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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 5861 | . . . . . . . . 9 | |
2 | oveq2 5861 | . . . . . . . . 9 | |
3 | 1, 2 | eleq12d 2241 | . . . . . . . 8 |
4 | 3 | imbi2d 229 | . . . . . . 7 |
5 | oveq2 5861 | . . . . . . . . 9 | |
6 | oveq2 5861 | . . . . . . . . 9 | |
7 | 5, 6 | eleq12d 2241 | . . . . . . . 8 |
8 | oveq2 5861 | . . . . . . . . 9 | |
9 | oveq2 5861 | . . . . . . . . 9 | |
10 | 8, 9 | eleq12d 2241 | . . . . . . . 8 |
11 | oveq2 5861 | . . . . . . . . 9 | |
12 | oveq2 5861 | . . . . . . . . 9 | |
13 | 11, 12 | eleq12d 2241 | . . . . . . . 8 |
14 | simpr 109 | . . . . . . . . 9 | |
15 | elnn 4590 | . . . . . . . . . . 11 | |
16 | 15 | ancoms 266 | . . . . . . . . . 10 |
17 | nna0 6453 | . . . . . . . . . 10 | |
18 | 16, 17 | syl 14 | . . . . . . . . 9 |
19 | nna0 6453 | . . . . . . . . . 10 | |
20 | 19 | adantr 274 | . . . . . . . . 9 |
21 | 14, 18, 20 | 3eltr4d 2254 | . . . . . . . 8 |
22 | simprl 526 | . . . . . . . . . . . . 13 | |
23 | simpl 108 | . . . . . . . . . . . . 13 | |
24 | nnacl 6459 | . . . . . . . . . . . . 13 | |
25 | 22, 23, 24 | syl2anc 409 | . . . . . . . . . . . 12 |
26 | nnsucelsuc 6470 | . . . . . . . . . . . 12 | |
27 | 25, 26 | syl 14 | . . . . . . . . . . 11 |
28 | 16 | adantl 275 | . . . . . . . . . . . . . 14 |
29 | nnon 4594 | . . . . . . . . . . . . . 14 | |
30 | 28, 29 | syl 14 | . . . . . . . . . . . . 13 |
31 | nnon 4594 | . . . . . . . . . . . . . 14 | |
32 | 31 | adantr 274 | . . . . . . . . . . . . 13 |
33 | oasuc 6443 | . . . . . . . . . . . . 13 | |
34 | 30, 32, 33 | syl2anc 409 | . . . . . . . . . . . 12 |
35 | nnon 4594 | . . . . . . . . . . . . . 14 | |
36 | 35 | ad2antrl 487 | . . . . . . . . . . . . 13 |
37 | oasuc 6443 | . . . . . . . . . . . . 13 | |
38 | 36, 32, 37 | syl2anc 409 | . . . . . . . . . . . 12 |
39 | 34, 38 | eleq12d 2241 | . . . . . . . . . . 11 |
40 | 27, 39 | bitr4d 190 | . . . . . . . . . 10 |
41 | 40 | biimpd 143 | . . . . . . . . 9 |
42 | 41 | ex 114 | . . . . . . . 8 |
43 | 7, 10, 13, 21, 42 | finds2 4585 | . . . . . . 7 |
44 | 4, 43 | vtoclga 2796 | . . . . . 6 |
45 | 44 | imp 123 | . . . . 5 |
46 | 16 | adantl 275 | . . . . . 6 |
47 | simpl 108 | . . . . . 6 | |
48 | nnacom 6463 | . . . . . 6 | |
49 | 46, 47, 48 | syl2anc 409 | . . . . 5 |
50 | nnacom 6463 | . . . . . . 7 | |
51 | 50 | ancoms 266 | . . . . . 6 |
52 | 51 | adantrr 476 | . . . . 5 |
53 | 45, 49, 52 | 3eltr3d 2253 | . . . 4 |
54 | 53 | 3impb 1194 | . . 3 |
55 | 54 | 3com12 1202 | . 2 |
56 | 55 | 3expia 1200 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1348 wcel 2141 c0 3414 con0 4348 csuc 4350 com 4574 (class class class)co 5853 coa 6392 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-oadd 6399 |
This theorem is referenced by: nnaord 6488 nnmordi 6495 addclpi 7289 addnidpig 7298 archnqq 7379 prarloclemarch2 7381 prarloclemlt 7455 |
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