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Mirrors > Home > ILE Home > Th. List > xrlttri3 | Unicode version |
Description: Extended real version of lttri3 7999. (Contributed by NM, 9-Feb-2006.) |
Ref | Expression |
---|---|
xrlttri3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 9733 | . 2 | |
2 | elxr 9733 | . 2 | |
3 | lttri3 7999 | . . . . . 6 | |
4 | 3 | ancoms 266 | . . . . 5 |
5 | renepnf 7967 | . . . . . . . . . 10 | |
6 | 5 | adantr 274 | . . . . . . . . 9 |
7 | neeq2 2354 | . . . . . . . . . 10 | |
8 | 7 | adantl 275 | . . . . . . . . 9 |
9 | 6, 8 | mpbird 166 | . . . . . . . 8 |
10 | 9 | necomd 2426 | . . . . . . 7 |
11 | 10 | neneqd 2361 | . . . . . 6 |
12 | ltpnf 9737 | . . . . . . . . 9 | |
13 | 12 | adantr 274 | . . . . . . . 8 |
14 | breq2 3993 | . . . . . . . . 9 | |
15 | 14 | adantl 275 | . . . . . . . 8 |
16 | 13, 15 | mpbird 166 | . . . . . . 7 |
17 | notnot 624 | . . . . . . . . 9 | |
18 | 17 | olcs 731 | . . . . . . . 8 |
19 | ioran 747 | . . . . . . . 8 | |
20 | 18, 19 | sylnib 671 | . . . . . . 7 |
21 | 16, 20 | syl 14 | . . . . . 6 |
22 | 11, 21 | 2falsed 697 | . . . . 5 |
23 | renemnf 7968 | . . . . . . . . . 10 | |
24 | 23 | adantr 274 | . . . . . . . . 9 |
25 | neeq2 2354 | . . . . . . . . . 10 | |
26 | 25 | adantl 275 | . . . . . . . . 9 |
27 | 24, 26 | mpbird 166 | . . . . . . . 8 |
28 | 27 | necomd 2426 | . . . . . . 7 |
29 | 28 | neneqd 2361 | . . . . . 6 |
30 | mnflt 9740 | . . . . . . . . 9 | |
31 | 30 | adantr 274 | . . . . . . . 8 |
32 | breq1 3992 | . . . . . . . . 9 | |
33 | 32 | adantl 275 | . . . . . . . 8 |
34 | 31, 33 | mpbird 166 | . . . . . . 7 |
35 | orc 707 | . . . . . . 7 | |
36 | oranim 776 | . . . . . . 7 | |
37 | 34, 35, 36 | 3syl 17 | . . . . . 6 |
38 | 29, 37 | 2falsed 697 | . . . . 5 |
39 | 4, 22, 38 | 3jaodan 1301 | . . . 4 |
40 | 39 | ancoms 266 | . . 3 |
41 | renepnf 7967 | . . . . . . . . 9 | |
42 | 41 | adantl 275 | . . . . . . . 8 |
43 | neeq2 2354 | . . . . . . . . 9 | |
44 | 43 | adantr 274 | . . . . . . . 8 |
45 | 42, 44 | mpbird 166 | . . . . . . 7 |
46 | 45 | neneqd 2361 | . . . . . 6 |
47 | ltpnf 9737 | . . . . . . . . 9 | |
48 | 47 | adantl 275 | . . . . . . . 8 |
49 | breq2 3993 | . . . . . . . . 9 | |
50 | 49 | adantr 274 | . . . . . . . 8 |
51 | 48, 50 | mpbird 166 | . . . . . . 7 |
52 | 51, 35, 36 | 3syl 17 | . . . . . 6 |
53 | 46, 52 | 2falsed 697 | . . . . 5 |
54 | eqtr3 2190 | . . . . . . 7 | |
55 | 54 | eqcomd 2176 | . . . . . 6 |
56 | pnfxr 7972 | . . . . . . . . 9 | |
57 | xrltnr 9736 | . . . . . . . . 9 | |
58 | 56, 57 | ax-mp 5 | . . . . . . . 8 |
59 | breq12 3994 | . . . . . . . . 9 | |
60 | 59 | ancoms 266 | . . . . . . . 8 |
61 | 58, 60 | mtbiri 670 | . . . . . . 7 |
62 | breq12 3994 | . . . . . . . 8 | |
63 | 58, 62 | mtbiri 670 | . . . . . . 7 |
64 | 61, 63 | jca 304 | . . . . . 6 |
65 | 55, 64 | 2thd 174 | . . . . 5 |
66 | mnfnepnf 7975 | . . . . . . . . 9 | |
67 | eqeq12 2183 | . . . . . . . . . 10 | |
68 | 67 | necon3bid 2381 | . . . . . . . . 9 |
69 | 66, 68 | mpbiri 167 | . . . . . . . 8 |
70 | 69 | ancoms 266 | . . . . . . 7 |
71 | 70 | neneqd 2361 | . . . . . 6 |
72 | mnfltpnf 9742 | . . . . . . . . 9 | |
73 | breq12 3994 | . . . . . . . . 9 | |
74 | 72, 73 | mpbiri 167 | . . . . . . . 8 |
75 | 74 | ancoms 266 | . . . . . . 7 |
76 | 75, 35, 36 | 3syl 17 | . . . . . 6 |
77 | 71, 76 | 2falsed 697 | . . . . 5 |
78 | 53, 65, 77 | 3jaodan 1301 | . . . 4 |
79 | 78 | ancoms 266 | . . 3 |
80 | renemnf 7968 | . . . . . . . . 9 | |
81 | 80 | adantl 275 | . . . . . . . 8 |
82 | neeq2 2354 | . . . . . . . . 9 | |
83 | 82 | adantr 274 | . . . . . . . 8 |
84 | 81, 83 | mpbird 166 | . . . . . . 7 |
85 | 84 | neneqd 2361 | . . . . . 6 |
86 | mnflt 9740 | . . . . . . . . 9 | |
87 | 86 | adantl 275 | . . . . . . . 8 |
88 | breq1 3992 | . . . . . . . . 9 | |
89 | 88 | adantr 274 | . . . . . . . 8 |
90 | 87, 89 | mpbird 166 | . . . . . . 7 |
91 | 90, 20 | syl 14 | . . . . . 6 |
92 | 85, 91 | 2falsed 697 | . . . . 5 |
93 | 66 | neii 2342 | . . . . . . . . . 10 |
94 | eqeq12 2183 | . . . . . . . . . 10 | |
95 | 93, 94 | mtbiri 670 | . . . . . . . . 9 |
96 | 95 | neneqad 2419 | . . . . . . . 8 |
97 | 96 | necomd 2426 | . . . . . . 7 |
98 | 97 | neneqd 2361 | . . . . . 6 |
99 | breq12 3994 | . . . . . . . 8 | |
100 | 72, 99 | mpbiri 167 | . . . . . . 7 |
101 | 100, 20 | syl 14 | . . . . . 6 |
102 | 98, 101 | 2falsed 697 | . . . . 5 |
103 | eqtr3 2190 | . . . . . . 7 | |
104 | 103 | ancoms 266 | . . . . . 6 |
105 | mnfxr 7976 | . . . . . . . . 9 | |
106 | xrltnr 9736 | . . . . . . . . 9 | |
107 | 105, 106 | ax-mp 5 | . . . . . . . 8 |
108 | breq12 3994 | . . . . . . . . 9 | |
109 | 108 | ancoms 266 | . . . . . . . 8 |
110 | 107, 109 | mtbiri 670 | . . . . . . 7 |
111 | breq12 3994 | . . . . . . . 8 | |
112 | 107, 111 | mtbiri 670 | . . . . . . 7 |
113 | 110, 112 | jca 304 | . . . . . 6 |
114 | 104, 113 | 2thd 174 | . . . . 5 |
115 | 92, 102, 114 | 3jaodan 1301 | . . . 4 |
116 | 115 | ancoms 266 | . . 3 |
117 | 40, 79, 116 | 3jaodan 1301 | . 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 703 w3o 972 wceq 1348 wcel 2141 wne 2340 class class class wbr 3989 cr 7773 cpnf 7951 cmnf 7952 cxr 7953 clt 7954 |
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-pre-ltirr 7886 ax-pre-apti 7889 |
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-rab 2457 df-v 2732 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-br 3990 df-opab 4051 df-xp 4617 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 |
This theorem is referenced by: xrletri3 9761 iccid 9882 xrmaxleim 11207 xrmaxif 11214 xrmaxaddlem 11223 infxrnegsupex 11226 bdxmet 13295 |
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