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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 5832 | . . . . . . . . 9 | |
2 | oveq2 5832 | . . . . . . . . 9 | |
3 | 1, 2 | eleq12d 2228 | . . . . . . . 8 |
4 | 3 | imbi2d 229 | . . . . . . 7 |
5 | oveq2 5832 | . . . . . . . . 9 | |
6 | oveq2 5832 | . . . . . . . . 9 | |
7 | 5, 6 | eleq12d 2228 | . . . . . . . 8 |
8 | oveq2 5832 | . . . . . . . . 9 | |
9 | oveq2 5832 | . . . . . . . . 9 | |
10 | 8, 9 | eleq12d 2228 | . . . . . . . 8 |
11 | oveq2 5832 | . . . . . . . . 9 | |
12 | oveq2 5832 | . . . . . . . . 9 | |
13 | 11, 12 | eleq12d 2228 | . . . . . . . 8 |
14 | simpr 109 | . . . . . . . . 9 | |
15 | elnn 4565 | . . . . . . . . . . 11 | |
16 | 15 | ancoms 266 | . . . . . . . . . 10 |
17 | nna0 6421 | . . . . . . . . . 10 | |
18 | 16, 17 | syl 14 | . . . . . . . . 9 |
19 | nna0 6421 | . . . . . . . . . 10 | |
20 | 19 | adantr 274 | . . . . . . . . 9 |
21 | 14, 18, 20 | 3eltr4d 2241 | . . . . . . . 8 |
22 | simprl 521 | . . . . . . . . . . . . 13 | |
23 | simpl 108 | . . . . . . . . . . . . 13 | |
24 | nnacl 6427 | . . . . . . . . . . . . 13 | |
25 | 22, 23, 24 | syl2anc 409 | . . . . . . . . . . . 12 |
26 | nnsucelsuc 6438 | . . . . . . . . . . . 12 | |
27 | 25, 26 | syl 14 | . . . . . . . . . . 11 |
28 | 16 | adantl 275 | . . . . . . . . . . . . . 14 |
29 | nnon 4569 | . . . . . . . . . . . . . 14 | |
30 | 28, 29 | syl 14 | . . . . . . . . . . . . 13 |
31 | nnon 4569 | . . . . . . . . . . . . . 14 | |
32 | 31 | adantr 274 | . . . . . . . . . . . . 13 |
33 | oasuc 6411 | . . . . . . . . . . . . 13 | |
34 | 30, 32, 33 | syl2anc 409 | . . . . . . . . . . . 12 |
35 | nnon 4569 | . . . . . . . . . . . . . 14 | |
36 | 35 | ad2antrl 482 | . . . . . . . . . . . . 13 |
37 | oasuc 6411 | . . . . . . . . . . . . 13 | |
38 | 36, 32, 37 | syl2anc 409 | . . . . . . . . . . . 12 |
39 | 34, 38 | eleq12d 2228 | . . . . . . . . . . 11 |
40 | 27, 39 | bitr4d 190 | . . . . . . . . . 10 |
41 | 40 | biimpd 143 | . . . . . . . . 9 |
42 | 41 | ex 114 | . . . . . . . 8 |
43 | 7, 10, 13, 21, 42 | finds2 4560 | . . . . . . 7 |
44 | 4, 43 | vtoclga 2778 | . . . . . 6 |
45 | 44 | imp 123 | . . . . 5 |
46 | 16 | adantl 275 | . . . . . 6 |
47 | simpl 108 | . . . . . 6 | |
48 | nnacom 6431 | . . . . . 6 | |
49 | 46, 47, 48 | syl2anc 409 | . . . . 5 |
50 | nnacom 6431 | . . . . . . 7 | |
51 | 50 | ancoms 266 | . . . . . 6 |
52 | 51 | adantrr 471 | . . . . 5 |
53 | 45, 49, 52 | 3eltr3d 2240 | . . . 4 |
54 | 53 | 3impb 1181 | . . 3 |
55 | 54 | 3com12 1189 | . 2 |
56 | 55 | 3expia 1187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1335 wcel 2128 c0 3394 con0 4323 csuc 4325 com 4549 (class class class)co 5824 coa 6360 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-recs 6252 df-irdg 6317 df-oadd 6367 |
This theorem is referenced by: nnaord 6456 nnmordi 6463 addclpi 7247 addnidpig 7256 archnqq 7337 prarloclemarch2 7339 prarloclemlt 7413 |
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