Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 5540 | . . . . . 6 | |
2 | oveq2 5540 | . . . . . . 7 | |
3 | 2 | oveq2d 5548 | . . . . . 6 |
4 | 1, 3 | eqeq12d 2095 | . . . . 5 |
5 | 4 | imbi2d 228 | . . . 4 |
6 | oveq2 5540 | . . . . . 6 | |
7 | oveq2 5540 | . . . . . . 7 | |
8 | 7 | oveq2d 5548 | . . . . . 6 |
9 | 6, 8 | eqeq12d 2095 | . . . . 5 |
10 | oveq2 5540 | . . . . . 6 | |
11 | oveq2 5540 | . . . . . . 7 | |
12 | 11 | oveq2d 5548 | . . . . . 6 |
13 | 10, 12 | eqeq12d 2095 | . . . . 5 |
14 | oveq2 5540 | . . . . . 6 | |
15 | oveq2 5540 | . . . . . . 7 | |
16 | 15 | oveq2d 5548 | . . . . . 6 |
17 | 14, 16 | eqeq12d 2095 | . . . . 5 |
18 | nnacl 6082 | . . . . . . 7 | |
19 | nna0 6076 | . . . . . . 7 | |
20 | 18, 19 | syl 14 | . . . . . 6 |
21 | nna0 6076 | . . . . . . . 8 | |
22 | 21 | oveq2d 5548 | . . . . . . 7 |
23 | 22 | adantl 271 | . . . . . 6 |
24 | 20, 23 | eqtr4d 2116 | . . . . 5 |
25 | suceq 4157 | . . . . . . 7 | |
26 | nnasuc 6078 | . . . . . . . . 9 | |
27 | 18, 26 | sylan 277 | . . . . . . . 8 |
28 | nnasuc 6078 | . . . . . . . . . . . 12 | |
29 | 28 | oveq2d 5548 | . . . . . . . . . . 11 |
30 | 29 | adantl 271 | . . . . . . . . . 10 |
31 | nnacl 6082 | . . . . . . . . . . 11 | |
32 | nnasuc 6078 | . . . . . . . . . . 11 | |
33 | 31, 32 | sylan2 280 | . . . . . . . . . 10 |
34 | 30, 33 | eqtrd 2113 | . . . . . . . . 9 |
35 | 34 | anassrs 392 | . . . . . . . 8 |
36 | 27, 35 | eqeq12d 2095 | . . . . . . 7 |
37 | 25, 36 | syl5ibr 154 | . . . . . 6 |
38 | 37 | expcom 114 | . . . . 5 |
39 | 9, 13, 17, 24, 38 | finds2 4342 | . . . 4 |
40 | 5, 39 | vtoclga 2664 | . . 3 |
41 | 40 | com12 30 | . 2 |
42 | 41 | 3impia 1135 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 w3a 919 wceq 1284 wcel 1433 c0 3251 csuc 4120 com 4331 (class class class)co 5532 coa 6021 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-oadd 6028 |
This theorem is referenced by: nndi 6088 nnmsucr 6090 addasspig 6520 addassnq0 6652 prarloclemlo 6684 |
Copyright terms: Public domain | W3C validator |