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Mirrors > Home > ILE Home > Th. List > xleaddadd | Unicode version |
Description: Cancelling a factor of two in (expressed as addition rather than as a factor to avoid extended real multiplication). (Contributed by Jim Kingdon, 18-Apr-2023.) |
Ref | Expression |
---|---|
xleaddadd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recn 7877 | . . . . . . 7 | |
2 | 1 | adantl 275 | . . . . . 6 |
3 | 2 | 2timesd 9090 | . . . . 5 |
4 | recn 7877 | . . . . . . 7 | |
5 | 4 | ad2antlr 481 | . . . . . 6 |
6 | 5 | 2timesd 9090 | . . . . 5 |
7 | 3, 6 | breq12d 3989 | . . . 4 |
8 | simpr 109 | . . . . 5 | |
9 | simplr 520 | . . . . 5 | |
10 | 2re 8918 | . . . . . 6 | |
11 | 10 | a1i 9 | . . . . 5 |
12 | 2pos 8939 | . . . . . 6 | |
13 | 12 | a1i 9 | . . . . 5 |
14 | lemul2 8743 | . . . . 5 | |
15 | 8, 9, 11, 13, 14 | syl112anc 1231 | . . . 4 |
16 | 8, 8 | rexaddd 9781 | . . . . 5 |
17 | 9, 9 | rexaddd 9781 | . . . . 5 |
18 | 16, 17 | breq12d 3989 | . . . 4 |
19 | 7, 15, 18 | 3bitr4d 219 | . . 3 |
20 | renepnf 7937 | . . . . . . . 8 | |
21 | 20 | neneqd 2355 | . . . . . . 7 |
22 | 21 | ad2antlr 481 | . . . . . 6 |
23 | xgepnf 9743 | . . . . . . 7 | |
24 | 23 | ad3antlr 485 | . . . . . 6 |
25 | 22, 24 | mtbird 663 | . . . . 5 |
26 | breq1 3979 | . . . . . 6 | |
27 | 26 | adantl 275 | . . . . 5 |
28 | 25, 27 | mtbird 663 | . . . 4 |
29 | simplr 520 | . . . . . . . . 9 | |
30 | 29, 29 | rexaddd 9781 | . . . . . . . 8 |
31 | 29, 29 | readdcld 7919 | . . . . . . . 8 |
32 | 30, 31 | eqeltrd 2241 | . . . . . . 7 |
33 | renepnf 7937 | . . . . . . . 8 | |
34 | 33 | neneqd 2355 | . . . . . . 7 |
35 | 32, 34 | syl 14 | . . . . . 6 |
36 | simpllr 524 | . . . . . . . 8 | |
37 | 36, 36 | xaddcld 9811 | . . . . . . 7 |
38 | xgepnf 9743 | . . . . . . 7 | |
39 | 37, 38 | syl 14 | . . . . . 6 |
40 | 35, 39 | mtbird 663 | . . . . 5 |
41 | simpr 109 | . . . . . . . 8 | |
42 | 41, 41 | oveq12d 5854 | . . . . . . 7 |
43 | pnfxr 7942 | . . . . . . . 8 | |
44 | pnfnemnf 7944 | . . . . . . . 8 | |
45 | xaddpnf2 9774 | . . . . . . . 8 | |
46 | 43, 44, 45 | mp2an 423 | . . . . . . 7 |
47 | 42, 46 | eqtrdi 2213 | . . . . . 6 |
48 | 47 | breq1d 3986 | . . . . 5 |
49 | 40, 48 | mtbird 663 | . . . 4 |
50 | 28, 49 | 2falsed 692 | . . 3 |
51 | mnfle 9719 | . . . . . 6 | |
52 | 51 | ad3antlr 485 | . . . . 5 |
53 | breq1 3979 | . . . . . 6 | |
54 | 53 | adantl 275 | . . . . 5 |
55 | 52, 54 | mpbird 166 | . . . 4 |
56 | oveq1 5843 | . . . . . . 7 | |
57 | 56 | adantl 275 | . . . . . 6 |
58 | simplll 523 | . . . . . . 7 | |
59 | mnfnepnf 7945 | . . . . . . . . 9 | |
60 | neeq1 2347 | . . . . . . . . 9 | |
61 | 59, 60 | mpbiri 167 | . . . . . . . 8 |
62 | 61 | adantl 275 | . . . . . . 7 |
63 | xaddmnf2 9776 | . . . . . . 7 | |
64 | 58, 62, 63 | syl2anc 409 | . . . . . 6 |
65 | 57, 64 | eqtrd 2197 | . . . . 5 |
66 | simpr 109 | . . . . . . . 8 | |
67 | 66, 66 | xaddcld 9811 | . . . . . . 7 |
68 | 67 | ad2antrr 480 | . . . . . 6 |
69 | mnfle 9719 | . . . . . 6 | |
70 | 68, 69 | syl 14 | . . . . 5 |
71 | 65, 70 | eqbrtrd 3998 | . . . 4 |
72 | 55, 71 | 2thd 174 | . . 3 |
73 | elxr 9703 | . . . . 5 | |
74 | 73 | biimpi 119 | . . . 4 |
75 | 74 | ad2antrr 480 | . . 3 |
76 | 19, 50, 72, 75 | mpjao3dan 1296 | . 2 |
77 | pnfge 9716 | . . . . 5 | |
78 | 77 | ad2antrr 480 | . . . 4 |
79 | breq2 3980 | . . . . 5 | |
80 | 79 | adantl 275 | . . . 4 |
81 | 78, 80 | mpbird 166 | . . 3 |
82 | simpll 519 | . . . . . 6 | |
83 | 82, 82 | xaddcld 9811 | . . . . 5 |
84 | pnfge 9716 | . . . . 5 | |
85 | 83, 84 | syl 14 | . . . 4 |
86 | oveq1 5843 | . . . . . 6 | |
87 | eleq1 2227 | . . . . . . . 8 | |
88 | 43, 87 | mpbiri 167 | . . . . . . 7 |
89 | neeq1 2347 | . . . . . . . 8 | |
90 | 44, 89 | mpbiri 167 | . . . . . . 7 |
91 | xaddpnf2 9774 | . . . . . . 7 | |
92 | 88, 90, 91 | syl2anc 409 | . . . . . 6 |
93 | 86, 92 | eqtrd 2197 | . . . . 5 |
94 | 93 | adantl 275 | . . . 4 |
95 | 85, 94 | breqtrrd 4004 | . . 3 |
96 | 81, 95 | 2thd 174 | . 2 |
97 | simpr 109 | . . . . . . 7 | |
98 | 97 | renemnfd 7941 | . . . . . 6 |
99 | 98 | neneqd 2355 | . . . . 5 |
100 | ngtmnft 9744 | . . . . . . . . 9 | |
101 | mnfxr 7946 | . . . . . . . . . 10 | |
102 | xrlenlt 7954 | . . . . . . . . . 10 | |
103 | 101, 102 | mpan2 422 | . . . . . . . . 9 |
104 | 100, 103 | bitr4d 190 | . . . . . . . 8 |
105 | 104 | ad2antrr 480 | . . . . . . 7 |
106 | breq2 3980 | . . . . . . . 8 | |
107 | 106 | adantl 275 | . . . . . . 7 |
108 | 105, 107 | bitr4d 190 | . . . . . 6 |
109 | 108 | adantr 274 | . . . . 5 |
110 | 99, 109 | mtbid 662 | . . . 4 |
111 | 97, 97 | rexaddd 9781 | . . . . . . . 8 |
112 | 97, 97 | readdcld 7919 | . . . . . . . 8 |
113 | 111, 112 | eqeltrd 2241 | . . . . . . 7 |
114 | 113 | renemnfd 7941 | . . . . . 6 |
115 | 114 | neneqd 2355 | . . . . 5 |
116 | simpll 519 | . . . . . . . . 9 | |
117 | 116, 116 | xaddcld 9811 | . . . . . . . 8 |
118 | xrlenlt 7954 | . . . . . . . . 9 | |
119 | 101, 118 | mpan2 422 | . . . . . . . 8 |
120 | 117, 119 | syl 14 | . . . . . . 7 |
121 | oveq2 5844 | . . . . . . . . . 10 | |
122 | eleq1 2227 | . . . . . . . . . . . 12 | |
123 | 101, 122 | mpbiri 167 | . . . . . . . . . . 11 |
124 | 90 | necon2i 2390 | . . . . . . . . . . 11 |
125 | xaddmnf1 9775 | . . . . . . . . . . 11 | |
126 | 123, 124, 125 | syl2anc 409 | . . . . . . . . . 10 |
127 | 121, 126 | eqtrd 2197 | . . . . . . . . 9 |
128 | 127 | adantl 275 | . . . . . . . 8 |
129 | 128 | breq2d 3988 | . . . . . . 7 |
130 | ngtmnft 9744 | . . . . . . . 8 | |
131 | 117, 130 | syl 14 | . . . . . . 7 |
132 | 120, 129, 131 | 3bitr4rd 220 | . . . . . 6 |
133 | 132 | adantr 274 | . . . . 5 |
134 | 115, 133 | mtbid 662 | . . . 4 |
135 | 110, 134 | 2falsed 692 | . . 3 |
136 | 44 | neii 2336 | . . . . . 6 |
137 | eqeq1 2171 | . . . . . . 7 | |
138 | 137 | adantl 275 | . . . . . 6 |
139 | 136, 138 | mtbiri 665 | . . . . 5 |
140 | 108 | adantr 274 | . . . . 5 |
141 | 139, 140 | mtbid 662 | . . . 4 |
142 | simplll 523 | . . . . . . 7 | |
143 | 139 | neqned 2341 | . . . . . . 7 |
144 | xaddnemnf 9784 | . . . . . . 7 | |
145 | 142, 143, 142, 143, 144 | syl22anc 1228 | . . . . . 6 |
146 | 145 | neneqd 2355 | . . . . 5 |
147 | 132 | adantr 274 | . . . . 5 |
148 | 146, 147 | mtbid 662 | . . . 4 |
149 | 141, 148 | 2falsed 692 | . . 3 |
150 | 108 | biimpa 294 | . . . 4 |
151 | simplll 523 | . . . . . . 7 | |
152 | 151, 151 | xaddcld 9811 | . . . . . 6 |
153 | 152 | xrleidd 9728 | . . . . 5 |
154 | simpr 109 | . . . . . . 7 | |
155 | simplr 520 | . . . . . . 7 | |
156 | 154, 155 | eqtr4d 2200 | . . . . . 6 |
157 | 156, 156 | oveq12d 5854 | . . . . 5 |
158 | 153, 157 | breqtrd 4002 | . . . 4 |
159 | 150, 158 | 2thd 174 | . . 3 |
160 | 74 | ad2antrr 480 | . . 3 |
161 | 135, 149, 159, 160 | mpjao3dan 1296 | . 2 |
162 | elxr 9703 | . . . 4 | |
163 | 162 | biimpi 119 | . . 3 |
164 | 163 | adantl 275 | . 2 |
165 | 76, 96, 161, 164 | mpjao3dan 1296 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 w3o 966 wceq 1342 wcel 2135 wne 2334 class class class wbr 3976 (class class class)co 5836 cc 7742 cr 7743 cc0 7744 caddc 7747 cmul 7749 cpnf 7921 cmnf 7922 cxr 7923 clt 7924 cle 7925 c2 8899 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-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-lttrn 7858 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 |
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-csb 3041 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-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-2 8907 df-xadd 9700 |
This theorem is referenced by: psmetge0 12872 xmetge0 12906 |
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