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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 3993 | . . . . . 6 | |
2 | oveq1 5860 | . . . . . . 7 | |
3 | 2 | eleq1d 2239 | . . . . . 6 |
4 | 1, 3 | imbi12d 233 | . . . . 5 |
5 | 4 | ralbidv 2470 | . . . 4 |
6 | breq2 3993 | . . . . . 6 | |
7 | oveq1 5860 | . . . . . . 7 | |
8 | 7 | eleq1d 2239 | . . . . . 6 |
9 | 6, 8 | imbi12d 233 | . . . . 5 |
10 | 9 | ralbidv 2470 | . . . 4 |
11 | breq2 3993 | . . . . . 6 | |
12 | oveq1 5860 | . . . . . . 7 | |
13 | 12 | eleq1d 2239 | . . . . . 6 |
14 | 11, 13 | imbi12d 233 | . . . . 5 |
15 | 14 | ralbidv 2470 | . . . 4 |
16 | breq2 3993 | . . . . . 6 | |
17 | oveq1 5860 | . . . . . . 7 | |
18 | 17 | eleq1d 2239 | . . . . . 6 |
19 | 16, 18 | imbi12d 233 | . . . . 5 |
20 | 19 | ralbidv 2470 | . . . 4 |
21 | nnnlt1 8904 | . . . . . 6 | |
22 | 21 | pm2.21d 614 | . . . . 5 |
23 | 22 | rgen 2523 | . . . 4 |
24 | breq1 3992 | . . . . . . 7 | |
25 | oveq2 5861 | . . . . . . . 8 | |
26 | 25 | eleq1d 2239 | . . . . . . 7 |
27 | 24, 26 | imbi12d 233 | . . . . . 6 |
28 | 27 | cbvralv 2696 | . . . . 5 |
29 | nncn 8886 | . . . . . . . . . . . . 13 | |
30 | 29 | adantr 274 | . . . . . . . . . . . 12 |
31 | ax-1cn 7867 | . . . . . . . . . . . 12 | |
32 | pncan 8125 | . . . . . . . . . . . 12 | |
33 | 30, 31, 32 | sylancl 411 | . . . . . . . . . . 11 |
34 | simpl 108 | . . . . . . . . . . 11 | |
35 | 33, 34 | eqeltrd 2247 | . . . . . . . . . 10 |
36 | oveq2 5861 | . . . . . . . . . . 11 | |
37 | 36 | eleq1d 2239 | . . . . . . . . . 10 |
38 | 35, 37 | syl5ibrcom 156 | . . . . . . . . 9 |
39 | 38 | a1dd 48 | . . . . . . . 8 |
40 | 39 | a1dd 48 | . . . . . . 7 |
41 | breq1 3992 | . . . . . . . . . 10 | |
42 | oveq2 5861 | . . . . . . . . . . 11 | |
43 | 42 | eleq1d 2239 | . . . . . . . . . 10 |
44 | 41, 43 | imbi12d 233 | . . . . . . . . 9 |
45 | 44 | rspcv 2830 | . . . . . . . 8 |
46 | nnre 8885 | . . . . . . . . . . 11 | |
47 | nnre 8885 | . . . . . . . . . . 11 | |
48 | 1re 7919 | . . . . . . . . . . . 12 | |
49 | ltsubadd 8351 | . . . . . . . . . . . 12 | |
50 | 48, 49 | mp3an2 1320 | . . . . . . . . . . 11 |
51 | 46, 47, 50 | syl2anr 288 | . . . . . . . . . 10 |
52 | nncn 8886 | . . . . . . . . . . . 12 | |
53 | subsub3 8151 | . . . . . . . . . . . . 13 | |
54 | 31, 53 | mp3an3 1321 | . . . . . . . . . . . 12 |
55 | 29, 52, 54 | syl2an 287 | . . . . . . . . . . 11 |
56 | 55 | eleq1d 2239 | . . . . . . . . . 10 |
57 | 51, 56 | imbi12d 233 | . . . . . . . . 9 |
58 | 57 | biimpd 143 | . . . . . . . 8 |
59 | 45, 58 | syl9r 73 | . . . . . . 7 |
60 | nn1m1nn 8896 | . . . . . . . 8 | |
61 | 60 | adantl 275 | . . . . . . 7 |
62 | 40, 59, 61 | mpjaod 713 | . . . . . 6 |
63 | 62 | ralrimdva 2550 | . . . . 5 |
64 | 28, 63 | syl5bi 151 | . . . 4 |
65 | 5, 10, 15, 20, 23, 64 | nnind 8894 | . . 3 |
66 | breq1 3992 | . . . . 5 | |
67 | oveq2 5861 | . . . . . 6 | |
68 | 67 | eleq1d 2239 | . . . . 5 |
69 | 66, 68 | imbi12d 233 | . . . 4 |
70 | 69 | rspcva 2832 | . . 3 |
71 | 65, 70 | sylan2 284 | . 2 |
72 | nngt0 8903 | . . 3 | |
73 | nnre 8885 | . . . 4 | |
74 | nnre 8885 | . . . 4 | |
75 | posdif 8374 | . . . 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 703 wceq 1348 wcel 2141 wral 2448 class class class wbr 3989 (class class class)co 5853 cc 7772 cr 7773 cc0 7774 c1 7775 caddc 7777 clt 7954 cmin 8090 cn 8878 |
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-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-ltadd 7890 |
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-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-inn 8879 |
This theorem is referenced by: nnsubi 8918 uz3m2nn 9532 pythagtriplem13 12230 |
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