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Mirrors > Home > ILE Home > Th. List > nnsub | Unicode version |
Description: Subtraction of positive integers. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 16-May-2014.) |
Ref | Expression |
---|---|
nnsub |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 3969 | . . . . . 6 | |
2 | oveq1 5831 | . . . . . . 7 | |
3 | 2 | eleq1d 2226 | . . . . . 6 |
4 | 1, 3 | imbi12d 233 | . . . . 5 |
5 | 4 | ralbidv 2457 | . . . 4 |
6 | breq2 3969 | . . . . . 6 | |
7 | oveq1 5831 | . . . . . . 7 | |
8 | 7 | eleq1d 2226 | . . . . . 6 |
9 | 6, 8 | imbi12d 233 | . . . . 5 |
10 | 9 | ralbidv 2457 | . . . 4 |
11 | breq2 3969 | . . . . . 6 | |
12 | oveq1 5831 | . . . . . . 7 | |
13 | 12 | eleq1d 2226 | . . . . . 6 |
14 | 11, 13 | imbi12d 233 | . . . . 5 |
15 | 14 | ralbidv 2457 | . . . 4 |
16 | breq2 3969 | . . . . . 6 | |
17 | oveq1 5831 | . . . . . . 7 | |
18 | 17 | eleq1d 2226 | . . . . . 6 |
19 | 16, 18 | imbi12d 233 | . . . . 5 |
20 | 19 | ralbidv 2457 | . . . 4 |
21 | nnnlt1 8859 | . . . . . 6 | |
22 | 21 | pm2.21d 609 | . . . . 5 |
23 | 22 | rgen 2510 | . . . 4 |
24 | breq1 3968 | . . . . . . 7 | |
25 | oveq2 5832 | . . . . . . . 8 | |
26 | 25 | eleq1d 2226 | . . . . . . 7 |
27 | 24, 26 | imbi12d 233 | . . . . . 6 |
28 | 27 | cbvralv 2680 | . . . . 5 |
29 | nncn 8841 | . . . . . . . . . . . . 13 | |
30 | 29 | adantr 274 | . . . . . . . . . . . 12 |
31 | ax-1cn 7825 | . . . . . . . . . . . 12 | |
32 | pncan 8081 | . . . . . . . . . . . 12 | |
33 | 30, 31, 32 | sylancl 410 | . . . . . . . . . . 11 |
34 | simpl 108 | . . . . . . . . . . 11 | |
35 | 33, 34 | eqeltrd 2234 | . . . . . . . . . 10 |
36 | oveq2 5832 | . . . . . . . . . . 11 | |
37 | 36 | eleq1d 2226 | . . . . . . . . . 10 |
38 | 35, 37 | syl5ibrcom 156 | . . . . . . . . 9 |
39 | 38 | a1dd 48 | . . . . . . . 8 |
40 | 39 | a1dd 48 | . . . . . . 7 |
41 | breq1 3968 | . . . . . . . . . 10 | |
42 | oveq2 5832 | . . . . . . . . . . 11 | |
43 | 42 | eleq1d 2226 | . . . . . . . . . 10 |
44 | 41, 43 | imbi12d 233 | . . . . . . . . 9 |
45 | 44 | rspcv 2812 | . . . . . . . 8 |
46 | nnre 8840 | . . . . . . . . . . 11 | |
47 | nnre 8840 | . . . . . . . . . . 11 | |
48 | 1re 7877 | . . . . . . . . . . . 12 | |
49 | ltsubadd 8307 | . . . . . . . . . . . 12 | |
50 | 48, 49 | mp3an2 1307 | . . . . . . . . . . 11 |
51 | 46, 47, 50 | syl2anr 288 | . . . . . . . . . 10 |
52 | nncn 8841 | . . . . . . . . . . . 12 | |
53 | subsub3 8107 | . . . . . . . . . . . . 13 | |
54 | 31, 53 | mp3an3 1308 | . . . . . . . . . . . 12 |
55 | 29, 52, 54 | syl2an 287 | . . . . . . . . . . 11 |
56 | 55 | eleq1d 2226 | . . . . . . . . . 10 |
57 | 51, 56 | imbi12d 233 | . . . . . . . . 9 |
58 | 57 | biimpd 143 | . . . . . . . 8 |
59 | 45, 58 | syl9r 73 | . . . . . . 7 |
60 | nn1m1nn 8851 | . . . . . . . 8 | |
61 | 60 | adantl 275 | . . . . . . 7 |
62 | 40, 59, 61 | mpjaod 708 | . . . . . 6 |
63 | 62 | ralrimdva 2537 | . . . . 5 |
64 | 28, 63 | syl5bi 151 | . . . 4 |
65 | 5, 10, 15, 20, 23, 64 | nnind 8849 | . . 3 |
66 | breq1 3968 | . . . . 5 | |
67 | oveq2 5832 | . . . . . 6 | |
68 | 67 | eleq1d 2226 | . . . . 5 |
69 | 66, 68 | imbi12d 233 | . . . 4 |
70 | 69 | rspcva 2814 | . . 3 |
71 | 65, 70 | sylan2 284 | . 2 |
72 | nngt0 8858 | . . 3 | |
73 | nnre 8840 | . . . 4 | |
74 | nnre 8840 | . . . 4 | |
75 | posdif 8330 | . . . 4 | |
76 | 73, 74, 75 | syl2an 287 | . . 3 |
77 | 72, 76 | syl5ibr 155 | . 2 |
78 | 71, 77 | impbid 128 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 698 wceq 1335 wcel 2128 wral 2435 class class class wbr 3965 (class class class)co 5824 cc 7730 cr 7731 cc0 7732 c1 7733 caddc 7735 clt 7912 cmin 8046 cn 8833 |
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-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-addcom 7832 ax-addass 7834 ax-distr 7836 ax-i2m1 7837 ax-0lt1 7838 ax-0id 7840 ax-rnegex 7841 ax-cnre 7843 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 ax-pre-ltadd 7848 |
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-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-br 3966 df-opab 4026 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-iota 5135 df-fun 5172 df-fv 5178 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 df-sub 8048 df-neg 8049 df-inn 8834 |
This theorem is referenced by: nnsubi 8873 uz3m2nn 9484 |
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