Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 3933 | . . . . . 6 | |
2 | oveq1 5781 | . . . . . . 7 | |
3 | 2 | eleq1d 2208 | . . . . . 6 |
4 | 1, 3 | imbi12d 233 | . . . . 5 |
5 | 4 | ralbidv 2437 | . . . 4 |
6 | breq2 3933 | . . . . . 6 | |
7 | oveq1 5781 | . . . . . . 7 | |
8 | 7 | eleq1d 2208 | . . . . . 6 |
9 | 6, 8 | imbi12d 233 | . . . . 5 |
10 | 9 | ralbidv 2437 | . . . 4 |
11 | breq2 3933 | . . . . . 6 | |
12 | oveq1 5781 | . . . . . . 7 | |
13 | 12 | eleq1d 2208 | . . . . . 6 |
14 | 11, 13 | imbi12d 233 | . . . . 5 |
15 | 14 | ralbidv 2437 | . . . 4 |
16 | breq2 3933 | . . . . . 6 | |
17 | oveq1 5781 | . . . . . . 7 | |
18 | 17 | eleq1d 2208 | . . . . . 6 |
19 | 16, 18 | imbi12d 233 | . . . . 5 |
20 | 19 | ralbidv 2437 | . . . 4 |
21 | nnnlt1 8746 | . . . . . 6 | |
22 | 21 | pm2.21d 608 | . . . . 5 |
23 | 22 | rgen 2485 | . . . 4 |
24 | breq1 3932 | . . . . . . 7 | |
25 | oveq2 5782 | . . . . . . . 8 | |
26 | 25 | eleq1d 2208 | . . . . . . 7 |
27 | 24, 26 | imbi12d 233 | . . . . . 6 |
28 | 27 | cbvralv 2654 | . . . . 5 |
29 | nncn 8728 | . . . . . . . . . . . . 13 | |
30 | 29 | adantr 274 | . . . . . . . . . . . 12 |
31 | ax-1cn 7713 | . . . . . . . . . . . 12 | |
32 | pncan 7968 | . . . . . . . . . . . 12 | |
33 | 30, 31, 32 | sylancl 409 | . . . . . . . . . . 11 |
34 | simpl 108 | . . . . . . . . . . 11 | |
35 | 33, 34 | eqeltrd 2216 | . . . . . . . . . 10 |
36 | oveq2 5782 | . . . . . . . . . . 11 | |
37 | 36 | eleq1d 2208 | . . . . . . . . . 10 |
38 | 35, 37 | syl5ibrcom 156 | . . . . . . . . 9 |
39 | 38 | a1dd 48 | . . . . . . . 8 |
40 | 39 | a1dd 48 | . . . . . . 7 |
41 | breq1 3932 | . . . . . . . . . 10 | |
42 | oveq2 5782 | . . . . . . . . . . 11 | |
43 | 42 | eleq1d 2208 | . . . . . . . . . 10 |
44 | 41, 43 | imbi12d 233 | . . . . . . . . 9 |
45 | 44 | rspcv 2785 | . . . . . . . 8 |
46 | nnre 8727 | . . . . . . . . . . 11 | |
47 | nnre 8727 | . . . . . . . . . . 11 | |
48 | 1re 7765 | . . . . . . . . . . . 12 | |
49 | ltsubadd 8194 | . . . . . . . . . . . 12 | |
50 | 48, 49 | mp3an2 1303 | . . . . . . . . . . 11 |
51 | 46, 47, 50 | syl2anr 288 | . . . . . . . . . 10 |
52 | nncn 8728 | . . . . . . . . . . . 12 | |
53 | subsub3 7994 | . . . . . . . . . . . . 13 | |
54 | 31, 53 | mp3an3 1304 | . . . . . . . . . . . 12 |
55 | 29, 52, 54 | syl2an 287 | . . . . . . . . . . 11 |
56 | 55 | eleq1d 2208 | . . . . . . . . . 10 |
57 | 51, 56 | imbi12d 233 | . . . . . . . . 9 |
58 | 57 | biimpd 143 | . . . . . . . 8 |
59 | 45, 58 | syl9r 73 | . . . . . . 7 |
60 | nn1m1nn 8738 | . . . . . . . 8 | |
61 | 60 | adantl 275 | . . . . . . 7 |
62 | 40, 59, 61 | mpjaod 707 | . . . . . 6 |
63 | 62 | ralrimdva 2512 | . . . . 5 |
64 | 28, 63 | syl5bi 151 | . . . 4 |
65 | 5, 10, 15, 20, 23, 64 | nnind 8736 | . . 3 |
66 | breq1 3932 | . . . . 5 | |
67 | oveq2 5782 | . . . . . 6 | |
68 | 67 | eleq1d 2208 | . . . . 5 |
69 | 66, 68 | imbi12d 233 | . . . 4 |
70 | 69 | rspcva 2787 | . . 3 |
71 | 65, 70 | sylan2 284 | . 2 |
72 | nngt0 8745 | . . 3 | |
73 | nnre 8727 | . . . 4 | |
74 | nnre 8727 | . . . 4 | |
75 | posdif 8217 | . . . 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 697 wceq 1331 wcel 1480 wral 2416 class class class wbr 3929 (class class class)co 5774 cc 7618 cr 7619 cc0 7620 c1 7621 caddc 7623 clt 7800 cmin 7933 cn 8720 |
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-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
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-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 |
This theorem is referenced by: nnsubi 8760 uz3m2nn 9368 |
Copyright terms: Public domain | W3C validator |