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Mirrors > Home > ILE Home > Th. List > xnegdi | Unicode version |
Description: Extended real version of negdi 8146. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xnegdi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 9703 | . 2 | |
2 | elxr 9703 | . . . 4 | |
3 | recn 7877 | . . . . . . . 8 | |
4 | recn 7877 | . . . . . . . 8 | |
5 | negdi 8146 | . . . . . . . 8 | |
6 | 3, 4, 5 | syl2an 287 | . . . . . . 7 |
7 | readdcl 7870 | . . . . . . . 8 | |
8 | rexneg 9757 | . . . . . . . 8 | |
9 | 7, 8 | syl 14 | . . . . . . 7 |
10 | renegcl 8150 | . . . . . . . 8 | |
11 | renegcl 8150 | . . . . . . . 8 | |
12 | rexadd 9779 | . . . . . . . 8 | |
13 | 10, 11, 12 | syl2an 287 | . . . . . . 7 |
14 | 6, 9, 13 | 3eqtr4d 2207 | . . . . . 6 |
15 | rexadd 9779 | . . . . . . 7 | |
16 | xnegeq 9754 | . . . . . . 7 | |
17 | 15, 16 | syl 14 | . . . . . 6 |
18 | rexneg 9757 | . . . . . . 7 | |
19 | rexneg 9757 | . . . . . . 7 | |
20 | 18, 19 | oveqan12d 5855 | . . . . . 6 |
21 | 14, 17, 20 | 3eqtr4d 2207 | . . . . 5 |
22 | xnegpnf 9755 | . . . . . 6 | |
23 | oveq2 5844 | . . . . . . . 8 | |
24 | rexr 7935 | . . . . . . . . 9 | |
25 | renemnf 7938 | . . . . . . . . 9 | |
26 | xaddpnf1 9773 | . . . . . . . . 9 | |
27 | 24, 25, 26 | syl2anc 409 | . . . . . . . 8 |
28 | 23, 27 | sylan9eqr 2219 | . . . . . . 7 |
29 | xnegeq 9754 | . . . . . . 7 | |
30 | 28, 29 | syl 14 | . . . . . 6 |
31 | xnegeq 9754 | . . . . . . . . 9 | |
32 | 31, 22 | eqtrdi 2213 | . . . . . . . 8 |
33 | 32 | oveq2d 5852 | . . . . . . 7 |
34 | 18, 10 | eqeltrd 2241 | . . . . . . . 8 |
35 | rexr 7935 | . . . . . . . . 9 | |
36 | renepnf 7937 | . . . . . . . . 9 | |
37 | xaddmnf1 9775 | . . . . . . . . 9 | |
38 | 35, 36, 37 | syl2anc 409 | . . . . . . . 8 |
39 | 34, 38 | syl 14 | . . . . . . 7 |
40 | 33, 39 | sylan9eqr 2219 | . . . . . 6 |
41 | 22, 30, 40 | 3eqtr4a 2223 | . . . . 5 |
42 | xnegmnf 9756 | . . . . . 6 | |
43 | oveq2 5844 | . . . . . . . 8 | |
44 | renepnf 7937 | . . . . . . . . 9 | |
45 | xaddmnf1 9775 | . . . . . . . . 9 | |
46 | 24, 44, 45 | syl2anc 409 | . . . . . . . 8 |
47 | 43, 46 | sylan9eqr 2219 | . . . . . . 7 |
48 | xnegeq 9754 | . . . . . . 7 | |
49 | 47, 48 | syl 14 | . . . . . 6 |
50 | xnegeq 9754 | . . . . . . . . 9 | |
51 | 50, 42 | eqtrdi 2213 | . . . . . . . 8 |
52 | 51 | oveq2d 5852 | . . . . . . 7 |
53 | renemnf 7938 | . . . . . . . . 9 | |
54 | xaddpnf1 9773 | . . . . . . . . 9 | |
55 | 35, 53, 54 | syl2anc 409 | . . . . . . . 8 |
56 | 34, 55 | syl 14 | . . . . . . 7 |
57 | 52, 56 | sylan9eqr 2219 | . . . . . 6 |
58 | 42, 49, 57 | 3eqtr4a 2223 | . . . . 5 |
59 | 21, 41, 58 | 3jaodan 1295 | . . . 4 |
60 | 2, 59 | sylan2b 285 | . . 3 |
61 | xneg0 9758 | . . . . . . 7 | |
62 | simpr 109 | . . . . . . . . . 10 | |
63 | 62 | oveq2d 5852 | . . . . . . . . 9 |
64 | pnfaddmnf 9777 | . . . . . . . . 9 | |
65 | 63, 64 | eqtrdi 2213 | . . . . . . . 8 |
66 | xnegeq 9754 | . . . . . . . 8 | |
67 | 65, 66 | syl 14 | . . . . . . 7 |
68 | 51 | adantl 275 | . . . . . . . . 9 |
69 | 68 | oveq2d 5852 | . . . . . . . 8 |
70 | mnfaddpnf 9778 | . . . . . . . 8 | |
71 | 69, 70 | eqtrdi 2213 | . . . . . . 7 |
72 | 61, 67, 71 | 3eqtr4a 2223 | . . . . . 6 |
73 | xaddpnf2 9774 | . . . . . . . 8 | |
74 | xnegeq 9754 | . . . . . . . 8 | |
75 | 73, 74 | syl 14 | . . . . . . 7 |
76 | xnegcl 9759 | . . . . . . . 8 | |
77 | xnegeq 9754 | . . . . . . . . . . . 12 | |
78 | 77, 22 | eqtrdi 2213 | . . . . . . . . . . 11 |
79 | xnegneg 9760 | . . . . . . . . . . . 12 | |
80 | 79 | eqeq1d 2173 | . . . . . . . . . . 11 |
81 | 78, 80 | syl5ib 153 | . . . . . . . . . 10 |
82 | 81 | necon3d 2378 | . . . . . . . . 9 |
83 | 82 | imp 123 | . . . . . . . 8 |
84 | xaddmnf2 9776 | . . . . . . . 8 | |
85 | 76, 83, 84 | syl2an2r 585 | . . . . . . 7 |
86 | 22, 75, 85 | 3eqtr4a 2223 | . . . . . 6 |
87 | xrmnfdc 9770 | . . . . . . . 8 DECID | |
88 | exmiddc 826 | . . . . . . . 8 DECID | |
89 | 87, 88 | syl 14 | . . . . . . 7 |
90 | df-ne 2335 | . . . . . . . 8 | |
91 | 90 | orbi2i 752 | . . . . . . 7 |
92 | 89, 91 | sylibr 133 | . . . . . 6 |
93 | 72, 86, 92 | mpjaodan 788 | . . . . 5 |
94 | 93 | adantl 275 | . . . 4 |
95 | simpl 108 | . . . . . 6 | |
96 | 95 | oveq1d 5851 | . . . . 5 |
97 | xnegeq 9754 | . . . . 5 | |
98 | 96, 97 | syl 14 | . . . 4 |
99 | xnegeq 9754 | . . . . . . 7 | |
100 | 99 | adantr 274 | . . . . . 6 |
101 | 100, 22 | eqtrdi 2213 | . . . . 5 |
102 | 101 | oveq1d 5851 | . . . 4 |
103 | 94, 98, 102 | 3eqtr4d 2207 | . . 3 |
104 | simpr 109 | . . . . . . . . . 10 | |
105 | 104 | oveq2d 5852 | . . . . . . . . 9 |
106 | 105, 70 | eqtrdi 2213 | . . . . . . . 8 |
107 | xnegeq 9754 | . . . . . . . 8 | |
108 | 106, 107 | syl 14 | . . . . . . 7 |
109 | 32 | adantl 275 | . . . . . . . . 9 |
110 | 109 | oveq2d 5852 | . . . . . . . 8 |
111 | 110, 64 | eqtrdi 2213 | . . . . . . 7 |
112 | 61, 108, 111 | 3eqtr4a 2223 | . . . . . 6 |
113 | xaddmnf2 9776 | . . . . . . . 8 | |
114 | xnegeq 9754 | . . . . . . . 8 | |
115 | 113, 114 | syl 14 | . . . . . . 7 |
116 | xnegeq 9754 | . . . . . . . . . . . 12 | |
117 | 116, 42 | eqtrdi 2213 | . . . . . . . . . . 11 |
118 | 79 | eqeq1d 2173 | . . . . . . . . . . 11 |
119 | 117, 118 | syl5ib 153 | . . . . . . . . . 10 |
120 | 119 | necon3d 2378 | . . . . . . . . 9 |
121 | 120 | imp 123 | . . . . . . . 8 |
122 | xaddpnf2 9774 | . . . . . . . 8 | |
123 | 76, 121, 122 | syl2an2r 585 | . . . . . . 7 |
124 | 42, 115, 123 | 3eqtr4a 2223 | . . . . . 6 |
125 | xrpnfdc 9769 | . . . . . . . 8 DECID | |
126 | exmiddc 826 | . . . . . . . 8 DECID | |
127 | 125, 126 | syl 14 | . . . . . . 7 |
128 | df-ne 2335 | . . . . . . . 8 | |
129 | 128 | orbi2i 752 | . . . . . . 7 |
130 | 127, 129 | sylibr 133 | . . . . . 6 |
131 | 112, 124, 130 | mpjaodan 788 | . . . . 5 |
132 | 131 | adantl 275 | . . . 4 |
133 | simpl 108 | . . . . . 6 | |
134 | 133 | oveq1d 5851 | . . . . 5 |
135 | xnegeq 9754 | . . . . 5 | |
136 | 134, 135 | syl 14 | . . . 4 |
137 | xnegeq 9754 | . . . . . . 7 | |
138 | 137 | adantr 274 | . . . . . 6 |
139 | 138, 42 | eqtrdi 2213 | . . . . 5 |
140 | 139 | oveq1d 5851 | . . . 4 |
141 | 132, 136, 140 | 3eqtr4d 2207 | . . 3 |
142 | 60, 103, 141 | 3jaoian 1294 | . 2 |
143 | 1, 142 | sylanb 282 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 698 DECID wdc 824 w3o 966 wceq 1342 wcel 2135 wne 2334 (class class class)co 5836 cc 7742 cr 7743 cc0 7744 caddc 7747 cpnf 7921 cmnf 7922 cxr 7923 cneg 8061 cxne 9696 cxad 9697 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-sub 8062 df-neg 8063 df-xneg 9699 df-xadd 9700 |
This theorem is referenced by: xaddass2 9797 xrminadd 11202 |
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