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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 5782 | . . . . . . . . 9 | |
2 | oveq2 5782 | . . . . . . . . 9 | |
3 | 1, 2 | eleq12d 2210 | . . . . . . . 8 |
4 | 3 | imbi2d 229 | . . . . . . 7 |
5 | oveq2 5782 | . . . . . . . . 9 | |
6 | oveq2 5782 | . . . . . . . . 9 | |
7 | 5, 6 | eleq12d 2210 | . . . . . . . 8 |
8 | oveq2 5782 | . . . . . . . . 9 | |
9 | oveq2 5782 | . . . . . . . . 9 | |
10 | 8, 9 | eleq12d 2210 | . . . . . . . 8 |
11 | oveq2 5782 | . . . . . . . . 9 | |
12 | oveq2 5782 | . . . . . . . . 9 | |
13 | 11, 12 | eleq12d 2210 | . . . . . . . 8 |
14 | simpr 109 | . . . . . . . . 9 | |
15 | elnn 4519 | . . . . . . . . . . 11 | |
16 | 15 | ancoms 266 | . . . . . . . . . 10 |
17 | nna0 6370 | . . . . . . . . . 10 | |
18 | 16, 17 | syl 14 | . . . . . . . . 9 |
19 | nna0 6370 | . . . . . . . . . 10 | |
20 | 19 | adantr 274 | . . . . . . . . 9 |
21 | 14, 18, 20 | 3eltr4d 2223 | . . . . . . . 8 |
22 | simprl 520 | . . . . . . . . . . . . 13 | |
23 | simpl 108 | . . . . . . . . . . . . 13 | |
24 | nnacl 6376 | . . . . . . . . . . . . 13 | |
25 | 22, 23, 24 | syl2anc 408 | . . . . . . . . . . . 12 |
26 | nnsucelsuc 6387 | . . . . . . . . . . . 12 | |
27 | 25, 26 | syl 14 | . . . . . . . . . . 11 |
28 | 16 | adantl 275 | . . . . . . . . . . . . . 14 |
29 | nnon 4523 | . . . . . . . . . . . . . 14 | |
30 | 28, 29 | syl 14 | . . . . . . . . . . . . 13 |
31 | nnon 4523 | . . . . . . . . . . . . . 14 | |
32 | 31 | adantr 274 | . . . . . . . . . . . . 13 |
33 | oasuc 6360 | . . . . . . . . . . . . 13 | |
34 | 30, 32, 33 | syl2anc 408 | . . . . . . . . . . . 12 |
35 | nnon 4523 | . . . . . . . . . . . . . 14 | |
36 | 35 | ad2antrl 481 | . . . . . . . . . . . . 13 |
37 | oasuc 6360 | . . . . . . . . . . . . 13 | |
38 | 36, 32, 37 | syl2anc 408 | . . . . . . . . . . . 12 |
39 | 34, 38 | eleq12d 2210 | . . . . . . . . . . 11 |
40 | 27, 39 | bitr4d 190 | . . . . . . . . . 10 |
41 | 40 | biimpd 143 | . . . . . . . . 9 |
42 | 41 | ex 114 | . . . . . . . 8 |
43 | 7, 10, 13, 21, 42 | finds2 4515 | . . . . . . 7 |
44 | 4, 43 | vtoclga 2752 | . . . . . 6 |
45 | 44 | imp 123 | . . . . 5 |
46 | 16 | adantl 275 | . . . . . 6 |
47 | simpl 108 | . . . . . 6 | |
48 | nnacom 6380 | . . . . . 6 | |
49 | 46, 47, 48 | syl2anc 408 | . . . . 5 |
50 | nnacom 6380 | . . . . . . 7 | |
51 | 50 | ancoms 266 | . . . . . 6 |
52 | 51 | adantrr 470 | . . . . 5 |
53 | 45, 49, 52 | 3eltr3d 2222 | . . . 4 |
54 | 53 | 3impb 1177 | . . 3 |
55 | 54 | 3com12 1185 | . 2 |
56 | 55 | 3expia 1183 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 c0 3363 con0 4285 csuc 4287 com 4504 (class class class)co 5774 coa 6310 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-oadd 6317 |
This theorem is referenced by: nnaord 6405 nnmordi 6412 addclpi 7135 addnidpig 7144 archnqq 7225 prarloclemarch2 7227 prarloclemlt 7301 |
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