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Mirrors > Home > ILE Home > Th. List > xrbdtri | Unicode version |
Description: Triangle inequality for bounded values. (Contributed by Jim Kingdon, 15-May-2023.) |
Ref | Expression |
---|---|
xrbdtri | inf inf inf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 | . . . . . 6 | |
2 | simp1r 1006 | . . . . . . 7 | |
3 | 2 | ad3antrrr 483 | . . . . . 6 |
4 | simplr 519 | . . . . . 6 | |
5 | simp2r 1008 | . . . . . . 7 | |
6 | 5 | ad3antrrr 483 | . . . . . 6 |
7 | simpllr 523 | . . . . . . 7 | |
8 | simp3r 1010 | . . . . . . . 8 | |
9 | 8 | ad3antrrr 483 | . . . . . . 7 |
10 | 7, 9 | elrpd 9481 | . . . . . 6 |
11 | bdtri 11011 | . . . . . 6 inf inf inf | |
12 | 1, 3, 4, 6, 10, 11 | syl221anc 1227 | . . . . 5 inf inf inf |
13 | 1, 4 | rexaddd 9637 | . . . . . . . 8 |
14 | 13 | preq1d 3606 | . . . . . . 7 |
15 | 14 | infeq1d 6899 | . . . . . 6 inf inf |
16 | 1, 4 | readdcld 7795 | . . . . . . 7 |
17 | xrminrecl 11042 | . . . . . . 7 inf inf | |
18 | 16, 7, 17 | syl2anc 408 | . . . . . 6 inf inf |
19 | 15, 18 | eqtrd 2172 | . . . . 5 inf inf |
20 | xrminrecl 11042 | . . . . . . . 8 inf inf | |
21 | 1, 7, 20 | syl2anc 408 | . . . . . . 7 inf inf |
22 | xrminrecl 11042 | . . . . . . . 8 inf inf | |
23 | 4, 7, 22 | syl2anc 408 | . . . . . . 7 inf inf |
24 | 21, 23 | oveq12d 5792 | . . . . . 6 inf inf inf inf |
25 | mincl 11002 | . . . . . . . 8 inf | |
26 | 1, 7, 25 | syl2anc 408 | . . . . . . 7 inf |
27 | mincl 11002 | . . . . . . . 8 inf | |
28 | 4, 7, 27 | syl2anc 408 | . . . . . . 7 inf |
29 | 26, 28 | rexaddd 9637 | . . . . . 6 inf inf inf inf |
30 | 24, 29 | eqtrd 2172 | . . . . 5 inf inf inf inf |
31 | 12, 19, 30 | 3brtr4d 3960 | . . . 4 inf inf inf |
32 | simp3l 1009 | . . . . . . . 8 | |
33 | 32 | xaddid1d 9647 | . . . . . . 7 |
34 | 32 | xrleidd 9587 | . . . . . . . 8 |
35 | 0xr 7812 | . . . . . . . . . . 11 | |
36 | 35 | a1i 9 | . . . . . . . . . 10 |
37 | 36, 32, 8 | xrltled 9585 | . . . . . . . . 9 |
38 | simp2l 1007 | . . . . . . . . . 10 | |
39 | xrlemininf 11040 | . . . . . . . . . 10 inf | |
40 | 36, 38, 32, 39 | syl3anc 1216 | . . . . . . . . 9 inf |
41 | 5, 37, 40 | mpbir2and 928 | . . . . . . . 8 inf |
42 | xrmincl 11035 | . . . . . . . . . 10 inf | |
43 | 38, 32, 42 | syl2anc 408 | . . . . . . . . 9 inf |
44 | xle2add 9662 | . . . . . . . . 9 inf inf inf | |
45 | 32, 36, 32, 43, 44 | syl22anc 1217 | . . . . . . . 8 inf inf |
46 | 34, 41, 45 | mp2and 429 | . . . . . . 7 inf |
47 | 33, 46 | eqbrtrrd 3952 | . . . . . 6 inf |
48 | 47 | ad3antrrr 483 | . . . . 5 inf |
49 | simp1l 1005 | . . . . . . . 8 | |
50 | 49, 38 | xaddcld 9667 | . . . . . . 7 |
51 | 50 | ad3antrrr 483 | . . . . . 6 |
52 | 32 | ad3antrrr 483 | . . . . . 6 |
53 | pnfge 9575 | . . . . . . . 8 | |
54 | 52, 53 | syl 14 | . . . . . . 7 |
55 | simpr 109 | . . . . . . . . 9 | |
56 | 55 | oveq1d 5789 | . . . . . . . 8 |
57 | simpl2l 1034 | . . . . . . . . . 10 | |
58 | 57 | ad2antrr 479 | . . . . . . . . 9 |
59 | simplr 519 | . . . . . . . . . 10 | |
60 | 59 | renemnfd 7817 | . . . . . . . . 9 |
61 | xaddpnf2 9630 | . . . . . . . . 9 | |
62 | 58, 60, 61 | syl2anc 408 | . . . . . . . 8 |
63 | 56, 62 | eqtrd 2172 | . . . . . . 7 |
64 | 54, 63 | breqtrrd 3956 | . . . . . 6 |
65 | xrmineqinf 11038 | . . . . . 6 inf | |
66 | 51, 52, 64, 65 | syl3anc 1216 | . . . . 5 inf |
67 | 49 | ad3antrrr 483 | . . . . . . 7 |
68 | 54, 55 | breqtrrd 3956 | . . . . . . 7 |
69 | xrmineqinf 11038 | . . . . . . 7 inf | |
70 | 67, 52, 68, 69 | syl3anc 1216 | . . . . . 6 inf |
71 | 70 | oveq1d 5789 | . . . . 5 inf inf inf |
72 | 48, 66, 71 | 3brtr4d 3960 | . . . 4 inf inf inf |
73 | simpr 109 | . . . . 5 | |
74 | ge0nemnf 9607 | . . . . . . 7 | |
75 | 49, 2, 74 | syl2anc 408 | . . . . . 6 |
76 | 75 | ad3antrrr 483 | . . . . 5 |
77 | 73, 76 | pm2.21ddne 2391 | . . . 4 inf inf inf |
78 | elxr 9563 | . . . . . 6 | |
79 | 49, 78 | sylib 121 | . . . . 5 |
80 | 79 | ad2antrr 479 | . . . 4 |
81 | 31, 72, 77, 80 | mpjao3dan 1285 | . . 3 inf inf inf |
82 | xrlemininf 11040 | . . . . . . . 8 inf | |
83 | 36, 49, 32, 82 | syl3anc 1216 | . . . . . . 7 inf |
84 | 2, 37, 83 | mpbir2and 928 | . . . . . 6 inf |
85 | xrmincl 11035 | . . . . . . . 8 inf | |
86 | 49, 32, 85 | syl2anc 408 | . . . . . . 7 inf |
87 | xle2add 9662 | . . . . . . 7 inf inf inf | |
88 | 36, 32, 86, 32, 87 | syl22anc 1217 | . . . . . 6 inf inf |
89 | 84, 34, 88 | mp2and 429 | . . . . 5 inf |
90 | 89 | ad2antrr 479 | . . . 4 inf |
91 | 50 | ad2antrr 479 | . . . . . 6 |
92 | 32 | ad2antrr 479 | . . . . . 6 |
93 | 92, 53 | syl 14 | . . . . . . 7 |
94 | simpr 109 | . . . . . . . . 9 | |
95 | 94 | oveq2d 5790 | . . . . . . . 8 |
96 | xaddpnf1 9629 | . . . . . . . . . 10 | |
97 | 49, 75, 96 | syl2anc 408 | . . . . . . . . 9 |
98 | 97 | ad2antrr 479 | . . . . . . . 8 |
99 | 95, 98 | eqtrd 2172 | . . . . . . 7 |
100 | 93, 99 | breqtrrd 3956 | . . . . . 6 |
101 | 91, 92, 100, 65 | syl3anc 1216 | . . . . 5 inf |
102 | xaddid2 9646 | . . . . . 6 | |
103 | 92, 102 | syl 14 | . . . . 5 |
104 | 101, 103 | eqtr4d 2175 | . . . 4 inf |
105 | 57 | adantr 274 | . . . . . 6 |
106 | 93, 94 | breqtrrd 3956 | . . . . . 6 |
107 | xrmineqinf 11038 | . . . . . 6 inf | |
108 | 105, 92, 106, 107 | syl3anc 1216 | . . . . 5 inf |
109 | 108 | oveq2d 5790 | . . . 4 inf inf inf |
110 | 90, 104, 109 | 3brtr4d 3960 | . . 3 inf inf inf |
111 | simpr 109 | . . . 4 | |
112 | 57 | adantr 274 | . . . . 5 |
113 | 5 | ad2antrr 479 | . . . . 5 |
114 | ge0nemnf 9607 | . . . . 5 | |
115 | 112, 113, 114 | syl2anc 408 | . . . 4 |
116 | 111, 115 | pm2.21ddne 2391 | . . 3 inf inf inf |
117 | elxr 9563 | . . . 4 | |
118 | 57, 117 | sylib 121 | . . 3 |
119 | 81, 110, 116, 118 | mpjao3dan 1285 | . 2 inf inf inf |
120 | 50 | adantr 274 | . . . 4 |
121 | 120 | xrleidd 9587 | . . 3 |
122 | prcom 3599 | . . . . 5 | |
123 | 122 | infeq1i 6900 | . . . 4 inf inf |
124 | 32 | adantr 274 | . . . . 5 |
125 | pnfge 9575 | . . . . . . 7 | |
126 | 120, 125 | syl 14 | . . . . . 6 |
127 | simpr 109 | . . . . . 6 | |
128 | 126, 127 | breqtrrd 3956 | . . . . 5 |
129 | xrmineqinf 11038 | . . . . 5 inf | |
130 | 124, 120, 128, 129 | syl3anc 1216 | . . . 4 inf |
131 | 123, 130 | syl5eqr 2186 | . . 3 inf |
132 | prcom 3599 | . . . . . 6 | |
133 | 132 | infeq1i 6900 | . . . . 5 inf inf |
134 | 49 | adantr 274 | . . . . . 6 |
135 | pnfge 9575 | . . . . . . . 8 | |
136 | 134, 135 | syl 14 | . . . . . . 7 |
137 | 136, 127 | breqtrrd 3956 | . . . . . 6 |
138 | xrmineqinf 11038 | . . . . . 6 inf | |
139 | 124, 134, 137, 138 | syl3anc 1216 | . . . . 5 inf |
140 | 133, 139 | syl5eqr 2186 | . . . 4 inf |
141 | prcom 3599 | . . . . . 6 | |
142 | 141 | infeq1i 6900 | . . . . 5 inf inf |
143 | 38 | adantr 274 | . . . . . 6 |
144 | pnfge 9575 | . . . . . . . 8 | |
145 | 143, 144 | syl 14 | . . . . . . 7 |
146 | 145, 127 | breqtrrd 3956 | . . . . . 6 |
147 | xrmineqinf 11038 | . . . . . 6 inf | |
148 | 124, 143, 146, 147 | syl3anc 1216 | . . . . 5 inf |
149 | 142, 148 | syl5eqr 2186 | . . . 4 inf |
150 | 140, 149 | oveq12d 5792 | . . 3 inf inf |
151 | 121, 131, 150 | 3brtr4d 3960 | . 2 inf inf inf |
152 | simpl3r 1037 | . . 3 | |
153 | nltmnf 9574 | . . . . . 6 | |
154 | 35, 153 | ax-mp 5 | . . . . 5 |
155 | breq2 3933 | . . . . 5 | |
156 | 154, 155 | mtbiri 664 | . . . 4 |
157 | 156 | adantl 275 | . . 3 |
158 | 152, 157 | pm2.21dd 609 | . 2 inf inf inf |
159 | elxr 9563 | . . 3 | |
160 | 32, 159 | sylib 121 | . 2 |
161 | 119, 151, 158, 160 | mpjao3dan 1285 | 1 inf inf inf |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 w3o 961 w3a 962 wceq 1331 wcel 1480 wne 2308 cpr 3528 class class class wbr 3929 (class class class)co 5774 infcinf 6870 cr 7619 cc0 7620 caddc 7623 cpnf 7797 cmnf 7798 cxr 7799 clt 7800 cle 7801 crp 9441 cxad 9557 |
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-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 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-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 ax-caucvg 7740 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 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-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-sup 6871 df-inf 6872 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-rp 9442 df-xneg 9559 df-xadd 9560 df-seqfrec 10219 df-exp 10293 df-cj 10614 df-re 10615 df-im 10616 df-rsqrt 10770 df-abs 10771 |
This theorem is referenced by: bdxmet 12670 |
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