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Mirrors > Home > ILE Home > Th. List > nnaordi | Unicode version |
Description: Ordering property of addition. Proposition 8.4 of [TakeutiZaring] p. 58, limited to natural numbers. (Contributed by NM, 3-Feb-1996.) (Revised by Mario Carneiro, 15-Nov-2014.) |
Ref | Expression |
---|---|
nnaordi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 5520 | . . . . . . . . 9 | |
2 | oveq2 5520 | . . . . . . . . 9 | |
3 | 1, 2 | eleq12d 2108 | . . . . . . . 8 |
4 | 3 | imbi2d 219 | . . . . . . 7 |
5 | oveq2 5520 | . . . . . . . . 9 | |
6 | oveq2 5520 | . . . . . . . . 9 | |
7 | 5, 6 | eleq12d 2108 | . . . . . . . 8 |
8 | oveq2 5520 | . . . . . . . . 9 | |
9 | oveq2 5520 | . . . . . . . . 9 | |
10 | 8, 9 | eleq12d 2108 | . . . . . . . 8 |
11 | oveq2 5520 | . . . . . . . . 9 | |
12 | oveq2 5520 | . . . . . . . . 9 | |
13 | 11, 12 | eleq12d 2108 | . . . . . . . 8 |
14 | simpr 103 | . . . . . . . . 9 | |
15 | elnn 4328 | . . . . . . . . . . 11 | |
16 | 15 | ancoms 255 | . . . . . . . . . 10 |
17 | nna0 6053 | . . . . . . . . . 10 | |
18 | 16, 17 | syl 14 | . . . . . . . . 9 |
19 | nna0 6053 | . . . . . . . . . 10 | |
20 | 19 | adantr 261 | . . . . . . . . 9 |
21 | 14, 18, 20 | 3eltr4d 2121 | . . . . . . . 8 |
22 | simprl 483 | . . . . . . . . . . . . 13 | |
23 | simpl 102 | . . . . . . . . . . . . 13 | |
24 | nnacl 6059 | . . . . . . . . . . . . 13 | |
25 | 22, 23, 24 | syl2anc 391 | . . . . . . . . . . . 12 |
26 | nnsucelsuc 6070 | . . . . . . . . . . . 12 | |
27 | 25, 26 | syl 14 | . . . . . . . . . . 11 |
28 | 16 | adantl 262 | . . . . . . . . . . . . . 14 |
29 | nnon 4332 | . . . . . . . . . . . . . 14 | |
30 | 28, 29 | syl 14 | . . . . . . . . . . . . 13 |
31 | nnon 4332 | . . . . . . . . . . . . . 14 | |
32 | 31 | adantr 261 | . . . . . . . . . . . . 13 |
33 | oasuc 6044 | . . . . . . . . . . . . 13 | |
34 | 30, 32, 33 | syl2anc 391 | . . . . . . . . . . . 12 |
35 | nnon 4332 | . . . . . . . . . . . . . 14 | |
36 | 35 | ad2antrl 459 | . . . . . . . . . . . . 13 |
37 | oasuc 6044 | . . . . . . . . . . . . 13 | |
38 | 36, 32, 37 | syl2anc 391 | . . . . . . . . . . . 12 |
39 | 34, 38 | eleq12d 2108 | . . . . . . . . . . 11 |
40 | 27, 39 | bitr4d 180 | . . . . . . . . . 10 |
41 | 40 | biimpd 132 | . . . . . . . . 9 |
42 | 41 | ex 108 | . . . . . . . 8 |
43 | 7, 10, 13, 21, 42 | finds2 4324 | . . . . . . 7 |
44 | 4, 43 | vtoclga 2619 | . . . . . 6 |
45 | 44 | imp 115 | . . . . 5 |
46 | 16 | adantl 262 | . . . . . 6 |
47 | simpl 102 | . . . . . 6 | |
48 | nnacom 6063 | . . . . . 6 | |
49 | 46, 47, 48 | syl2anc 391 | . . . . 5 |
50 | nnacom 6063 | . . . . . . 7 | |
51 | 50 | ancoms 255 | . . . . . 6 |
52 | 51 | adantrr 448 | . . . . 5 |
53 | 45, 49, 52 | 3eltr3d 2120 | . . . 4 |
54 | 53 | 3impb 1100 | . . 3 |
55 | 54 | 3com12 1108 | . 2 |
56 | 55 | 3expia 1106 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wcel 1393 c0 3224 con0 4100 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: nnaord 6082 nnmordi 6089 addclpi 6425 addnidpig 6434 archnqq 6515 prarloclemarch2 6517 prarloclemlt 6591 |
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