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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 8299 | . . . . . 6 |
4 | eqid 2170 | . . . . . 6 | |
5 | 3, 4 | fmptd 5650 | . . . . 5 |
6 | 5 | ffvelrnda 5631 | . . . 4 |
7 | monoord2.3 | . . . . . . . . 9 | |
8 | 7 | ralrimiva 2543 | . . . . . . . 8 |
9 | oveq1 5860 | . . . . . . . . . . 11 | |
10 | 9 | fveq2d 5500 | . . . . . . . . . 10 |
11 | fveq2 5496 | . . . . . . . . . 10 | |
12 | 10, 11 | breq12d 4002 | . . . . . . . . 9 |
13 | 12 | cbvralv 2696 | . . . . . . . 8 |
14 | 8, 13 | sylib 121 | . . . . . . 7 |
15 | 14 | r19.21bi 2558 | . . . . . 6 |
16 | fveq2 5496 | . . . . . . . . 9 | |
17 | 16 | eleq1d 2239 | . . . . . . . 8 |
18 | 2 | ralrimiva 2543 | . . . . . . . . 9 |
19 | 18 | adantr 274 | . . . . . . . 8 |
20 | fzp1elp1 10031 | . . . . . . . . . 10 | |
21 | 20 | adantl 275 | . . . . . . . . 9 |
22 | eluzelz 9496 | . . . . . . . . . . . . . 14 | |
23 | 1, 22 | syl 14 | . . . . . . . . . . . . 13 |
24 | 23 | zcnd 9335 | . . . . . . . . . . . 12 |
25 | ax-1cn 7867 | . . . . . . . . . . . 12 | |
26 | npcan 8128 | . . . . . . . . . . . 12 | |
27 | 24, 25, 26 | sylancl 411 | . . . . . . . . . . 11 |
28 | 27 | oveq2d 5869 | . . . . . . . . . 10 |
29 | 28 | adantr 274 | . . . . . . . . 9 |
30 | 21, 29 | eleqtrd 2249 | . . . . . . . 8 |
31 | 17, 19, 30 | rspcdva 2839 | . . . . . . 7 |
32 | 11 | eleq1d 2239 | . . . . . . . 8 |
33 | fzssp1 10023 | . . . . . . . . . 10 | |
34 | 33, 28 | sseqtrid 3197 | . . . . . . . . 9 |
35 | 34 | sselda 3147 | . . . . . . . 8 |
36 | 32, 19, 35 | rspcdva 2839 | . . . . . . 7 |
37 | 31, 36 | lenegd 8443 | . . . . . 6 |
38 | 15, 37 | mpbid 146 | . . . . 5 |
39 | 36 | renegcld 8299 | . . . . . 6 |
40 | 11 | negeqd 8114 | . . . . . . 7 |
41 | 40, 4 | fvmptg 5572 | . . . . . 6 |
42 | 35, 39, 41 | syl2anc 409 | . . . . 5 |
43 | 31 | renegcld 8299 | . . . . . 6 |
44 | 16 | negeqd 8114 | . . . . . . 7 |
45 | 44, 4 | fvmptg 5572 | . . . . . 6 |
46 | 30, 43, 45 | syl2anc 409 | . . . . 5 |
47 | 38, 42, 46 | 3brtr4d 4021 | . . . 4 |
48 | 1, 6, 47 | monoord 10432 | . . 3 |
49 | eluzfz1 9987 | . . . . 5 | |
50 | 1, 49 | syl 14 | . . . 4 |
51 | fveq2 5496 | . . . . . . 7 | |
52 | 51 | eleq1d 2239 | . . . . . 6 |
53 | 52, 18, 50 | rspcdva 2839 | . . . . 5 |
54 | 53 | renegcld 8299 | . . . 4 |
55 | 51 | negeqd 8114 | . . . . 5 |
56 | 55, 4 | fvmptg 5572 | . . . 4 |
57 | 50, 54, 56 | syl2anc 409 | . . 3 |
58 | eluzfz2 9988 | . . . . 5 | |
59 | 1, 58 | syl 14 | . . . 4 |
60 | fveq2 5496 | . . . . . . 7 | |
61 | 60 | eleq1d 2239 | . . . . . 6 |
62 | 61, 18, 59 | rspcdva 2839 | . . . . 5 |
63 | 62 | renegcld 8299 | . . . 4 |
64 | 60 | negeqd 8114 | . . . . 5 |
65 | 64, 4 | fvmptg 5572 | . . . 4 |
66 | 59, 63, 65 | syl2anc 409 | . . 3 |
67 | 48, 57, 66 | 3brtr3d 4020 | . 2 |
68 | 62, 53 | lenegd 8443 | . 2 |
69 | 67, 68 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wcel 2141 wral 2448 class class class wbr 3989 cmpt 4050 cfv 5198 (class class class)co 5853 cc 7772 cr 7773 c1 7775 caddc 7777 cle 7955 cmin 8090 cneg 8091 cz 9212 cuz 9487 cfz 9965 |
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-3or 974 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-mpt 4052 df-id 4278 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-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 df-n0 9136 df-z 9213 df-uz 9488 df-fz 9966 |
This theorem is referenced by: (None) |
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