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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 7721 | . . . . . . 7 | |
2 | 1 | adantl 275 | . . . . . 6 |
3 | 2 | 2timesd 8930 | . . . . 5 |
4 | recn 7721 | . . . . . . 7 | |
5 | 4 | ad2antlr 480 | . . . . . 6 |
6 | 5 | 2timesd 8930 | . . . . 5 |
7 | 3, 6 | breq12d 3912 | . . . 4 |
8 | simpr 109 | . . . . 5 | |
9 | simplr 504 | . . . . 5 | |
10 | 2re 8758 | . . . . . 6 | |
11 | 10 | a1i 9 | . . . . 5 |
12 | 2pos 8779 | . . . . . 6 | |
13 | 12 | a1i 9 | . . . . 5 |
14 | lemul2 8583 | . . . . 5 | |
15 | 8, 9, 11, 13, 14 | syl112anc 1205 | . . . 4 |
16 | 8, 8 | rexaddd 9605 | . . . . 5 |
17 | 9, 9 | rexaddd 9605 | . . . . 5 |
18 | 16, 17 | breq12d 3912 | . . . 4 |
19 | 7, 15, 18 | 3bitr4d 219 | . . 3 |
20 | renepnf 7781 | . . . . . . . 8 | |
21 | 20 | neneqd 2306 | . . . . . . 7 |
22 | 21 | ad2antlr 480 | . . . . . 6 |
23 | xgepnf 9567 | . . . . . . 7 | |
24 | 23 | ad3antlr 484 | . . . . . 6 |
25 | 22, 24 | mtbird 647 | . . . . 5 |
26 | breq1 3902 | . . . . . 6 | |
27 | 26 | adantl 275 | . . . . 5 |
28 | 25, 27 | mtbird 647 | . . . 4 |
29 | simplr 504 | . . . . . . . . 9 | |
30 | 29, 29 | rexaddd 9605 | . . . . . . . 8 |
31 | 29, 29 | readdcld 7763 | . . . . . . . 8 |
32 | 30, 31 | eqeltrd 2194 | . . . . . . 7 |
33 | renepnf 7781 | . . . . . . . 8 | |
34 | 33 | neneqd 2306 | . . . . . . 7 |
35 | 32, 34 | syl 14 | . . . . . 6 |
36 | simpllr 508 | . . . . . . . 8 | |
37 | 36, 36 | xaddcld 9635 | . . . . . . 7 |
38 | xgepnf 9567 | . . . . . . 7 | |
39 | 37, 38 | syl 14 | . . . . . 6 |
40 | 35, 39 | mtbird 647 | . . . . 5 |
41 | simpr 109 | . . . . . . . 8 | |
42 | 41, 41 | oveq12d 5760 | . . . . . . 7 |
43 | pnfxr 7786 | . . . . . . . 8 | |
44 | pnfnemnf 7788 | . . . . . . . 8 | |
45 | xaddpnf2 9598 | . . . . . . . 8 | |
46 | 43, 44, 45 | mp2an 422 | . . . . . . 7 |
47 | 42, 46 | syl6eq 2166 | . . . . . 6 |
48 | 47 | breq1d 3909 | . . . . 5 |
49 | 40, 48 | mtbird 647 | . . . 4 |
50 | 28, 49 | 2falsed 676 | . . 3 |
51 | mnfle 9546 | . . . . . 6 | |
52 | 51 | ad3antlr 484 | . . . . 5 |
53 | breq1 3902 | . . . . . 6 | |
54 | 53 | adantl 275 | . . . . 5 |
55 | 52, 54 | mpbird 166 | . . . 4 |
56 | oveq1 5749 | . . . . . . 7 | |
57 | 56 | adantl 275 | . . . . . 6 |
58 | simplll 507 | . . . . . . 7 | |
59 | mnfnepnf 7789 | . . . . . . . . 9 | |
60 | neeq1 2298 | . . . . . . . . 9 | |
61 | 59, 60 | mpbiri 167 | . . . . . . . 8 |
62 | 61 | adantl 275 | . . . . . . 7 |
63 | xaddmnf2 9600 | . . . . . . 7 | |
64 | 58, 62, 63 | syl2anc 408 | . . . . . 6 |
65 | 57, 64 | eqtrd 2150 | . . . . 5 |
66 | simpr 109 | . . . . . . . 8 | |
67 | 66, 66 | xaddcld 9635 | . . . . . . 7 |
68 | 67 | ad2antrr 479 | . . . . . 6 |
69 | mnfle 9546 | . . . . . 6 | |
70 | 68, 69 | syl 14 | . . . . 5 |
71 | 65, 70 | eqbrtrd 3920 | . . . 4 |
72 | 55, 71 | 2thd 174 | . . 3 |
73 | elxr 9531 | . . . . 5 | |
74 | 73 | biimpi 119 | . . . 4 |
75 | 74 | ad2antrr 479 | . . 3 |
76 | 19, 50, 72, 75 | mpjao3dan 1270 | . 2 |
77 | pnfge 9543 | . . . . 5 | |
78 | 77 | ad2antrr 479 | . . . 4 |
79 | breq2 3903 | . . . . 5 | |
80 | 79 | adantl 275 | . . . 4 |
81 | 78, 80 | mpbird 166 | . . 3 |
82 | simpll 503 | . . . . . 6 | |
83 | 82, 82 | xaddcld 9635 | . . . . 5 |
84 | pnfge 9543 | . . . . 5 | |
85 | 83, 84 | syl 14 | . . . 4 |
86 | oveq1 5749 | . . . . . 6 | |
87 | eleq1 2180 | . . . . . . . 8 | |
88 | 43, 87 | mpbiri 167 | . . . . . . 7 |
89 | neeq1 2298 | . . . . . . . 8 | |
90 | 44, 89 | mpbiri 167 | . . . . . . 7 |
91 | xaddpnf2 9598 | . . . . . . 7 | |
92 | 88, 90, 91 | syl2anc 408 | . . . . . 6 |
93 | 86, 92 | eqtrd 2150 | . . . . 5 |
94 | 93 | adantl 275 | . . . 4 |
95 | 85, 94 | breqtrrd 3926 | . . 3 |
96 | 81, 95 | 2thd 174 | . 2 |
97 | simpr 109 | . . . . . . 7 | |
98 | 97 | renemnfd 7785 | . . . . . 6 |
99 | 98 | neneqd 2306 | . . . . 5 |
100 | ngtmnft 9568 | . . . . . . . . 9 | |
101 | mnfxr 7790 | . . . . . . . . . 10 | |
102 | xrlenlt 7797 | . . . . . . . . . 10 | |
103 | 101, 102 | mpan2 421 | . . . . . . . . 9 |
104 | 100, 103 | bitr4d 190 | . . . . . . . 8 |
105 | 104 | ad2antrr 479 | . . . . . . 7 |
106 | breq2 3903 | . . . . . . . 8 | |
107 | 106 | adantl 275 | . . . . . . 7 |
108 | 105, 107 | bitr4d 190 | . . . . . 6 |
109 | 108 | adantr 274 | . . . . 5 |
110 | 99, 109 | mtbid 646 | . . . 4 |
111 | 97, 97 | rexaddd 9605 | . . . . . . . 8 |
112 | 97, 97 | readdcld 7763 | . . . . . . . 8 |
113 | 111, 112 | eqeltrd 2194 | . . . . . . 7 |
114 | 113 | renemnfd 7785 | . . . . . 6 |
115 | 114 | neneqd 2306 | . . . . 5 |
116 | simpll 503 | . . . . . . . . 9 | |
117 | 116, 116 | xaddcld 9635 | . . . . . . . 8 |
118 | xrlenlt 7797 | . . . . . . . . 9 | |
119 | 101, 118 | mpan2 421 | . . . . . . . 8 |
120 | 117, 119 | syl 14 | . . . . . . 7 |
121 | oveq2 5750 | . . . . . . . . . 10 | |
122 | eleq1 2180 | . . . . . . . . . . . 12 | |
123 | 101, 122 | mpbiri 167 | . . . . . . . . . . 11 |
124 | 90 | necon2i 2341 | . . . . . . . . . . 11 |
125 | xaddmnf1 9599 | . . . . . . . . . . 11 | |
126 | 123, 124, 125 | syl2anc 408 | . . . . . . . . . 10 |
127 | 121, 126 | eqtrd 2150 | . . . . . . . . 9 |
128 | 127 | adantl 275 | . . . . . . . 8 |
129 | 128 | breq2d 3911 | . . . . . . 7 |
130 | ngtmnft 9568 | . . . . . . . 8 | |
131 | 117, 130 | syl 14 | . . . . . . 7 |
132 | 120, 129, 131 | 3bitr4rd 220 | . . . . . 6 |
133 | 132 | adantr 274 | . . . . 5 |
134 | 115, 133 | mtbid 646 | . . . 4 |
135 | 110, 134 | 2falsed 676 | . . 3 |
136 | 44 | neii 2287 | . . . . . 6 |
137 | eqeq1 2124 | . . . . . . 7 | |
138 | 137 | adantl 275 | . . . . . 6 |
139 | 136, 138 | mtbiri 649 | . . . . 5 |
140 | 108 | adantr 274 | . . . . 5 |
141 | 139, 140 | mtbid 646 | . . . 4 |
142 | simplll 507 | . . . . . . 7 | |
143 | 139 | neqned 2292 | . . . . . . 7 |
144 | xaddnemnf 9608 | . . . . . . 7 | |
145 | 142, 143, 142, 143, 144 | syl22anc 1202 | . . . . . 6 |
146 | 145 | neneqd 2306 | . . . . 5 |
147 | 132 | adantr 274 | . . . . 5 |
148 | 146, 147 | mtbid 646 | . . . 4 |
149 | 141, 148 | 2falsed 676 | . . 3 |
150 | 108 | biimpa 294 | . . . 4 |
151 | simplll 507 | . . . . . . 7 | |
152 | 151, 151 | xaddcld 9635 | . . . . . 6 |
153 | 152 | xrleidd 9555 | . . . . 5 |
154 | simpr 109 | . . . . . . 7 | |
155 | simplr 504 | . . . . . . 7 | |
156 | 154, 155 | eqtr4d 2153 | . . . . . 6 |
157 | 156, 156 | oveq12d 5760 | . . . . 5 |
158 | 153, 157 | breqtrd 3924 | . . . 4 |
159 | 150, 158 | 2thd 174 | . . 3 |
160 | 74 | ad2antrr 479 | . . 3 |
161 | 135, 149, 159, 160 | mpjao3dan 1270 | . 2 |
162 | elxr 9531 | . . . 4 | |
163 | 162 | biimpi 119 | . . 3 |
164 | 163 | adantl 275 | . 2 |
165 | 76, 96, 161, 164 | mpjao3dan 1270 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 w3o 946 wceq 1316 wcel 1465 wne 2285 class class class wbr 3899 (class class class)co 5742 cc 7586 cr 7587 cc0 7588 caddc 7591 cmul 7593 cpnf 7765 cmnf 7766 cxr 7767 clt 7768 cle 7769 c2 8739 cxad 9525 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-lttrn 7702 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-2 8747 df-xadd 9528 |
This theorem is referenced by: psmetge0 12427 xmetge0 12461 |
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