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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 5540 | . . . . . . . . 9 | |
2 | oveq2 5540 | . . . . . . . . 9 | |
3 | 1, 2 | eleq12d 2149 | . . . . . . . 8 |
4 | 3 | imbi2d 228 | . . . . . . 7 |
5 | oveq2 5540 | . . . . . . . . 9 | |
6 | oveq2 5540 | . . . . . . . . 9 | |
7 | 5, 6 | eleq12d 2149 | . . . . . . . 8 |
8 | oveq2 5540 | . . . . . . . . 9 | |
9 | oveq2 5540 | . . . . . . . . 9 | |
10 | 8, 9 | eleq12d 2149 | . . . . . . . 8 |
11 | oveq2 5540 | . . . . . . . . 9 | |
12 | oveq2 5540 | . . . . . . . . 9 | |
13 | 11, 12 | eleq12d 2149 | . . . . . . . 8 |
14 | simpr 108 | . . . . . . . . 9 | |
15 | elnn 4346 | . . . . . . . . . . 11 | |
16 | 15 | ancoms 264 | . . . . . . . . . 10 |
17 | nna0 6076 | . . . . . . . . . 10 | |
18 | 16, 17 | syl 14 | . . . . . . . . 9 |
19 | nna0 6076 | . . . . . . . . . 10 | |
20 | 19 | adantr 270 | . . . . . . . . 9 |
21 | 14, 18, 20 | 3eltr4d 2162 | . . . . . . . 8 |
22 | simprl 497 | . . . . . . . . . . . . 13 | |
23 | simpl 107 | . . . . . . . . . . . . 13 | |
24 | nnacl 6082 | . . . . . . . . . . . . 13 | |
25 | 22, 23, 24 | syl2anc 403 | . . . . . . . . . . . 12 |
26 | nnsucelsuc 6093 | . . . . . . . . . . . 12 | |
27 | 25, 26 | syl 14 | . . . . . . . . . . 11 |
28 | 16 | adantl 271 | . . . . . . . . . . . . . 14 |
29 | nnon 4350 | . . . . . . . . . . . . . 14 | |
30 | 28, 29 | syl 14 | . . . . . . . . . . . . 13 |
31 | nnon 4350 | . . . . . . . . . . . . . 14 | |
32 | 31 | adantr 270 | . . . . . . . . . . . . 13 |
33 | oasuc 6067 | . . . . . . . . . . . . 13 | |
34 | 30, 32, 33 | syl2anc 403 | . . . . . . . . . . . 12 |
35 | nnon 4350 | . . . . . . . . . . . . . 14 | |
36 | 35 | ad2antrl 473 | . . . . . . . . . . . . 13 |
37 | oasuc 6067 | . . . . . . . . . . . . 13 | |
38 | 36, 32, 37 | syl2anc 403 | . . . . . . . . . . . 12 |
39 | 34, 38 | eleq12d 2149 | . . . . . . . . . . 11 |
40 | 27, 39 | bitr4d 189 | . . . . . . . . . 10 |
41 | 40 | biimpd 142 | . . . . . . . . 9 |
42 | 41 | ex 113 | . . . . . . . 8 |
43 | 7, 10, 13, 21, 42 | finds2 4342 | . . . . . . 7 |
44 | 4, 43 | vtoclga 2664 | . . . . . 6 |
45 | 44 | imp 122 | . . . . 5 |
46 | 16 | adantl 271 | . . . . . 6 |
47 | simpl 107 | . . . . . 6 | |
48 | nnacom 6086 | . . . . . 6 | |
49 | 46, 47, 48 | syl2anc 403 | . . . . 5 |
50 | nnacom 6086 | . . . . . . 7 | |
51 | 50 | ancoms 264 | . . . . . 6 |
52 | 51 | adantrr 462 | . . . . 5 |
53 | 45, 49, 52 | 3eltr3d 2161 | . . . 4 |
54 | 53 | 3impb 1134 | . . 3 |
55 | 54 | 3com12 1142 | . 2 |
56 | 55 | 3expia 1140 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wcel 1433 c0 3251 con0 4118 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: nnaord 6105 nnmordi 6112 addclpi 6517 addnidpig 6526 archnqq 6607 prarloclemarch2 6609 prarloclemlt 6683 |
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