Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > axpaschlem | Structured version Visualization version Unicode version |
Description: Lemma for axpasch 25821. Set up coefficents used in the proof. (Contributed by Scott Fenton, 5-Jun-2013.) |
Ref | Expression |
---|---|
axpaschlem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1re 10039 | . . . . . . . 8 | |
2 | 0re 10040 | . . . . . . . . . . 11 | |
3 | 2, 1 | elicc2i 12239 | . . . . . . . . . 10 |
4 | 3 | simp1bi 1076 | . . . . . . . . 9 |
5 | 4 | ad2antrl 764 | . . . . . . . 8 |
6 | resubcl 10345 | . . . . . . . 8 | |
7 | 1, 5, 6 | sylancr 695 | . . . . . . 7 |
8 | 7 | recnd 10068 | . . . . . 6 |
9 | 8 | mul02d 10234 | . . . . 5 |
10 | 9 | eqcomd 2628 | . . . 4 |
11 | 2, 1 | elicc2i 12239 | . . . . . . . . . 10 |
12 | 11 | simp1bi 1076 | . . . . . . . . 9 |
13 | 12 | ad2antll 765 | . . . . . . . 8 |
14 | resubcl 10345 | . . . . . . . 8 | |
15 | 1, 13, 14 | sylancr 695 | . . . . . . 7 |
16 | 15 | recnd 10068 | . . . . . 6 |
17 | 16 | mulid2d 10058 | . . . . 5 |
18 | oveq2 6658 | . . . . . . 7 | |
19 | 18 | adantr 481 | . . . . . 6 |
20 | 1m0e1 11131 | . . . . . 6 | |
21 | 19, 20 | syl6eq 2672 | . . . . 5 |
22 | 17, 21 | eqtr2d 2657 | . . . 4 |
23 | 5 | recnd 10068 | . . . . . 6 |
24 | 23 | mul02d 10234 | . . . . 5 |
25 | oveq2 6658 | . . . . . . 7 | |
26 | 25 | adantr 481 | . . . . . 6 |
27 | ax-1cn 9994 | . . . . . . 7 | |
28 | 27 | mul01i 10226 | . . . . . 6 |
29 | 26, 28 | syl6eq 2672 | . . . . 5 |
30 | 24, 29 | eqtr4d 2659 | . . . 4 |
31 | 1elunit 12291 | . . . . 5 | |
32 | 0elunit 12290 | . . . . 5 | |
33 | oveq2 6658 | . . . . . . . . . 10 | |
34 | 1m1e0 11089 | . . . . . . . . . 10 | |
35 | 33, 34 | syl6eq 2672 | . . . . . . . . 9 |
36 | 35 | oveq1d 6665 | . . . . . . . 8 |
37 | 36 | eqeq2d 2632 | . . . . . . 7 |
38 | eqeq1 2626 | . . . . . . 7 | |
39 | 35 | oveq1d 6665 | . . . . . . . 8 |
40 | 39 | eqeq1d 2624 | . . . . . . 7 |
41 | 37, 38, 40 | 3anbi123d 1399 | . . . . . 6 |
42 | eqeq1 2626 | . . . . . . 7 | |
43 | oveq2 6658 | . . . . . . . . . 10 | |
44 | 43, 20 | syl6eq 2672 | . . . . . . . . 9 |
45 | 44 | oveq1d 6665 | . . . . . . . 8 |
46 | 45 | eqeq2d 2632 | . . . . . . 7 |
47 | 44 | oveq1d 6665 | . . . . . . . 8 |
48 | 47 | eqeq2d 2632 | . . . . . . 7 |
49 | 42, 46, 48 | 3anbi123d 1399 | . . . . . 6 |
50 | 41, 49 | rspc2ev 3324 | . . . . 5 |
51 | 31, 32, 50 | mp3an12 1414 | . . . 4 |
52 | 10, 22, 30, 51 | syl3anc 1326 | . . 3 |
53 | 52 | ex 450 | . 2 |
54 | 4 | ad2antrl 764 | . . . . . . 7 |
55 | 12 | ad2antll 765 | . . . . . . . 8 |
56 | 55, 54 | remulcld 10070 | . . . . . . 7 |
57 | 54, 56 | resubcld 10458 | . . . . . 6 |
58 | 55, 54 | readdcld 10069 | . . . . . . 7 |
59 | 58, 56 | resubcld 10458 | . . . . . 6 |
60 | 1red 10055 | . . . . . . . . . . 11 | |
61 | 3 | simp2bi 1077 | . . . . . . . . . . . 12 |
62 | 61 | ad2antrl 764 | . . . . . . . . . . 11 |
63 | 11 | simp3bi 1078 | . . . . . . . . . . . 12 |
64 | 63 | ad2antll 765 | . . . . . . . . . . 11 |
65 | 55, 60, 54, 62, 64 | lemul1ad 10963 | . . . . . . . . . 10 |
66 | 54 | recnd 10068 | . . . . . . . . . . 11 |
67 | 66 | mulid2d 10058 | . . . . . . . . . 10 |
68 | 65, 67 | breqtrd 4679 | . . . . . . . . 9 |
69 | 11 | simp2bi 1077 | . . . . . . . . . . . 12 |
70 | 69 | ad2antll 765 | . . . . . . . . . . 11 |
71 | simpl 473 | . . . . . . . . . . 11 | |
72 | 55, 70, 71 | ne0gt0d 10174 | . . . . . . . . . 10 |
73 | 55, 54 | ltaddpos2d 10612 | . . . . . . . . . 10 |
74 | 72, 73 | mpbid 222 | . . . . . . . . 9 |
75 | 56, 54, 58, 68, 74 | lelttrd 10195 | . . . . . . . 8 |
76 | 56, 58 | posdifd 10614 | . . . . . . . 8 |
77 | 75, 76 | mpbid 222 | . . . . . . 7 |
78 | 77 | gt0ne0d 10592 | . . . . . 6 |
79 | 57, 59, 78 | redivcld 10853 | . . . . 5 |
80 | 54, 56 | subge0d 10617 | . . . . . . 7 |
81 | 68, 80 | mpbird 247 | . . . . . 6 |
82 | divge0 10892 | . . . . . 6 | |
83 | 57, 81, 59, 77, 82 | syl22anc 1327 | . . . . 5 |
84 | 54, 58, 74 | ltled 10185 | . . . . . . . 8 |
85 | 54, 58, 56, 84 | lesub1dd 10643 | . . . . . . 7 |
86 | 59 | recnd 10068 | . . . . . . . 8 |
87 | 86 | mulid2d 10058 | . . . . . . 7 |
88 | 85, 87 | breqtrrd 4681 | . . . . . 6 |
89 | ledivmul2 10902 | . . . . . . 7 | |
90 | 57, 60, 59, 77, 89 | syl112anc 1330 | . . . . . 6 |
91 | 88, 90 | mpbird 247 | . . . . 5 |
92 | 2, 1 | elicc2i 12239 | . . . . 5 |
93 | 79, 83, 91, 92 | syl3anbrc 1246 | . . . 4 |
94 | 55, 56 | resubcld 10458 | . . . . . 6 |
95 | 94, 59, 78 | redivcld 10853 | . . . . 5 |
96 | 3 | simp3bi 1078 | . . . . . . . . . 10 |
97 | 96 | ad2antrl 764 | . . . . . . . . 9 |
98 | 54, 60, 55, 70, 97 | lemul2ad 10964 | . . . . . . . 8 |
99 | 55 | recnd 10068 | . . . . . . . . 9 |
100 | 99 | mulid1d 10057 | . . . . . . . 8 |
101 | 98, 100 | breqtrd 4679 | . . . . . . 7 |
102 | 55, 56 | subge0d 10617 | . . . . . . 7 |
103 | 101, 102 | mpbird 247 | . . . . . 6 |
104 | divge0 10892 | . . . . . 6 | |
105 | 94, 103, 59, 77, 104 | syl22anc 1327 | . . . . 5 |
106 | 55, 54 | addge01d 10615 | . . . . . . . . 9 |
107 | 62, 106 | mpbid 222 | . . . . . . . 8 |
108 | 55, 58, 56, 107 | lesub1dd 10643 | . . . . . . 7 |
109 | 108, 87 | breqtrrd 4681 | . . . . . 6 |
110 | ledivmul2 10902 | . . . . . . 7 | |
111 | 94, 60, 59, 77, 110 | syl112anc 1330 | . . . . . 6 |
112 | 109, 111 | mpbird 247 | . . . . 5 |
113 | 2, 1 | elicc2i 12239 | . . . . 5 |
114 | 95, 105, 112, 113 | syl3anbrc 1246 | . . . 4 |
115 | 1, 54, 6 | sylancr 695 | . . . . . . 7 |
116 | 115 | recnd 10068 | . . . . . 6 |
117 | 99, 116, 86, 78 | div23d 10838 | . . . . 5 |
118 | subdi 10463 | . . . . . . . . 9 | |
119 | 27, 118 | mp3an2 1412 | . . . . . . . 8 |
120 | 99, 66, 119 | syl2anc 693 | . . . . . . 7 |
121 | 100 | oveq1d 6665 | . . . . . . 7 |
122 | 120, 121 | eqtrd 2656 | . . . . . 6 |
123 | 122 | oveq1d 6665 | . . . . 5 |
124 | 57 | recnd 10068 | . . . . . . . 8 |
125 | 86, 124, 86, 78 | divsubdird 10840 | . . . . . . 7 |
126 | 58 | recnd 10068 | . . . . . . . . . 10 |
127 | 56 | recnd 10068 | . . . . . . . . . 10 |
128 | 126, 66, 127 | nnncan2d 10427 | . . . . . . . . 9 |
129 | 99, 66 | pncand 10393 | . . . . . . . . 9 |
130 | 128, 129 | eqtrd 2656 | . . . . . . . 8 |
131 | 130 | oveq1d 6665 | . . . . . . 7 |
132 | 86, 78 | dividd 10799 | . . . . . . . 8 |
133 | 132 | oveq1d 6665 | . . . . . . 7 |
134 | 125, 131, 133 | 3eqtr3d 2664 | . . . . . 6 |
135 | 134 | oveq1d 6665 | . . . . 5 |
136 | 117, 123, 135 | 3eqtr3d 2664 | . . . 4 |
137 | 1, 55, 14 | sylancr 695 | . . . . . . 7 |
138 | 137 | recnd 10068 | . . . . . 6 |
139 | 66, 138, 86, 78 | div23d 10838 | . . . . 5 |
140 | subdi 10463 | . . . . . . . . 9 | |
141 | 27, 140 | mp3an2 1412 | . . . . . . . 8 |
142 | 66, 99, 141 | syl2anc 693 | . . . . . . 7 |
143 | 66 | mulid1d 10057 | . . . . . . . 8 |
144 | 66, 99 | mulcomd 10061 | . . . . . . . 8 |
145 | 143, 144 | oveq12d 6668 | . . . . . . 7 |
146 | 142, 145 | eqtrd 2656 | . . . . . 6 |
147 | 146 | oveq1d 6665 | . . . . 5 |
148 | 94 | recnd 10068 | . . . . . . . 8 |
149 | 86, 148, 86, 78 | divsubdird 10840 | . . . . . . 7 |
150 | 126, 99, 127 | nnncan2d 10427 | . . . . . . . . 9 |
151 | 99, 66 | pncan2d 10394 | . . . . . . . . 9 |
152 | 150, 151 | eqtrd 2656 | . . . . . . . 8 |
153 | 152 | oveq1d 6665 | . . . . . . 7 |
154 | 132 | oveq1d 6665 | . . . . . . 7 |
155 | 149, 153, 154 | 3eqtr3d 2664 | . . . . . 6 |
156 | 155 | oveq1d 6665 | . . . . 5 |
157 | 139, 147, 156 | 3eqtr3d 2664 | . . . 4 |
158 | 99, 66 | mulcomd 10061 | . . . . . 6 |
159 | 158 | oveq1d 6665 | . . . . 5 |
160 | 99, 66, 86, 78 | div23d 10838 | . . . . . 6 |
161 | 134 | oveq1d 6665 | . . . . . 6 |
162 | 160, 161 | eqtrd 2656 | . . . . 5 |
163 | 66, 99, 86, 78 | div23d 10838 | . . . . . 6 |
164 | 155 | oveq1d 6665 | . . . . . 6 |
165 | 163, 164 | eqtrd 2656 | . . . . 5 |
166 | 159, 162, 165 | 3eqtr3d 2664 | . . . 4 |
167 | oveq2 6658 | . . . . . . . 8 | |
168 | 167 | oveq1d 6665 | . . . . . . 7 |
169 | 168 | eqeq2d 2632 | . . . . . 6 |
170 | eqeq1 2626 | . . . . . 6 | |
171 | 167 | oveq1d 6665 | . . . . . . 7 |
172 | 171 | eqeq1d 2624 | . . . . . 6 |
173 | 169, 170, 172 | 3anbi123d 1399 | . . . . 5 |
174 | eqeq1 2626 | . . . . . 6 | |
175 | oveq2 6658 | . . . . . . . 8 | |
176 | 175 | oveq1d 6665 | . . . . . . 7 |
177 | 176 | eqeq2d 2632 | . . . . . 6 |
178 | 175 | oveq1d 6665 | . . . . . . 7 |
179 | 178 | eqeq2d 2632 | . . . . . 6 |
180 | 174, 177, 179 | 3anbi123d 1399 | . . . . 5 |
181 | 173, 180 | rspc2ev 3324 | . . . 4 |
182 | 93, 114, 136, 157, 166, 181 | syl113anc 1338 | . . 3 |
183 | 182 | ex 450 | . 2 |
184 | 53, 183 | pm2.61ine 2877 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wne 2794 wrex 2913 class class class wbr 4653 (class class class)co 6650 cc 9934 cr 9935 cc0 9936 c1 9937 caddc 9939 cmul 9941 clt 10074 cle 10075 cmin 10266 cdiv 10684 cicc 12178 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-icc 12182 |
This theorem is referenced by: axpasch 25821 |
Copyright terms: Public domain | W3C validator |