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Mirrors > Home > ILE Home > Th. List > monoord2 | Unicode version |
Description: Ordering relation for a monotonic sequence, decreasing case. (Contributed by Mario Carneiro, 18-Jul-2014.) |
Ref | Expression |
---|---|
monoord2.1 | |
monoord2.2 | |
monoord2.3 |
Ref | Expression |
---|---|
monoord2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | monoord2.1 | . . . 4 | |
2 | monoord2.2 | . . . . . . 7 | |
3 | 2 | renegcld 8256 | . . . . . 6 |
4 | eqid 2157 | . . . . . 6 | |
5 | 3, 4 | fmptd 5622 | . . . . 5 |
6 | 5 | ffvelrnda 5603 | . . . 4 |
7 | monoord2.3 | . . . . . . . . 9 | |
8 | 7 | ralrimiva 2530 | . . . . . . . 8 |
9 | oveq1 5832 | . . . . . . . . . . 11 | |
10 | 9 | fveq2d 5473 | . . . . . . . . . 10 |
11 | fveq2 5469 | . . . . . . . . . 10 | |
12 | 10, 11 | breq12d 3979 | . . . . . . . . 9 |
13 | 12 | cbvralv 2680 | . . . . . . . 8 |
14 | 8, 13 | sylib 121 | . . . . . . 7 |
15 | 14 | r19.21bi 2545 | . . . . . 6 |
16 | fveq2 5469 | . . . . . . . . 9 | |
17 | 16 | eleq1d 2226 | . . . . . . . 8 |
18 | 2 | ralrimiva 2530 | . . . . . . . . 9 |
19 | 18 | adantr 274 | . . . . . . . 8 |
20 | fzp1elp1 9978 | . . . . . . . . . 10 | |
21 | 20 | adantl 275 | . . . . . . . . 9 |
22 | eluzelz 9449 | . . . . . . . . . . . . . 14 | |
23 | 1, 22 | syl 14 | . . . . . . . . . . . . 13 |
24 | 23 | zcnd 9288 | . . . . . . . . . . . 12 |
25 | ax-1cn 7826 | . . . . . . . . . . . 12 | |
26 | npcan 8085 | . . . . . . . . . . . 12 | |
27 | 24, 25, 26 | sylancl 410 | . . . . . . . . . . 11 |
28 | 27 | oveq2d 5841 | . . . . . . . . . 10 |
29 | 28 | adantr 274 | . . . . . . . . 9 |
30 | 21, 29 | eleqtrd 2236 | . . . . . . . 8 |
31 | 17, 19, 30 | rspcdva 2821 | . . . . . . 7 |
32 | 11 | eleq1d 2226 | . . . . . . . 8 |
33 | fzssp1 9970 | . . . . . . . . . 10 | |
34 | 33, 28 | sseqtrid 3178 | . . . . . . . . 9 |
35 | 34 | sselda 3128 | . . . . . . . 8 |
36 | 32, 19, 35 | rspcdva 2821 | . . . . . . 7 |
37 | 31, 36 | lenegd 8400 | . . . . . 6 |
38 | 15, 37 | mpbid 146 | . . . . 5 |
39 | 36 | renegcld 8256 | . . . . . 6 |
40 | 11 | negeqd 8071 | . . . . . . 7 |
41 | 40, 4 | fvmptg 5545 | . . . . . 6 |
42 | 35, 39, 41 | syl2anc 409 | . . . . 5 |
43 | 31 | renegcld 8256 | . . . . . 6 |
44 | 16 | negeqd 8071 | . . . . . . 7 |
45 | 44, 4 | fvmptg 5545 | . . . . . 6 |
46 | 30, 43, 45 | syl2anc 409 | . . . . 5 |
47 | 38, 42, 46 | 3brtr4d 3997 | . . . 4 |
48 | 1, 6, 47 | monoord 10379 | . . 3 |
49 | eluzfz1 9934 | . . . . 5 | |
50 | 1, 49 | syl 14 | . . . 4 |
51 | fveq2 5469 | . . . . . . 7 | |
52 | 51 | eleq1d 2226 | . . . . . 6 |
53 | 52, 18, 50 | rspcdva 2821 | . . . . 5 |
54 | 53 | renegcld 8256 | . . . 4 |
55 | 51 | negeqd 8071 | . . . . 5 |
56 | 55, 4 | fvmptg 5545 | . . . 4 |
57 | 50, 54, 56 | syl2anc 409 | . . 3 |
58 | eluzfz2 9935 | . . . . 5 | |
59 | 1, 58 | syl 14 | . . . 4 |
60 | fveq2 5469 | . . . . . . 7 | |
61 | 60 | eleq1d 2226 | . . . . . 6 |
62 | 61, 18, 59 | rspcdva 2821 | . . . . 5 |
63 | 62 | renegcld 8256 | . . . 4 |
64 | 60 | negeqd 8071 | . . . . 5 |
65 | 64, 4 | fvmptg 5545 | . . . 4 |
66 | 59, 63, 65 | syl2anc 409 | . . 3 |
67 | 48, 57, 66 | 3brtr3d 3996 | . 2 |
68 | 62, 53 | lenegd 8400 | . 2 |
69 | 67, 68 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1335 wcel 2128 wral 2435 class class class wbr 3966 cmpt 4026 cfv 5171 (class class class)co 5825 cc 7731 cr 7732 c1 7734 caddc 7736 cle 7914 cmin 8047 cneg 8048 cz 9168 cuz 9440 cfz 9913 |
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 4083 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-cnex 7824 ax-resscn 7825 ax-1cn 7826 ax-1re 7827 ax-icn 7828 ax-addcl 7829 ax-addrcl 7830 ax-mulcl 7831 ax-addcom 7833 ax-addass 7835 ax-distr 7837 ax-i2m1 7838 ax-0lt1 7839 ax-0id 7841 ax-rnegex 7842 ax-cnre 7844 ax-pre-ltirr 7845 ax-pre-ltwlin 7846 ax-pre-lttrn 7847 ax-pre-ltadd 7849 |
This theorem depends on definitions: df-bi 116 df-3or 964 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 3774 df-int 3809 df-br 3967 df-opab 4027 df-mpt 4028 df-id 4254 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-fv 5179 df-riota 5781 df-ov 5828 df-oprab 5829 df-mpo 5830 df-pnf 7915 df-mnf 7916 df-xr 7917 df-ltxr 7918 df-le 7919 df-sub 8049 df-neg 8050 df-inn 8835 df-n0 9092 df-z 9169 df-uz 9441 df-fz 9914 |
This theorem is referenced by: (None) |
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