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Mirrors > Home > MPE Home > Th. List > nnaass | Structured version Visualization version 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 6658 | . . . . . 6 | |
2 | oveq2 6658 | . . . . . . 7 | |
3 | 2 | oveq2d 6666 | . . . . . 6 |
4 | 1, 3 | eqeq12d 2637 | . . . . 5 |
5 | 4 | imbi2d 330 | . . . 4 |
6 | oveq2 6658 | . . . . . 6 | |
7 | oveq2 6658 | . . . . . . 7 | |
8 | 7 | oveq2d 6666 | . . . . . 6 |
9 | 6, 8 | eqeq12d 2637 | . . . . 5 |
10 | oveq2 6658 | . . . . . 6 | |
11 | oveq2 6658 | . . . . . . 7 | |
12 | 11 | oveq2d 6666 | . . . . . 6 |
13 | 10, 12 | eqeq12d 2637 | . . . . 5 |
14 | oveq2 6658 | . . . . . 6 | |
15 | oveq2 6658 | . . . . . . 7 | |
16 | 15 | oveq2d 6666 | . . . . . 6 |
17 | 14, 16 | eqeq12d 2637 | . . . . 5 |
18 | nnacl 7691 | . . . . . . 7 | |
19 | nna0 7684 | . . . . . . 7 | |
20 | 18, 19 | syl 17 | . . . . . 6 |
21 | nna0 7684 | . . . . . . . 8 | |
22 | 21 | oveq2d 6666 | . . . . . . 7 |
23 | 22 | adantl 482 | . . . . . 6 |
24 | 20, 23 | eqtr4d 2659 | . . . . 5 |
25 | suceq 5790 | . . . . . . 7 | |
26 | nnasuc 7686 | . . . . . . . . 9 | |
27 | 18, 26 | sylan 488 | . . . . . . . 8 |
28 | nnasuc 7686 | . . . . . . . . . . . 12 | |
29 | 28 | oveq2d 6666 | . . . . . . . . . . 11 |
30 | 29 | adantl 482 | . . . . . . . . . 10 |
31 | nnacl 7691 | . . . . . . . . . . 11 | |
32 | nnasuc 7686 | . . . . . . . . . . 11 | |
33 | 31, 32 | sylan2 491 | . . . . . . . . . 10 |
34 | 30, 33 | eqtrd 2656 | . . . . . . . . 9 |
35 | 34 | anassrs 680 | . . . . . . . 8 |
36 | 27, 35 | eqeq12d 2637 | . . . . . . 7 |
37 | 25, 36 | syl5ibr 236 | . . . . . 6 |
38 | 37 | expcom 451 | . . . . 5 |
39 | 9, 13, 17, 24, 38 | finds2 7094 | . . . 4 |
40 | 5, 39 | vtoclga 3272 | . . 3 |
41 | 40 | com12 32 | . 2 |
42 | 41 | 3impia 1261 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 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: nndi 7703 nnmsucr 7705 omopthlem1 7735 omopthlem2 7736 addasspi 9717 |
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