Proof of Theorem 4atlem11
Step | Hyp | Ref
| Expression |
1 | | 3anass 1042 |
. . . 4
|
2 | | simpl11 1136 |
. . . . . . 7
|
3 | | hllat 34650 |
. . . . . . 7
|
4 | 2, 3 | syl 17 |
. . . . . 6
|
5 | | simpl2l 1114 |
. . . . . . 7
|
6 | | eqid 2622 |
. . . . . . . 8
|
7 | | 4at.a |
. . . . . . . 8
|
8 | 6, 7 | atbase 34576 |
. . . . . . 7
|
9 | 5, 8 | syl 17 |
. . . . . 6
|
10 | | simpl2r 1115 |
. . . . . . 7
|
11 | 6, 7 | atbase 34576 |
. . . . . . 7
|
12 | 10, 11 | syl 17 |
. . . . . 6
|
13 | | simpl12 1137 |
. . . . . . . 8
|
14 | | simpl31 1142 |
. . . . . . . 8
|
15 | | 4at.j |
. . . . . . . . 9
|
16 | 6, 15, 7 | hlatjcl 34653 |
. . . . . . . 8
|
17 | 2, 13, 14, 16 | syl3anc 1326 |
. . . . . . 7
|
18 | | simpl32 1143 |
. . . . . . . 8
|
19 | | simpl33 1144 |
. . . . . . . 8
|
20 | 6, 15, 7 | hlatjcl 34653 |
. . . . . . . 8
|
21 | 2, 18, 19, 20 | syl3anc 1326 |
. . . . . . 7
|
22 | 6, 15 | latjcl 17051 |
. . . . . . 7
|
23 | 4, 17, 21, 22 | syl3anc 1326 |
. . . . . 6
|
24 | | 4at.l |
. . . . . . 7
|
25 | 6, 24, 15 | latjle12 17062 |
. . . . . 6
|
26 | 4, 9, 12, 23, 25 | syl13anc 1328 |
. . . . 5
|
27 | 26 | anbi2d 740 |
. . . 4
|
28 | 1, 27 | syl5bb 272 |
. . 3
|
29 | | simpl13 1138 |
. . . . 5
|
30 | 6, 7 | atbase 34576 |
. . . . 5
|
31 | 29, 30 | syl 17 |
. . . 4
|
32 | 6, 15, 7 | hlatjcl 34653 |
. . . . 5
|
33 | 2, 5, 10, 32 | syl3anc 1326 |
. . . 4
|
34 | 6, 24, 15 | latjle12 17062 |
. . . 4
|
35 | 4, 31, 33, 23, 34 | syl13anc 1328 |
. . 3
|
36 | 28, 35 | bitrd 268 |
. 2
|
37 | | simpl1 1064 |
. . . 4
|
38 | | simpl2 1065 |
. . . 4
|
39 | 18, 19 | jca 554 |
. . . 4
|
40 | | simpr 477 |
. . . 4
|
41 | 24, 15, 7 | 4atlem3a 34883 |
. . . 4
|
42 | 37, 38, 39, 40, 41 | syl31anc 1329 |
. . 3
|
43 | | simp1l 1085 |
. . . . . 6
|
44 | | simp1r 1086 |
. . . . . 6
|
45 | | simp2 1062 |
. . . . . 6
|
46 | | simp3 1063 |
. . . . . 6
|
47 | 24, 15, 7 | 4atlem11b 34894 |
. . . . . 6
|
48 | 43, 44, 45, 46, 47 | syl121anc 1331 |
. . . . 5
|
49 | 48 | 3exp 1264 |
. . . 4
|
50 | 2 | 3ad2ant1 1082 |
. . . . . . 7
|
51 | 13 | 3ad2ant1 1082 |
. . . . . . 7
|
52 | 29 | 3ad2ant1 1082 |
. . . . . . 7
|
53 | 5 | 3ad2ant1 1082 |
. . . . . . 7
|
54 | 10 | 3ad2ant1 1082 |
. . . . . . 7
|
55 | 15, 7 | hlatj4 34660 |
. . . . . . 7
|
56 | 50, 51, 52, 53, 54, 55 | syl122anc 1335 |
. . . . . 6
|
57 | 50, 51, 53 | 3jca 1242 |
. . . . . . 7
|
58 | 52, 54 | jca 554 |
. . . . . . 7
|
59 | | simp1l3 1156 |
. . . . . . 7
|
60 | | simp1r2 1158 |
. . . . . . . . 9
|
61 | 24, 15, 7 | 4atlem0be 34881 |
. . . . . . . . 9
|
62 | 50, 51, 52, 53, 60, 61 | syl131anc 1339 |
. . . . . . . 8
|
63 | | simp1r1 1157 |
. . . . . . . . 9
|
64 | 24, 15, 7 | 4atlem0ae 34880 |
. . . . . . . . 9
|
65 | 50, 51, 52, 53, 63, 60, 64 | syl132anc 1344 |
. . . . . . . 8
|
66 | | simp1r3 1159 |
. . . . . . . . 9
|
67 | 15, 7 | hlatj32 34658 |
. . . . . . . . . . 11
|
68 | 50, 51, 52, 53, 67 | syl13anc 1328 |
. . . . . . . . . 10
|
69 | 68 | breq2d 4665 |
. . . . . . . . 9
|
70 | 66, 69 | mtbid 314 |
. . . . . . . 8
|
71 | 62, 65, 70 | 3jca 1242 |
. . . . . . 7
|
72 | | simp2 1062 |
. . . . . . 7
|
73 | | simp32 1098 |
. . . . . . 7
|
74 | | simp31 1097 |
. . . . . . 7
|
75 | | simp33 1099 |
. . . . . . 7
|
76 | 24, 15, 7 | 4atlem11b 34894 |
. . . . . . 7
|
77 | 57, 58, 59, 71, 72, 73, 74, 75, 76 | syl323anc 1356 |
. . . . . 6
|
78 | 56, 77 | eqtrd 2656 |
. . . . 5
|
79 | 78 | 3exp 1264 |
. . . 4
|
80 | 6, 7 | atbase 34576 |
. . . . . . . . . 10
|
81 | 13, 80 | syl 17 |
. . . . . . . . 9
|
82 | 6, 15 | latj4rot 17102 |
. . . . . . . . 9
|
83 | 4, 81, 31, 9, 12, 82 | syl122anc 1335 |
. . . . . . . 8
|
84 | 15, 7 | hlatjcom 34654 |
. . . . . . . . . 10
|
85 | 2, 10, 13, 84 | syl3anc 1326 |
. . . . . . . . 9
|
86 | 85 | oveq1d 6665 |
. . . . . . . 8
|
87 | 83, 86 | eqtrd 2656 |
. . . . . . 7
|
88 | 87 | 3ad2ant1 1082 |
. . . . . 6
|
89 | 2, 13, 10 | 3jca 1242 |
. . . . . . . . 9
|
90 | 29, 5 | jca 554 |
. . . . . . . . 9
|
91 | | simpl3 1066 |
. . . . . . . . 9
|
92 | 89, 90, 91 | 3jca 1242 |
. . . . . . . 8
|
93 | 92 | 3ad2ant1 1082 |
. . . . . . 7
|
94 | 24, 15, 7 | 4noncolr1 34741 |
. . . . . . . . . 10
|
95 | 37, 38, 40, 94 | syl3anc 1326 |
. . . . . . . . 9
|
96 | | necom 2847 |
. . . . . . . . . . 11
|
97 | 96 | a1i 11 |
. . . . . . . . . 10
|
98 | 85 | breq2d 4665 |
. . . . . . . . . . 11
|
99 | 98 | notbid 308 |
. . . . . . . . . 10
|
100 | 85 | oveq1d 6665 |
. . . . . . . . . . . 12
|
101 | 100 | breq2d 4665 |
. . . . . . . . . . 11
|
102 | 101 | notbid 308 |
. . . . . . . . . 10
|
103 | 97, 99, 102 | 3anbi123d 1399 |
. . . . . . . . 9
|
104 | 95, 103 | mpbid 222 |
. . . . . . . 8
|
105 | 104 | 3ad2ant1 1082 |
. . . . . . 7
|
106 | | simp2 1062 |
. . . . . . 7
|
107 | | simpr3 1069 |
. . . . . . . . 9
|
108 | | simpr1 1067 |
. . . . . . . . 9
|
109 | | simpr2 1068 |
. . . . . . . . 9
|
110 | 107, 108,
109 | 3jca 1242 |
. . . . . . . 8
|
111 | 110 | 3adant2 1080 |
. . . . . . 7
|
112 | 24, 15, 7 | 4atlem11b 34894 |
. . . . . . 7
|
113 | 93, 105, 106, 111, 112 | syl121anc 1331 |
. . . . . 6
|
114 | 88, 113 | eqtrd 2656 |
. . . . 5
|
115 | 114 | 3exp 1264 |
. . . 4
|
116 | 49, 79, 115 | 3jaod 1392 |
. . 3
|
117 | 42, 116 | mpd 15 |
. 2
|
118 | 36, 117 | sylbird 250 |
1
|