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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 7753 | . . . . . . 7 | |
2 | 1 | adantl 275 | . . . . . 6 |
3 | 2 | 2timesd 8962 | . . . . 5 |
4 | recn 7753 | . . . . . . 7 | |
5 | 4 | ad2antlr 480 | . . . . . 6 |
6 | 5 | 2timesd 8962 | . . . . 5 |
7 | 3, 6 | breq12d 3942 | . . . 4 |
8 | simpr 109 | . . . . 5 | |
9 | simplr 519 | . . . . 5 | |
10 | 2re 8790 | . . . . . 6 | |
11 | 10 | a1i 9 | . . . . 5 |
12 | 2pos 8811 | . . . . . 6 | |
13 | 12 | a1i 9 | . . . . 5 |
14 | lemul2 8615 | . . . . 5 | |
15 | 8, 9, 11, 13, 14 | syl112anc 1220 | . . . 4 |
16 | 8, 8 | rexaddd 9637 | . . . . 5 |
17 | 9, 9 | rexaddd 9637 | . . . . 5 |
18 | 16, 17 | breq12d 3942 | . . . 4 |
19 | 7, 15, 18 | 3bitr4d 219 | . . 3 |
20 | renepnf 7813 | . . . . . . . 8 | |
21 | 20 | neneqd 2329 | . . . . . . 7 |
22 | 21 | ad2antlr 480 | . . . . . 6 |
23 | xgepnf 9599 | . . . . . . 7 | |
24 | 23 | ad3antlr 484 | . . . . . 6 |
25 | 22, 24 | mtbird 662 | . . . . 5 |
26 | breq1 3932 | . . . . . 6 | |
27 | 26 | adantl 275 | . . . . 5 |
28 | 25, 27 | mtbird 662 | . . . 4 |
29 | simplr 519 | . . . . . . . . 9 | |
30 | 29, 29 | rexaddd 9637 | . . . . . . . 8 |
31 | 29, 29 | readdcld 7795 | . . . . . . . 8 |
32 | 30, 31 | eqeltrd 2216 | . . . . . . 7 |
33 | renepnf 7813 | . . . . . . . 8 | |
34 | 33 | neneqd 2329 | . . . . . . 7 |
35 | 32, 34 | syl 14 | . . . . . 6 |
36 | simpllr 523 | . . . . . . . 8 | |
37 | 36, 36 | xaddcld 9667 | . . . . . . 7 |
38 | xgepnf 9599 | . . . . . . 7 | |
39 | 37, 38 | syl 14 | . . . . . 6 |
40 | 35, 39 | mtbird 662 | . . . . 5 |
41 | simpr 109 | . . . . . . . 8 | |
42 | 41, 41 | oveq12d 5792 | . . . . . . 7 |
43 | pnfxr 7818 | . . . . . . . 8 | |
44 | pnfnemnf 7820 | . . . . . . . 8 | |
45 | xaddpnf2 9630 | . . . . . . . 8 | |
46 | 43, 44, 45 | mp2an 422 | . . . . . . 7 |
47 | 42, 46 | syl6eq 2188 | . . . . . 6 |
48 | 47 | breq1d 3939 | . . . . 5 |
49 | 40, 48 | mtbird 662 | . . . 4 |
50 | 28, 49 | 2falsed 691 | . . 3 |
51 | mnfle 9578 | . . . . . 6 | |
52 | 51 | ad3antlr 484 | . . . . 5 |
53 | breq1 3932 | . . . . . 6 | |
54 | 53 | adantl 275 | . . . . 5 |
55 | 52, 54 | mpbird 166 | . . . 4 |
56 | oveq1 5781 | . . . . . . 7 | |
57 | 56 | adantl 275 | . . . . . 6 |
58 | simplll 522 | . . . . . . 7 | |
59 | mnfnepnf 7821 | . . . . . . . . 9 | |
60 | neeq1 2321 | . . . . . . . . 9 | |
61 | 59, 60 | mpbiri 167 | . . . . . . . 8 |
62 | 61 | adantl 275 | . . . . . . 7 |
63 | xaddmnf2 9632 | . . . . . . 7 | |
64 | 58, 62, 63 | syl2anc 408 | . . . . . 6 |
65 | 57, 64 | eqtrd 2172 | . . . . 5 |
66 | simpr 109 | . . . . . . . 8 | |
67 | 66, 66 | xaddcld 9667 | . . . . . . 7 |
68 | 67 | ad2antrr 479 | . . . . . 6 |
69 | mnfle 9578 | . . . . . 6 | |
70 | 68, 69 | syl 14 | . . . . 5 |
71 | 65, 70 | eqbrtrd 3950 | . . . 4 |
72 | 55, 71 | 2thd 174 | . . 3 |
73 | elxr 9563 | . . . . 5 | |
74 | 73 | biimpi 119 | . . . 4 |
75 | 74 | ad2antrr 479 | . . 3 |
76 | 19, 50, 72, 75 | mpjao3dan 1285 | . 2 |
77 | pnfge 9575 | . . . . 5 | |
78 | 77 | ad2antrr 479 | . . . 4 |
79 | breq2 3933 | . . . . 5 | |
80 | 79 | adantl 275 | . . . 4 |
81 | 78, 80 | mpbird 166 | . . 3 |
82 | simpll 518 | . . . . . 6 | |
83 | 82, 82 | xaddcld 9667 | . . . . 5 |
84 | pnfge 9575 | . . . . 5 | |
85 | 83, 84 | syl 14 | . . . 4 |
86 | oveq1 5781 | . . . . . 6 | |
87 | eleq1 2202 | . . . . . . . 8 | |
88 | 43, 87 | mpbiri 167 | . . . . . . 7 |
89 | neeq1 2321 | . . . . . . . 8 | |
90 | 44, 89 | mpbiri 167 | . . . . . . 7 |
91 | xaddpnf2 9630 | . . . . . . 7 | |
92 | 88, 90, 91 | syl2anc 408 | . . . . . 6 |
93 | 86, 92 | eqtrd 2172 | . . . . 5 |
94 | 93 | adantl 275 | . . . 4 |
95 | 85, 94 | breqtrrd 3956 | . . 3 |
96 | 81, 95 | 2thd 174 | . 2 |
97 | simpr 109 | . . . . . . 7 | |
98 | 97 | renemnfd 7817 | . . . . . 6 |
99 | 98 | neneqd 2329 | . . . . 5 |
100 | ngtmnft 9600 | . . . . . . . . 9 | |
101 | mnfxr 7822 | . . . . . . . . . 10 | |
102 | xrlenlt 7829 | . . . . . . . . . 10 | |
103 | 101, 102 | mpan2 421 | . . . . . . . . 9 |
104 | 100, 103 | bitr4d 190 | . . . . . . . 8 |
105 | 104 | ad2antrr 479 | . . . . . . 7 |
106 | breq2 3933 | . . . . . . . 8 | |
107 | 106 | adantl 275 | . . . . . . 7 |
108 | 105, 107 | bitr4d 190 | . . . . . 6 |
109 | 108 | adantr 274 | . . . . 5 |
110 | 99, 109 | mtbid 661 | . . . 4 |
111 | 97, 97 | rexaddd 9637 | . . . . . . . 8 |
112 | 97, 97 | readdcld 7795 | . . . . . . . 8 |
113 | 111, 112 | eqeltrd 2216 | . . . . . . 7 |
114 | 113 | renemnfd 7817 | . . . . . 6 |
115 | 114 | neneqd 2329 | . . . . 5 |
116 | simpll 518 | . . . . . . . . 9 | |
117 | 116, 116 | xaddcld 9667 | . . . . . . . 8 |
118 | xrlenlt 7829 | . . . . . . . . 9 | |
119 | 101, 118 | mpan2 421 | . . . . . . . 8 |
120 | 117, 119 | syl 14 | . . . . . . 7 |
121 | oveq2 5782 | . . . . . . . . . 10 | |
122 | eleq1 2202 | . . . . . . . . . . . 12 | |
123 | 101, 122 | mpbiri 167 | . . . . . . . . . . 11 |
124 | 90 | necon2i 2364 | . . . . . . . . . . 11 |
125 | xaddmnf1 9631 | . . . . . . . . . . 11 | |
126 | 123, 124, 125 | syl2anc 408 | . . . . . . . . . 10 |
127 | 121, 126 | eqtrd 2172 | . . . . . . . . 9 |
128 | 127 | adantl 275 | . . . . . . . 8 |
129 | 128 | breq2d 3941 | . . . . . . 7 |
130 | ngtmnft 9600 | . . . . . . . 8 | |
131 | 117, 130 | syl 14 | . . . . . . 7 |
132 | 120, 129, 131 | 3bitr4rd 220 | . . . . . 6 |
133 | 132 | adantr 274 | . . . . 5 |
134 | 115, 133 | mtbid 661 | . . . 4 |
135 | 110, 134 | 2falsed 691 | . . 3 |
136 | 44 | neii 2310 | . . . . . 6 |
137 | eqeq1 2146 | . . . . . . 7 | |
138 | 137 | adantl 275 | . . . . . 6 |
139 | 136, 138 | mtbiri 664 | . . . . 5 |
140 | 108 | adantr 274 | . . . . 5 |
141 | 139, 140 | mtbid 661 | . . . 4 |
142 | simplll 522 | . . . . . . 7 | |
143 | 139 | neqned 2315 | . . . . . . 7 |
144 | xaddnemnf 9640 | . . . . . . 7 | |
145 | 142, 143, 142, 143, 144 | syl22anc 1217 | . . . . . 6 |
146 | 145 | neneqd 2329 | . . . . 5 |
147 | 132 | adantr 274 | . . . . 5 |
148 | 146, 147 | mtbid 661 | . . . 4 |
149 | 141, 148 | 2falsed 691 | . . 3 |
150 | 108 | biimpa 294 | . . . 4 |
151 | simplll 522 | . . . . . . 7 | |
152 | 151, 151 | xaddcld 9667 | . . . . . 6 |
153 | 152 | xrleidd 9587 | . . . . 5 |
154 | simpr 109 | . . . . . . 7 | |
155 | simplr 519 | . . . . . . 7 | |
156 | 154, 155 | eqtr4d 2175 | . . . . . 6 |
157 | 156, 156 | oveq12d 5792 | . . . . 5 |
158 | 153, 157 | breqtrd 3954 | . . . 4 |
159 | 150, 158 | 2thd 174 | . . 3 |
160 | 74 | ad2antrr 479 | . . 3 |
161 | 135, 149, 159, 160 | mpjao3dan 1285 | . 2 |
162 | elxr 9563 | . . . 4 | |
163 | 162 | biimpi 119 | . . 3 |
164 | 163 | adantl 275 | . 2 |
165 | 76, 96, 161, 164 | mpjao3dan 1285 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 w3o 961 wceq 1331 wcel 1480 wne 2308 class class class wbr 3929 (class class class)co 5774 cc 7618 cr 7619 cc0 7620 caddc 7623 cmul 7625 cpnf 7797 cmnf 7798 cxr 7799 clt 7800 cle 7801 c2 8771 cxad 9557 |
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 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-lttrn 7734 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 |
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-csb 3004 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-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-2 8779 df-xadd 9560 |
This theorem is referenced by: psmetge0 12500 xmetge0 12534 |
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