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Mirrors > Home > HSE Home > Th. List > norm-ii-i | Structured version Visualization version Unicode version |
Description: Triangle inequality for norms. Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 11-Aug-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
norm-ii.1 | |
norm-ii.2 |
Ref | Expression |
---|---|
norm-ii-i |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1re 10039 | . . . . . . . . . . 11 | |
2 | ax-1cn 9994 | . . . . . . . . . . . 12 | |
3 | 2 | cjrebi 13914 | . . . . . . . . . . 11 |
4 | 1, 3 | mpbi 220 | . . . . . . . . . 10 |
5 | 4 | oveq1i 6660 | . . . . . . . . 9 |
6 | norm-ii.2 | . . . . . . . . . . 11 | |
7 | norm-ii.1 | . . . . . . . . . . 11 | |
8 | 6, 7 | hicli 27938 | . . . . . . . . . 10 |
9 | 8 | mulid2i 10043 | . . . . . . . . 9 |
10 | 5, 9 | eqtri 2644 | . . . . . . . 8 |
11 | 7, 6 | hicli 27938 | . . . . . . . . 9 |
12 | 11 | mulid2i 10043 | . . . . . . . 8 |
13 | 10, 12 | oveq12i 6662 | . . . . . . 7 |
14 | abs1 14037 | . . . . . . . 8 | |
15 | 2, 6, 7, 14 | normlem7 27973 | . . . . . . 7 |
16 | 13, 15 | eqbrtrri 4676 | . . . . . 6 |
17 | eqid 2622 | . . . . . . . . . 10 | |
18 | 2, 6, 7, 17 | normlem2 27968 | . . . . . . . . 9 |
19 | 2 | cjcli 13909 | . . . . . . . . . . . 12 |
20 | 19, 8 | mulcli 10045 | . . . . . . . . . . 11 |
21 | 2, 11 | mulcli 10045 | . . . . . . . . . . 11 |
22 | 20, 21 | addcli 10044 | . . . . . . . . . 10 |
23 | 22 | negrebi 10355 | . . . . . . . . 9 |
24 | 18, 23 | mpbi 220 | . . . . . . . 8 |
25 | 13, 24 | eqeltrri 2698 | . . . . . . 7 |
26 | 2re 11090 | . . . . . . . 8 | |
27 | hiidge0 27955 | . . . . . . . . . . 11 | |
28 | 7, 27 | ax-mp 5 | . . . . . . . . . 10 |
29 | hiidrcl 27952 | . . . . . . . . . . . 12 | |
30 | 7, 29 | ax-mp 5 | . . . . . . . . . . 11 |
31 | 30 | sqrtcli 14111 | . . . . . . . . . 10 |
32 | 28, 31 | ax-mp 5 | . . . . . . . . 9 |
33 | hiidge0 27955 | . . . . . . . . . . 11 | |
34 | 6, 33 | ax-mp 5 | . . . . . . . . . 10 |
35 | hiidrcl 27952 | . . . . . . . . . . . 12 | |
36 | 6, 35 | ax-mp 5 | . . . . . . . . . . 11 |
37 | 36 | sqrtcli 14111 | . . . . . . . . . 10 |
38 | 34, 37 | ax-mp 5 | . . . . . . . . 9 |
39 | 32, 38 | remulcli 10054 | . . . . . . . 8 |
40 | 26, 39 | remulcli 10054 | . . . . . . 7 |
41 | 30, 36 | readdcli 10053 | . . . . . . 7 |
42 | 25, 40, 41 | leadd2i 10584 | . . . . . 6 |
43 | 16, 42 | mpbi 220 | . . . . 5 |
44 | 7, 6, 7, 6 | normlem8 27974 | . . . . . 6 |
45 | 11, 8 | addcomi 10227 | . . . . . . 7 |
46 | 45 | oveq2i 6661 | . . . . . 6 |
47 | 44, 46 | eqtri 2644 | . . . . 5 |
48 | 32 | recni 10052 | . . . . . . 7 |
49 | 38 | recni 10052 | . . . . . . 7 |
50 | 48, 49 | binom2i 12974 | . . . . . 6 |
51 | 48 | sqcli 12944 | . . . . . . 7 |
52 | 2cn 11091 | . . . . . . . 8 | |
53 | 48, 49 | mulcli 10045 | . . . . . . . 8 |
54 | 52, 53 | mulcli 10045 | . . . . . . 7 |
55 | 49 | sqcli 12944 | . . . . . . 7 |
56 | 51, 54, 55 | add32i 10259 | . . . . . 6 |
57 | 30 | sqsqrti 14115 | . . . . . . . . 9 |
58 | 28, 57 | ax-mp 5 | . . . . . . . 8 |
59 | 36 | sqsqrti 14115 | . . . . . . . . 9 |
60 | 34, 59 | ax-mp 5 | . . . . . . . 8 |
61 | 58, 60 | oveq12i 6662 | . . . . . . 7 |
62 | 61 | oveq1i 6660 | . . . . . 6 |
63 | 50, 56, 62 | 3eqtri 2648 | . . . . 5 |
64 | 43, 47, 63 | 3brtr4i 4683 | . . . 4 |
65 | 7, 6 | hvaddcli 27875 | . . . . . 6 |
66 | hiidge0 27955 | . . . . . 6 | |
67 | 65, 66 | ax-mp 5 | . . . . 5 |
68 | 32, 38 | readdcli 10053 | . . . . . 6 |
69 | 68 | sqge0i 12951 | . . . . 5 |
70 | hiidrcl 27952 | . . . . . . 7 | |
71 | 65, 70 | ax-mp 5 | . . . . . 6 |
72 | 68 | resqcli 12949 | . . . . . 6 |
73 | 71, 72 | sqrtlei 14128 | . . . . 5 |
74 | 67, 69, 73 | mp2an 708 | . . . 4 |
75 | 64, 74 | mpbi 220 | . . 3 |
76 | 30 | sqrtge0i 14116 | . . . . . 6 |
77 | 28, 76 | ax-mp 5 | . . . . 5 |
78 | 36 | sqrtge0i 14116 | . . . . . 6 |
79 | 34, 78 | ax-mp 5 | . . . . 5 |
80 | 32, 38 | addge0i 10568 | . . . . 5 |
81 | 77, 79, 80 | mp2an 708 | . . . 4 |
82 | 68 | sqrtsqi 14114 | . . . 4 |
83 | 81, 82 | ax-mp 5 | . . 3 |
84 | 75, 83 | breqtri 4678 | . 2 |
85 | normval 27981 | . . 3 | |
86 | 65, 85 | ax-mp 5 | . 2 |
87 | normval 27981 | . . . 4 | |
88 | 7, 87 | ax-mp 5 | . . 3 |
89 | normval 27981 | . . . 4 | |
90 | 6, 89 | ax-mp 5 | . . 3 |
91 | 88, 90 | oveq12i 6662 | . 2 |
92 | 84, 86, 91 | 3brtr4i 4683 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wb 196 wceq 1483 wcel 1990 class class class wbr 4653 cfv 5888 (class class class)co 6650 cr 9935 cc0 9936 c1 9937 caddc 9939 cmul 9941 cle 10075 cneg 10267 c2 11070 cexp 12860 ccj 13836 csqrt 13973 chil 27776 cva 27777 csp 27779 cno 27780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-hfvadd 27857 ax-hv0cl 27860 ax-hfvmul 27862 ax-hvmulass 27864 ax-hvmul0 27867 ax-hfi 27936 ax-his1 27939 ax-his2 27940 ax-his3 27941 ax-his4 27942 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-hnorm 27825 df-hvsub 27828 |
This theorem is referenced by: norm-ii 27995 norm3difi 28004 |
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