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Mirrors > Home > ILE Home > Th. List > xnegdi | Unicode version |
Description: Extended real version of negdi 8022. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xnegdi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 9566 | . 2 | |
2 | elxr 9566 | . . . 4 | |
3 | recn 7756 | . . . . . . . 8 | |
4 | recn 7756 | . . . . . . . 8 | |
5 | negdi 8022 | . . . . . . . 8 | |
6 | 3, 4, 5 | syl2an 287 | . . . . . . 7 |
7 | readdcl 7749 | . . . . . . . 8 | |
8 | rexneg 9616 | . . . . . . . 8 | |
9 | 7, 8 | syl 14 | . . . . . . 7 |
10 | renegcl 8026 | . . . . . . . 8 | |
11 | renegcl 8026 | . . . . . . . 8 | |
12 | rexadd 9638 | . . . . . . . 8 | |
13 | 10, 11, 12 | syl2an 287 | . . . . . . 7 |
14 | 6, 9, 13 | 3eqtr4d 2182 | . . . . . 6 |
15 | rexadd 9638 | . . . . . . 7 | |
16 | xnegeq 9613 | . . . . . . 7 | |
17 | 15, 16 | syl 14 | . . . . . 6 |
18 | rexneg 9616 | . . . . . . 7 | |
19 | rexneg 9616 | . . . . . . 7 | |
20 | 18, 19 | oveqan12d 5793 | . . . . . 6 |
21 | 14, 17, 20 | 3eqtr4d 2182 | . . . . 5 |
22 | xnegpnf 9614 | . . . . . 6 | |
23 | oveq2 5782 | . . . . . . . 8 | |
24 | rexr 7814 | . . . . . . . . 9 | |
25 | renemnf 7817 | . . . . . . . . 9 | |
26 | xaddpnf1 9632 | . . . . . . . . 9 | |
27 | 24, 25, 26 | syl2anc 408 | . . . . . . . 8 |
28 | 23, 27 | sylan9eqr 2194 | . . . . . . 7 |
29 | xnegeq 9613 | . . . . . . 7 | |
30 | 28, 29 | syl 14 | . . . . . 6 |
31 | xnegeq 9613 | . . . . . . . . 9 | |
32 | 31, 22 | syl6eq 2188 | . . . . . . . 8 |
33 | 32 | oveq2d 5790 | . . . . . . 7 |
34 | 18, 10 | eqeltrd 2216 | . . . . . . . 8 |
35 | rexr 7814 | . . . . . . . . 9 | |
36 | renepnf 7816 | . . . . . . . . 9 | |
37 | xaddmnf1 9634 | . . . . . . . . 9 | |
38 | 35, 36, 37 | syl2anc 408 | . . . . . . . 8 |
39 | 34, 38 | syl 14 | . . . . . . 7 |
40 | 33, 39 | sylan9eqr 2194 | . . . . . 6 |
41 | 22, 30, 40 | 3eqtr4a 2198 | . . . . 5 |
42 | xnegmnf 9615 | . . . . . 6 | |
43 | oveq2 5782 | . . . . . . . 8 | |
44 | renepnf 7816 | . . . . . . . . 9 | |
45 | xaddmnf1 9634 | . . . . . . . . 9 | |
46 | 24, 44, 45 | syl2anc 408 | . . . . . . . 8 |
47 | 43, 46 | sylan9eqr 2194 | . . . . . . 7 |
48 | xnegeq 9613 | . . . . . . 7 | |
49 | 47, 48 | syl 14 | . . . . . 6 |
50 | xnegeq 9613 | . . . . . . . . 9 | |
51 | 50, 42 | syl6eq 2188 | . . . . . . . 8 |
52 | 51 | oveq2d 5790 | . . . . . . 7 |
53 | renemnf 7817 | . . . . . . . . 9 | |
54 | xaddpnf1 9632 | . . . . . . . . 9 | |
55 | 35, 53, 54 | syl2anc 408 | . . . . . . . 8 |
56 | 34, 55 | syl 14 | . . . . . . 7 |
57 | 52, 56 | sylan9eqr 2194 | . . . . . 6 |
58 | 42, 49, 57 | 3eqtr4a 2198 | . . . . 5 |
59 | 21, 41, 58 | 3jaodan 1284 | . . . 4 |
60 | 2, 59 | sylan2b 285 | . . 3 |
61 | xneg0 9617 | . . . . . . 7 | |
62 | simpr 109 | . . . . . . . . . 10 | |
63 | 62 | oveq2d 5790 | . . . . . . . . 9 |
64 | pnfaddmnf 9636 | . . . . . . . . 9 | |
65 | 63, 64 | syl6eq 2188 | . . . . . . . 8 |
66 | xnegeq 9613 | . . . . . . . 8 | |
67 | 65, 66 | syl 14 | . . . . . . 7 |
68 | 51 | adantl 275 | . . . . . . . . 9 |
69 | 68 | oveq2d 5790 | . . . . . . . 8 |
70 | mnfaddpnf 9637 | . . . . . . . 8 | |
71 | 69, 70 | syl6eq 2188 | . . . . . . 7 |
72 | 61, 67, 71 | 3eqtr4a 2198 | . . . . . 6 |
73 | xaddpnf2 9633 | . . . . . . . 8 | |
74 | xnegeq 9613 | . . . . . . . 8 | |
75 | 73, 74 | syl 14 | . . . . . . 7 |
76 | xnegcl 9618 | . . . . . . . 8 | |
77 | xnegeq 9613 | . . . . . . . . . . . 12 | |
78 | 77, 22 | syl6eq 2188 | . . . . . . . . . . 11 |
79 | xnegneg 9619 | . . . . . . . . . . . 12 | |
80 | 79 | eqeq1d 2148 | . . . . . . . . . . 11 |
81 | 78, 80 | syl5ib 153 | . . . . . . . . . 10 |
82 | 81 | necon3d 2352 | . . . . . . . . 9 |
83 | 82 | imp 123 | . . . . . . . 8 |
84 | xaddmnf2 9635 | . . . . . . . 8 | |
85 | 76, 83, 84 | syl2an2r 584 | . . . . . . 7 |
86 | 22, 75, 85 | 3eqtr4a 2198 | . . . . . 6 |
87 | xrmnfdc 9629 | . . . . . . . 8 DECID | |
88 | exmiddc 821 | . . . . . . . 8 DECID | |
89 | 87, 88 | syl 14 | . . . . . . 7 |
90 | df-ne 2309 | . . . . . . . 8 | |
91 | 90 | orbi2i 751 | . . . . . . 7 |
92 | 89, 91 | sylibr 133 | . . . . . 6 |
93 | 72, 86, 92 | mpjaodan 787 | . . . . 5 |
94 | 93 | adantl 275 | . . . 4 |
95 | simpl 108 | . . . . . 6 | |
96 | 95 | oveq1d 5789 | . . . . 5 |
97 | xnegeq 9613 | . . . . 5 | |
98 | 96, 97 | syl 14 | . . . 4 |
99 | xnegeq 9613 | . . . . . . 7 | |
100 | 99 | adantr 274 | . . . . . 6 |
101 | 100, 22 | syl6eq 2188 | . . . . 5 |
102 | 101 | oveq1d 5789 | . . . 4 |
103 | 94, 98, 102 | 3eqtr4d 2182 | . . 3 |
104 | simpr 109 | . . . . . . . . . 10 | |
105 | 104 | oveq2d 5790 | . . . . . . . . 9 |
106 | 105, 70 | syl6eq 2188 | . . . . . . . 8 |
107 | xnegeq 9613 | . . . . . . . 8 | |
108 | 106, 107 | syl 14 | . . . . . . 7 |
109 | 32 | adantl 275 | . . . . . . . . 9 |
110 | 109 | oveq2d 5790 | . . . . . . . 8 |
111 | 110, 64 | syl6eq 2188 | . . . . . . 7 |
112 | 61, 108, 111 | 3eqtr4a 2198 | . . . . . 6 |
113 | xaddmnf2 9635 | . . . . . . . 8 | |
114 | xnegeq 9613 | . . . . . . . 8 | |
115 | 113, 114 | syl 14 | . . . . . . 7 |
116 | xnegeq 9613 | . . . . . . . . . . . 12 | |
117 | 116, 42 | syl6eq 2188 | . . . . . . . . . . 11 |
118 | 79 | eqeq1d 2148 | . . . . . . . . . . 11 |
119 | 117, 118 | syl5ib 153 | . . . . . . . . . 10 |
120 | 119 | necon3d 2352 | . . . . . . . . 9 |
121 | 120 | imp 123 | . . . . . . . 8 |
122 | xaddpnf2 9633 | . . . . . . . 8 | |
123 | 76, 121, 122 | syl2an2r 584 | . . . . . . 7 |
124 | 42, 115, 123 | 3eqtr4a 2198 | . . . . . 6 |
125 | xrpnfdc 9628 | . . . . . . . 8 DECID | |
126 | exmiddc 821 | . . . . . . . 8 DECID | |
127 | 125, 126 | syl 14 | . . . . . . 7 |
128 | df-ne 2309 | . . . . . . . 8 | |
129 | 128 | orbi2i 751 | . . . . . . 7 |
130 | 127, 129 | sylibr 133 | . . . . . 6 |
131 | 112, 124, 130 | mpjaodan 787 | . . . . 5 |
132 | 131 | adantl 275 | . . . 4 |
133 | simpl 108 | . . . . . 6 | |
134 | 133 | oveq1d 5789 | . . . . 5 |
135 | xnegeq 9613 | . . . . 5 | |
136 | 134, 135 | syl 14 | . . . 4 |
137 | xnegeq 9613 | . . . . . . 7 | |
138 | 137 | adantr 274 | . . . . . 6 |
139 | 138, 42 | syl6eq 2188 | . . . . 5 |
140 | 139 | oveq1d 5789 | . . . 4 |
141 | 132, 136, 140 | 3eqtr4d 2182 | . . 3 |
142 | 60, 103, 141 | 3jaoian 1283 | . 2 |
143 | 1, 142 | sylanb 282 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 697 DECID wdc 819 w3o 961 wceq 1331 wcel 1480 wne 2308 (class class class)co 5774 cc 7621 cr 7622 cc0 7623 caddc 7626 cpnf 7800 cmnf 7801 cxr 7802 cneg 7937 cxne 9559 cxad 9560 |
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-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7714 ax-resscn 7715 ax-1cn 7716 ax-1re 7717 ax-icn 7718 ax-addcl 7719 ax-addrcl 7720 ax-mulcl 7721 ax-addcom 7723 ax-addass 7725 ax-distr 7727 ax-i2m1 7728 ax-0id 7731 ax-rnegex 7732 ax-cnre 7734 |
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-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7805 df-mnf 7806 df-xr 7807 df-sub 7938 df-neg 7939 df-xneg 9562 df-xadd 9563 |
This theorem is referenced by: xaddass2 9656 xrminadd 11047 |
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