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Mirrors > Home > ILE Home > Th. List > xrlttri3 | Unicode version |
Description: Extended real version of lttri3 7957. (Contributed by NM, 9-Feb-2006.) |
Ref | Expression |
---|---|
xrlttri3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 9683 | . 2 | |
2 | elxr 9683 | . 2 | |
3 | lttri3 7957 | . . . . . 6 | |
4 | 3 | ancoms 266 | . . . . 5 |
5 | renepnf 7925 | . . . . . . . . . 10 | |
6 | 5 | adantr 274 | . . . . . . . . 9 |
7 | neeq2 2341 | . . . . . . . . . 10 | |
8 | 7 | adantl 275 | . . . . . . . . 9 |
9 | 6, 8 | mpbird 166 | . . . . . . . 8 |
10 | 9 | necomd 2413 | . . . . . . 7 |
11 | 10 | neneqd 2348 | . . . . . 6 |
12 | ltpnf 9687 | . . . . . . . . 9 | |
13 | 12 | adantr 274 | . . . . . . . 8 |
14 | breq2 3969 | . . . . . . . . 9 | |
15 | 14 | adantl 275 | . . . . . . . 8 |
16 | 13, 15 | mpbird 166 | . . . . . . 7 |
17 | notnot 619 | . . . . . . . . 9 | |
18 | 17 | olcs 726 | . . . . . . . 8 |
19 | ioran 742 | . . . . . . . 8 | |
20 | 18, 19 | sylnib 666 | . . . . . . 7 |
21 | 16, 20 | syl 14 | . . . . . 6 |
22 | 11, 21 | 2falsed 692 | . . . . 5 |
23 | renemnf 7926 | . . . . . . . . . 10 | |
24 | 23 | adantr 274 | . . . . . . . . 9 |
25 | neeq2 2341 | . . . . . . . . . 10 | |
26 | 25 | adantl 275 | . . . . . . . . 9 |
27 | 24, 26 | mpbird 166 | . . . . . . . 8 |
28 | 27 | necomd 2413 | . . . . . . 7 |
29 | 28 | neneqd 2348 | . . . . . 6 |
30 | mnflt 9690 | . . . . . . . . 9 | |
31 | 30 | adantr 274 | . . . . . . . 8 |
32 | breq1 3968 | . . . . . . . . 9 | |
33 | 32 | adantl 275 | . . . . . . . 8 |
34 | 31, 33 | mpbird 166 | . . . . . . 7 |
35 | orc 702 | . . . . . . 7 | |
36 | oranim 771 | . . . . . . 7 | |
37 | 34, 35, 36 | 3syl 17 | . . . . . 6 |
38 | 29, 37 | 2falsed 692 | . . . . 5 |
39 | 4, 22, 38 | 3jaodan 1288 | . . . 4 |
40 | 39 | ancoms 266 | . . 3 |
41 | renepnf 7925 | . . . . . . . . 9 | |
42 | 41 | adantl 275 | . . . . . . . 8 |
43 | neeq2 2341 | . . . . . . . . 9 | |
44 | 43 | adantr 274 | . . . . . . . 8 |
45 | 42, 44 | mpbird 166 | . . . . . . 7 |
46 | 45 | neneqd 2348 | . . . . . 6 |
47 | ltpnf 9687 | . . . . . . . . 9 | |
48 | 47 | adantl 275 | . . . . . . . 8 |
49 | breq2 3969 | . . . . . . . . 9 | |
50 | 49 | adantr 274 | . . . . . . . 8 |
51 | 48, 50 | mpbird 166 | . . . . . . 7 |
52 | 51, 35, 36 | 3syl 17 | . . . . . 6 |
53 | 46, 52 | 2falsed 692 | . . . . 5 |
54 | eqtr3 2177 | . . . . . . 7 | |
55 | 54 | eqcomd 2163 | . . . . . 6 |
56 | pnfxr 7930 | . . . . . . . . 9 | |
57 | xrltnr 9686 | . . . . . . . . 9 | |
58 | 56, 57 | ax-mp 5 | . . . . . . . 8 |
59 | breq12 3970 | . . . . . . . . 9 | |
60 | 59 | ancoms 266 | . . . . . . . 8 |
61 | 58, 60 | mtbiri 665 | . . . . . . 7 |
62 | breq12 3970 | . . . . . . . 8 | |
63 | 58, 62 | mtbiri 665 | . . . . . . 7 |
64 | 61, 63 | jca 304 | . . . . . 6 |
65 | 55, 64 | 2thd 174 | . . . . 5 |
66 | mnfnepnf 7933 | . . . . . . . . 9 | |
67 | eqeq12 2170 | . . . . . . . . . 10 | |
68 | 67 | necon3bid 2368 | . . . . . . . . 9 |
69 | 66, 68 | mpbiri 167 | . . . . . . . 8 |
70 | 69 | ancoms 266 | . . . . . . 7 |
71 | 70 | neneqd 2348 | . . . . . 6 |
72 | mnfltpnf 9692 | . . . . . . . . 9 | |
73 | breq12 3970 | . . . . . . . . 9 | |
74 | 72, 73 | mpbiri 167 | . . . . . . . 8 |
75 | 74 | ancoms 266 | . . . . . . 7 |
76 | 75, 35, 36 | 3syl 17 | . . . . . 6 |
77 | 71, 76 | 2falsed 692 | . . . . 5 |
78 | 53, 65, 77 | 3jaodan 1288 | . . . 4 |
79 | 78 | ancoms 266 | . . 3 |
80 | renemnf 7926 | . . . . . . . . 9 | |
81 | 80 | adantl 275 | . . . . . . . 8 |
82 | neeq2 2341 | . . . . . . . . 9 | |
83 | 82 | adantr 274 | . . . . . . . 8 |
84 | 81, 83 | mpbird 166 | . . . . . . 7 |
85 | 84 | neneqd 2348 | . . . . . 6 |
86 | mnflt 9690 | . . . . . . . . 9 | |
87 | 86 | adantl 275 | . . . . . . . 8 |
88 | breq1 3968 | . . . . . . . . 9 | |
89 | 88 | adantr 274 | . . . . . . . 8 |
90 | 87, 89 | mpbird 166 | . . . . . . 7 |
91 | 90, 20 | syl 14 | . . . . . 6 |
92 | 85, 91 | 2falsed 692 | . . . . 5 |
93 | 66 | neii 2329 | . . . . . . . . . 10 |
94 | eqeq12 2170 | . . . . . . . . . 10 | |
95 | 93, 94 | mtbiri 665 | . . . . . . . . 9 |
96 | 95 | neneqad 2406 | . . . . . . . 8 |
97 | 96 | necomd 2413 | . . . . . . 7 |
98 | 97 | neneqd 2348 | . . . . . 6 |
99 | breq12 3970 | . . . . . . . 8 | |
100 | 72, 99 | mpbiri 167 | . . . . . . 7 |
101 | 100, 20 | syl 14 | . . . . . 6 |
102 | 98, 101 | 2falsed 692 | . . . . 5 |
103 | eqtr3 2177 | . . . . . . 7 | |
104 | 103 | ancoms 266 | . . . . . 6 |
105 | mnfxr 7934 | . . . . . . . . 9 | |
106 | xrltnr 9686 | . . . . . . . . 9 | |
107 | 105, 106 | ax-mp 5 | . . . . . . . 8 |
108 | breq12 3970 | . . . . . . . . 9 | |
109 | 108 | ancoms 266 | . . . . . . . 8 |
110 | 107, 109 | mtbiri 665 | . . . . . . 7 |
111 | breq12 3970 | . . . . . . . 8 | |
112 | 107, 111 | mtbiri 665 | . . . . . . 7 |
113 | 110, 112 | jca 304 | . . . . . 6 |
114 | 104, 113 | 2thd 174 | . . . . 5 |
115 | 92, 102, 114 | 3jaodan 1288 | . . . 4 |
116 | 115 | ancoms 266 | . . 3 |
117 | 40, 79, 116 | 3jaodan 1288 | . 2 |
118 | 1, 2, 117 | syl2anb 289 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 698 w3o 962 wceq 1335 wcel 2128 wne 2327 class class class wbr 3965 cr 7731 cpnf 7909 cmnf 7910 cxr 7911 clt 7912 |
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 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-cnex 7823 ax-resscn 7824 ax-pre-ltirr 7844 ax-pre-apti 7847 |
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-rab 2444 df-v 2714 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 3773 df-br 3966 df-opab 4026 df-xp 4592 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 |
This theorem is referenced by: xrletri3 9709 iccid 9829 xrmaxleim 11141 xrmaxif 11148 xrmaxaddlem 11157 infxrnegsupex 11160 bdxmet 12912 |
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