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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 1007 | . . . . . . 7 | |
3 | 2 | ad3antrrr 484 | . . . . . 6 |
4 | simplr 520 | . . . . . 6 | |
5 | simp2r 1009 | . . . . . . 7 | |
6 | 5 | ad3antrrr 484 | . . . . . 6 |
7 | simpllr 524 | . . . . . . 7 | |
8 | simp3r 1011 | . . . . . . . 8 | |
9 | 8 | ad3antrrr 484 | . . . . . . 7 |
10 | 7, 9 | elrpd 9510 | . . . . . 6 |
11 | bdtri 11043 | . . . . . 6 inf inf inf | |
12 | 1, 3, 4, 6, 10, 11 | syl221anc 1228 | . . . . 5 inf inf inf |
13 | 1, 4 | rexaddd 9667 | . . . . . . . 8 |
14 | 13 | preq1d 3614 | . . . . . . 7 |
15 | 14 | infeq1d 6907 | . . . . . 6 inf inf |
16 | 1, 4 | readdcld 7819 | . . . . . . 7 |
17 | xrminrecl 11074 | . . . . . . 7 inf inf | |
18 | 16, 7, 17 | syl2anc 409 | . . . . . 6 inf inf |
19 | 15, 18 | eqtrd 2173 | . . . . 5 inf inf |
20 | xrminrecl 11074 | . . . . . . . 8 inf inf | |
21 | 1, 7, 20 | syl2anc 409 | . . . . . . 7 inf inf |
22 | xrminrecl 11074 | . . . . . . . 8 inf inf | |
23 | 4, 7, 22 | syl2anc 409 | . . . . . . 7 inf inf |
24 | 21, 23 | oveq12d 5800 | . . . . . 6 inf inf inf inf |
25 | mincl 11034 | . . . . . . . 8 inf | |
26 | 1, 7, 25 | syl2anc 409 | . . . . . . 7 inf |
27 | mincl 11034 | . . . . . . . 8 inf | |
28 | 4, 7, 27 | syl2anc 409 | . . . . . . 7 inf |
29 | 26, 28 | rexaddd 9667 | . . . . . 6 inf inf inf inf |
30 | 24, 29 | eqtrd 2173 | . . . . 5 inf inf inf inf |
31 | 12, 19, 30 | 3brtr4d 3968 | . . . 4 inf inf inf |
32 | simp3l 1010 | . . . . . . . 8 | |
33 | 32 | xaddid1d 9677 | . . . . . . 7 |
34 | 32 | xrleidd 9617 | . . . . . . . 8 |
35 | 0xr 7836 | . . . . . . . . . . 11 | |
36 | 35 | a1i 9 | . . . . . . . . . 10 |
37 | 36, 32, 8 | xrltled 9615 | . . . . . . . . 9 |
38 | simp2l 1008 | . . . . . . . . . 10 | |
39 | xrlemininf 11072 | . . . . . . . . . 10 inf | |
40 | 36, 38, 32, 39 | syl3anc 1217 | . . . . . . . . 9 inf |
41 | 5, 37, 40 | mpbir2and 929 | . . . . . . . 8 inf |
42 | xrmincl 11067 | . . . . . . . . . 10 inf | |
43 | 38, 32, 42 | syl2anc 409 | . . . . . . . . 9 inf |
44 | xle2add 9692 | . . . . . . . . 9 inf inf inf | |
45 | 32, 36, 32, 43, 44 | syl22anc 1218 | . . . . . . . 8 inf inf |
46 | 34, 41, 45 | mp2and 430 | . . . . . . 7 inf |
47 | 33, 46 | eqbrtrrd 3960 | . . . . . 6 inf |
48 | 47 | ad3antrrr 484 | . . . . 5 inf |
49 | simp1l 1006 | . . . . . . . 8 | |
50 | 49, 38 | xaddcld 9697 | . . . . . . 7 |
51 | 50 | ad3antrrr 484 | . . . . . 6 |
52 | 32 | ad3antrrr 484 | . . . . . 6 |
53 | pnfge 9605 | . . . . . . . 8 | |
54 | 52, 53 | syl 14 | . . . . . . 7 |
55 | simpr 109 | . . . . . . . . 9 | |
56 | 55 | oveq1d 5797 | . . . . . . . 8 |
57 | simpl2l 1035 | . . . . . . . . . 10 | |
58 | 57 | ad2antrr 480 | . . . . . . . . 9 |
59 | simplr 520 | . . . . . . . . . 10 | |
60 | 59 | renemnfd 7841 | . . . . . . . . 9 |
61 | xaddpnf2 9660 | . . . . . . . . 9 | |
62 | 58, 60, 61 | syl2anc 409 | . . . . . . . 8 |
63 | 56, 62 | eqtrd 2173 | . . . . . . 7 |
64 | 54, 63 | breqtrrd 3964 | . . . . . 6 |
65 | xrmineqinf 11070 | . . . . . 6 inf | |
66 | 51, 52, 64, 65 | syl3anc 1217 | . . . . 5 inf |
67 | 49 | ad3antrrr 484 | . . . . . . 7 |
68 | 54, 55 | breqtrrd 3964 | . . . . . . 7 |
69 | xrmineqinf 11070 | . . . . . . 7 inf | |
70 | 67, 52, 68, 69 | syl3anc 1217 | . . . . . 6 inf |
71 | 70 | oveq1d 5797 | . . . . 5 inf inf inf |
72 | 48, 66, 71 | 3brtr4d 3968 | . . . 4 inf inf inf |
73 | simpr 109 | . . . . 5 | |
74 | ge0nemnf 9637 | . . . . . . 7 | |
75 | 49, 2, 74 | syl2anc 409 | . . . . . 6 |
76 | 75 | ad3antrrr 484 | . . . . 5 |
77 | 73, 76 | pm2.21ddne 2392 | . . . 4 inf inf inf |
78 | elxr 9593 | . . . . . 6 | |
79 | 49, 78 | sylib 121 | . . . . 5 |
80 | 79 | ad2antrr 480 | . . . 4 |
81 | 31, 72, 77, 80 | mpjao3dan 1286 | . . 3 inf inf inf |
82 | xrlemininf 11072 | . . . . . . . 8 inf | |
83 | 36, 49, 32, 82 | syl3anc 1217 | . . . . . . 7 inf |
84 | 2, 37, 83 | mpbir2and 929 | . . . . . 6 inf |
85 | xrmincl 11067 | . . . . . . . 8 inf | |
86 | 49, 32, 85 | syl2anc 409 | . . . . . . 7 inf |
87 | xle2add 9692 | . . . . . . 7 inf inf inf | |
88 | 36, 32, 86, 32, 87 | syl22anc 1218 | . . . . . 6 inf inf |
89 | 84, 34, 88 | mp2and 430 | . . . . 5 inf |
90 | 89 | ad2antrr 480 | . . . 4 inf |
91 | 50 | ad2antrr 480 | . . . . . 6 |
92 | 32 | ad2antrr 480 | . . . . . 6 |
93 | 92, 53 | syl 14 | . . . . . . 7 |
94 | simpr 109 | . . . . . . . . 9 | |
95 | 94 | oveq2d 5798 | . . . . . . . 8 |
96 | xaddpnf1 9659 | . . . . . . . . . 10 | |
97 | 49, 75, 96 | syl2anc 409 | . . . . . . . . 9 |
98 | 97 | ad2antrr 480 | . . . . . . . 8 |
99 | 95, 98 | eqtrd 2173 | . . . . . . 7 |
100 | 93, 99 | breqtrrd 3964 | . . . . . 6 |
101 | 91, 92, 100, 65 | syl3anc 1217 | . . . . 5 inf |
102 | xaddid2 9676 | . . . . . 6 | |
103 | 92, 102 | syl 14 | . . . . 5 |
104 | 101, 103 | eqtr4d 2176 | . . . 4 inf |
105 | 57 | adantr 274 | . . . . . 6 |
106 | 93, 94 | breqtrrd 3964 | . . . . . 6 |
107 | xrmineqinf 11070 | . . . . . 6 inf | |
108 | 105, 92, 106, 107 | syl3anc 1217 | . . . . 5 inf |
109 | 108 | oveq2d 5798 | . . . 4 inf inf inf |
110 | 90, 104, 109 | 3brtr4d 3968 | . . 3 inf inf inf |
111 | simpr 109 | . . . 4 | |
112 | 57 | adantr 274 | . . . . 5 |
113 | 5 | ad2antrr 480 | . . . . 5 |
114 | ge0nemnf 9637 | . . . . 5 | |
115 | 112, 113, 114 | syl2anc 409 | . . . 4 |
116 | 111, 115 | pm2.21ddne 2392 | . . 3 inf inf inf |
117 | elxr 9593 | . . . 4 | |
118 | 57, 117 | sylib 121 | . . 3 |
119 | 81, 110, 116, 118 | mpjao3dan 1286 | . 2 inf inf inf |
120 | 50 | adantr 274 | . . . 4 |
121 | 120 | xrleidd 9617 | . . 3 |
122 | prcom 3607 | . . . . 5 | |
123 | 122 | infeq1i 6908 | . . . 4 inf inf |
124 | 32 | adantr 274 | . . . . 5 |
125 | pnfge 9605 | . . . . . . 7 | |
126 | 120, 125 | syl 14 | . . . . . 6 |
127 | simpr 109 | . . . . . 6 | |
128 | 126, 127 | breqtrrd 3964 | . . . . 5 |
129 | xrmineqinf 11070 | . . . . 5 inf | |
130 | 124, 120, 128, 129 | syl3anc 1217 | . . . 4 inf |
131 | 123, 130 | syl5eqr 2187 | . . 3 inf |
132 | prcom 3607 | . . . . . 6 | |
133 | 132 | infeq1i 6908 | . . . . 5 inf inf |
134 | 49 | adantr 274 | . . . . . 6 |
135 | pnfge 9605 | . . . . . . . 8 | |
136 | 134, 135 | syl 14 | . . . . . . 7 |
137 | 136, 127 | breqtrrd 3964 | . . . . . 6 |
138 | xrmineqinf 11070 | . . . . . 6 inf | |
139 | 124, 134, 137, 138 | syl3anc 1217 | . . . . 5 inf |
140 | 133, 139 | syl5eqr 2187 | . . . 4 inf |
141 | prcom 3607 | . . . . . 6 | |
142 | 141 | infeq1i 6908 | . . . . 5 inf inf |
143 | 38 | adantr 274 | . . . . . 6 |
144 | pnfge 9605 | . . . . . . . 8 | |
145 | 143, 144 | syl 14 | . . . . . . 7 |
146 | 145, 127 | breqtrrd 3964 | . . . . . 6 |
147 | xrmineqinf 11070 | . . . . . 6 inf | |
148 | 124, 143, 146, 147 | syl3anc 1217 | . . . . 5 inf |
149 | 142, 148 | syl5eqr 2187 | . . . 4 inf |
150 | 140, 149 | oveq12d 5800 | . . 3 inf inf |
151 | 121, 131, 150 | 3brtr4d 3968 | . 2 inf inf inf |
152 | simpl3r 1038 | . . 3 | |
153 | nltmnf 9604 | . . . . . 6 | |
154 | 35, 153 | ax-mp 5 | . . . . 5 |
155 | breq2 3941 | . . . . 5 | |
156 | 154, 155 | mtbiri 665 | . . . 4 |
157 | 156 | adantl 275 | . . 3 |
158 | 152, 157 | pm2.21dd 610 | . 2 inf inf inf |
159 | elxr 9593 | . . 3 | |
160 | 32, 159 | sylib 121 | . 2 |
161 | 119, 151, 158, 160 | mpjao3dan 1286 | 1 inf inf inf |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 w3o 962 w3a 963 wceq 1332 wcel 1481 wne 2309 cpr 3533 class class class wbr 3937 (class class class)co 5782 infcinf 6878 cr 7643 cc0 7644 caddc 7647 cpnf 7821 cmnf 7822 cxr 7823 clt 7824 cle 7825 crp 9470 cxad 9587 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 ax-arch 7763 ax-caucvg 7764 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-isom 5140 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-frec 6296 df-sup 6879 df-inf 6880 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-n0 9002 df-z 9079 df-uz 9351 df-rp 9471 df-xneg 9589 df-xadd 9590 df-seqfrec 10250 df-exp 10324 df-cj 10646 df-re 10647 df-im 10648 df-rsqrt 10802 df-abs 10803 |
This theorem is referenced by: bdxmet 12709 |
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