
This theorem is referenced by: nndi 6390 nnmsucr 6392 addasspig 7162 addassnq0 7294 prarloclemlo 7326 |
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