Proof of Theorem dalawlem12
Step | Hyp | Ref
| Expression |
1 | | eqid 2622 |
. . . 4
|
2 | | dalawlem.l |
. . . 4
|
3 | | simp11 1091 |
. . . . 5
|
4 | | hllat 34650 |
. . . . 5
|
5 | 3, 4 | syl 17 |
. . . 4
|
6 | | simp21 1094 |
. . . . . 6
|
7 | | simp22 1095 |
. . . . . 6
|
8 | | dalawlem.j |
. . . . . . 7
|
9 | | dalawlem.a |
. . . . . . 7
|
10 | 1, 8, 9 | hlatjcl 34653 |
. . . . . 6
|
11 | 3, 6, 7, 10 | syl3anc 1326 |
. . . . 5
|
12 | | simp31 1097 |
. . . . . 6
|
13 | | simp32 1098 |
. . . . . 6
|
14 | 1, 8, 9 | hlatjcl 34653 |
. . . . . 6
|
15 | 3, 12, 13, 14 | syl3anc 1326 |
. . . . 5
|
16 | | dalawlem.m |
. . . . . 6
|
17 | 1, 16 | latmcl 17052 |
. . . . 5
|
18 | 5, 11, 15, 17 | syl3anc 1326 |
. . . 4
|
19 | 1, 9 | atbase 34576 |
. . . . . . . 8
|
20 | 12, 19 | syl 17 |
. . . . . . 7
|
21 | 1, 8 | latjcl 17051 |
. . . . . . 7
|
22 | 5, 11, 20, 21 | syl3anc 1326 |
. . . . . 6
|
23 | 1, 9 | atbase 34576 |
. . . . . . 7
|
24 | 13, 23 | syl 17 |
. . . . . 6
|
25 | 1, 16 | latmcl 17052 |
. . . . . 6
|
26 | 5, 22, 24, 25 | syl3anc 1326 |
. . . . 5
|
27 | 1, 8 | latjcl 17051 |
. . . . 5
|
28 | 5, 26, 20, 27 | syl3anc 1326 |
. . . 4
|
29 | 1, 9 | atbase 34576 |
. . . . . . 7
|
30 | 7, 29 | syl 17 |
. . . . . 6
|
31 | | simp33 1099 |
. . . . . . 7
|
32 | 1, 8, 9 | hlatjcl 34653 |
. . . . . . 7
|
33 | 3, 13, 31, 32 | syl3anc 1326 |
. . . . . 6
|
34 | 1, 16 | latmcl 17052 |
. . . . . 6
|
35 | 5, 30, 33, 34 | syl3anc 1326 |
. . . . 5
|
36 | 1, 8, 9 | hlatjcl 34653 |
. . . . . 6
|
37 | 3, 31, 12, 36 | syl3anc 1326 |
. . . . 5
|
38 | 1, 8 | latjcl 17051 |
. . . . 5
|
39 | 5, 35, 37, 38 | syl3anc 1326 |
. . . 4
|
40 | 1, 2, 8 | latlej1 17060 |
. . . . . . 7
|
41 | 5, 11, 20, 40 | syl3anc 1326 |
. . . . . 6
|
42 | 1, 8, 9 | hlatjcl 34653 |
. . . . . . . 8
|
43 | 3, 13, 12, 42 | syl3anc 1326 |
. . . . . . 7
|
44 | 1, 2, 16 | latmlem1 17081 |
. . . . . . 7
|
45 | 5, 11, 22, 43, 44 | syl13anc 1328 |
. . . . . 6
|
46 | 41, 45 | mpd 15 |
. . . . 5
|
47 | 8, 9 | hlatjcom 34654 |
. . . . . . 7
|
48 | 3, 12, 13, 47 | syl3anc 1326 |
. . . . . 6
|
49 | 48 | oveq2d 6666 |
. . . . 5
|
50 | 1, 2, 8 | latlej2 17061 |
. . . . . . 7
|
51 | 5, 11, 20, 50 | syl3anc 1326 |
. . . . . 6
|
52 | 1, 2, 8, 16, 9 | atmod2i2 35148 |
. . . . . 6
|
53 | 3, 13, 22, 20, 51, 52 | syl131anc 1339 |
. . . . 5
|
54 | 46, 49, 53 | 3brtr4d 4685 |
. . . 4
|
55 | | hlol 34648 |
. . . . . . . . . . 11
|
56 | 3, 55 | syl 17 |
. . . . . . . . . 10
|
57 | 1, 8, 9 | hlatjcl 34653 |
. . . . . . . . . . . 12
|
58 | 3, 6, 12, 57 | syl3anc 1326 |
. . . . . . . . . . 11
|
59 | 1, 8 | latjcl 17051 |
. . . . . . . . . . 11
|
60 | 5, 30, 58, 59 | syl3anc 1326 |
. . . . . . . . . 10
|
61 | 1, 8, 9 | hlatjcl 34653 |
. . . . . . . . . . 11
|
62 | 3, 7, 13, 61 | syl3anc 1326 |
. . . . . . . . . 10
|
63 | 1, 16 | latmassOLD 34516 |
. . . . . . . . . 10
|
64 | 56, 60, 62, 24, 63 | syl13anc 1328 |
. . . . . . . . 9
|
65 | 8, 9 | hlatjass 34656 |
. . . . . . . . . . . 12
|
66 | 3, 6, 7, 12, 65 | syl13anc 1328 |
. . . . . . . . . . 11
|
67 | 8, 9 | hlatj12 34657 |
. . . . . . . . . . . 12
|
68 | 3, 6, 7, 12, 67 | syl13anc 1328 |
. . . . . . . . . . 11
|
69 | 66, 68 | eqtr2d 2657 |
. . . . . . . . . 10
|
70 | 2, 8, 9 | hlatlej2 34662 |
. . . . . . . . . . . 12
|
71 | 3, 7, 13, 70 | syl3anc 1326 |
. . . . . . . . . . 11
|
72 | 1, 2, 16 | latleeqm2 17080 |
. . . . . . . . . . . 12
|
73 | 5, 24, 62, 72 | syl3anc 1326 |
. . . . . . . . . . 11
|
74 | 71, 73 | mpbid 222 |
. . . . . . . . . 10
|
75 | 69, 74 | oveq12d 6668 |
. . . . . . . . 9
|
76 | 64, 75 | eqtr2d 2657 |
. . . . . . . 8
|
77 | 2, 8, 9 | hlatlej1 34661 |
. . . . . . . . . . . 12
|
78 | 3, 7, 13, 77 | syl3anc 1326 |
. . . . . . . . . . 11
|
79 | 1, 2, 8, 16, 9 | atmod1i1 35143 |
. . . . . . . . . . 11
|
80 | 3, 7, 58, 62, 78, 79 | syl131anc 1339 |
. . . . . . . . . 10
|
81 | 2, 8, 9 | hlatlej2 34662 |
. . . . . . . . . . . 12
|
82 | 3, 31, 7, 81 | syl3anc 1326 |
. . . . . . . . . . 11
|
83 | | simp13 1093 |
. . . . . . . . . . . 12
|
84 | | simp12 1092 |
. . . . . . . . . . . . . 14
|
85 | 84 | oveq1d 6665 |
. . . . . . . . . . . . 13
|
86 | 8, 9 | hlatjcom 34654 |
. . . . . . . . . . . . . 14
|
87 | 3, 7, 31, 86 | syl3anc 1326 |
. . . . . . . . . . . . 13
|
88 | 85, 87 | eqtr3d 2658 |
. . . . . . . . . . . 12
|
89 | 83, 88 | breqtrd 4679 |
. . . . . . . . . . 11
|
90 | 1, 16 | latmcl 17052 |
. . . . . . . . . . . . 13
|
91 | 5, 58, 62, 90 | syl3anc 1326 |
. . . . . . . . . . . 12
|
92 | 1, 8, 9 | hlatjcl 34653 |
. . . . . . . . . . . . 13
|
93 | 3, 31, 7, 92 | syl3anc 1326 |
. . . . . . . . . . . 12
|
94 | 1, 2, 8 | latjle12 17062 |
. . . . . . . . . . . 12
|
95 | 5, 30, 91, 93, 94 | syl13anc 1328 |
. . . . . . . . . . 11
|
96 | 82, 89, 95 | mpbi2and 956 |
. . . . . . . . . 10
|
97 | 80, 96 | eqbrtrrd 4677 |
. . . . . . . . 9
|
98 | 2, 8, 9 | hlatlej1 34661 |
. . . . . . . . . 10
|
99 | 3, 13, 31, 98 | syl3anc 1326 |
. . . . . . . . 9
|
100 | 1, 16 | latmcl 17052 |
. . . . . . . . . . 11
|
101 | 5, 60, 62, 100 | syl3anc 1326 |
. . . . . . . . . 10
|
102 | 1, 2, 16 | latmlem12 17083 |
. . . . . . . . . 10
|
103 | 5, 101, 93, 24, 33, 102 | syl122anc 1335 |
. . . . . . . . 9
|
104 | 97, 99, 103 | mp2and 715 |
. . . . . . . 8
|
105 | 76, 104 | eqbrtrd 4675 |
. . . . . . 7
|
106 | 2, 8, 9 | hlatlej2 34662 |
. . . . . . . . . 10
|
107 | 3, 13, 31, 106 | syl3anc 1326 |
. . . . . . . . 9
|
108 | 1, 2, 8, 16, 9 | atmod1i1 35143 |
. . . . . . . . 9
|
109 | 3, 31, 30, 33, 107, 108 | syl131anc 1339 |
. . . . . . . 8
|
110 | 1, 9 | atbase 34576 |
. . . . . . . . . 10
|
111 | 31, 110 | syl 17 |
. . . . . . . . 9
|
112 | 1, 8 | latjcom 17059 |
. . . . . . . . 9
|
113 | 5, 111, 35, 112 | syl3anc 1326 |
. . . . . . . 8
|
114 | 109, 113 | eqtr3d 2658 |
. . . . . . 7
|
115 | 105, 114 | breqtrd 4679 |
. . . . . 6
|
116 | 1, 8 | latjcl 17051 |
. . . . . . . 8
|
117 | 5, 35, 111, 116 | syl3anc 1326 |
. . . . . . 7
|
118 | 1, 2, 8 | latjlej1 17065 |
. . . . . . 7
|
119 | 5, 26, 117, 20, 118 | syl13anc 1328 |
. . . . . 6
|
120 | 115, 119 | mpd 15 |
. . . . 5
|
121 | 1, 8 | latjass 17095 |
. . . . . 6
|
122 | 5, 35, 111, 20, 121 | syl13anc 1328 |
. . . . 5
|
123 | 120, 122 | breqtrd 4679 |
. . . 4
|
124 | 1, 2, 5, 18, 28, 39, 54, 123 | lattrd 17058 |
. . 3
|
125 | 1, 2, 16 | latmle1 17076 |
. . . 4
|
126 | 5, 11, 15, 125 | syl3anc 1326 |
. . 3
|
127 | 1, 2, 16 | latlem12 17078 |
. . . 4
|
128 | 5, 18, 39, 11, 127 | syl13anc 1328 |
. . 3
|
129 | 124, 126,
128 | mpbi2and 956 |
. 2
|
130 | 1, 9 | atbase 34576 |
. . . . . 6
|
131 | 6, 130 | syl 17 |
. . . . 5
|
132 | 1, 2, 8, 16 | latmlej12 17091 |
. . . . 5
|
133 | 5, 30, 33, 131, 132 | syl13anc 1328 |
. . . 4
|
134 | 1, 2, 8, 16, 9 | llnmod1i2 35146 |
. . . 4
|
135 | 3, 35, 11, 31, 12, 133, 134 | syl321anc 1348 |
. . 3
|
136 | 8, 9 | hlatjidm 34655 |
. . . . . . 7
|
137 | 3, 7, 136 | syl2anc 693 |
. . . . . 6
|
138 | 84 | oveq2d 6666 |
. . . . . 6
|
139 | 137, 138 | eqtr3d 2658 |
. . . . 5
|
140 | 139 | oveq1d 6665 |
. . . 4
|
141 | 1, 16 | latmcom 17075 |
. . . . . 6
|
142 | 5, 37, 11, 141 | syl3anc 1326 |
. . . . 5
|
143 | 8, 9 | hlatjcom 34654 |
. . . . . . . 8
|
144 | 3, 6, 7, 143 | syl3anc 1326 |
. . . . . . 7
|
145 | 84 | oveq1d 6665 |
. . . . . . 7
|
146 | 144, 145 | eqtrd 2656 |
. . . . . 6
|
147 | 146 | oveq1d 6665 |
. . . . 5
|
148 | 142, 147 | eqtrd 2656 |
. . . 4
|
149 | 140, 148 | oveq12d 6668 |
. . 3
|
150 | 135, 149 | eqtr3d 2658 |
. 2
|
151 | 129, 150 | breqtrd 4679 |
1
|