Description: Decimal multiplication galore! (Contributed by Thierry Arnoux, 26-Dec-2021)
Ref | Expression | ||
---|---|---|---|
Assertion | hgt750lem2 | |- ( 3 x. ( ( ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) x. ( 1 . _ 4 _ 1 4 ) ) x. ( ( 1 . _ 4 _ 2 _ 6 3 ) x. ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) ) ) ) < ( 7 . _ 3 _ 4 8 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nn0 | |- 1 e. NN0 |
|
2 | 0re | |- 0 e. RR |
|
3 | 7re | |- 7 e. RR |
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4 | 9re | |- 9 e. RR |
|
5 | 5re | |- 5 e. RR |
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6 | 5 5 | pm3.2i | |- ( 5 e. RR /\ 5 e. RR ) |
7 | dp2cl | |- ( ( 5 e. RR /\ 5 e. RR ) -> _ 5 5 e. RR ) |
|
8 | 6 7 | ax-mp | |- _ 5 5 e. RR |
9 | 4 8 | pm3.2i | |- ( 9 e. RR /\ _ 5 5 e. RR ) |
10 | dp2cl | |- ( ( 9 e. RR /\ _ 5 5 e. RR ) -> _ 9 _ 5 5 e. RR ) |
|
11 | 9 10 | ax-mp | |- _ 9 _ 5 5 e. RR |
12 | 4 11 | pm3.2i | |- ( 9 e. RR /\ _ 9 _ 5 5 e. RR ) |
13 | dp2cl | |- ( ( 9 e. RR /\ _ 9 _ 5 5 e. RR ) -> _ 9 _ 9 _ 5 5 e. RR ) |
|
14 | 12 13 | ax-mp | |- _ 9 _ 9 _ 5 5 e. RR |
15 | 3 14 | pm3.2i | |- ( 7 e. RR /\ _ 9 _ 9 _ 5 5 e. RR ) |
16 | dp2cl | |- ( ( 7 e. RR /\ _ 9 _ 9 _ 5 5 e. RR ) -> _ 7 _ 9 _ 9 _ 5 5 e. RR ) |
|
17 | 15 16 | ax-mp | |- _ 7 _ 9 _ 9 _ 5 5 e. RR |
18 | 2 17 | pm3.2i | |- ( 0 e. RR /\ _ 7 _ 9 _ 9 _ 5 5 e. RR ) |
19 | dp2cl | |- ( ( 0 e. RR /\ _ 7 _ 9 _ 9 _ 5 5 e. RR ) -> _ 0 _ 7 _ 9 _ 9 _ 5 5 e. RR ) |
|
20 | 18 19 | ax-mp | |- _ 0 _ 7 _ 9 _ 9 _ 5 5 e. RR |
21 | dpcl | |- ( ( 1 e. NN0 /\ _ 0 _ 7 _ 9 _ 9 _ 5 5 e. RR ) -> ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) e. RR ) |
|
22 | 1 20 21 | mp2an | |- ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) e. RR |
23 | 22 | resqcli | |- ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) e. RR |
24 | 4nn0 | |- 4 e. NN0 |
|
25 | 4nn | |- 4 e. NN |
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26 | nnrp | |- ( 4 e. NN -> 4 e. RR+ ) |
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27 | 25 26 | ax-mp | |- 4 e. RR+ |
28 | 1 27 | rpdp2cl | |- _ 1 4 e. RR+ |
29 | 24 28 | rpdp2cl | |- _ 4 _ 1 4 e. RR+ |
30 | 1 29 | rpdpcl | |- ( 1 . _ 4 _ 1 4 ) e. RR+ |
31 | rpre | |- ( ( 1 . _ 4 _ 1 4 ) e. RR+ -> ( 1 . _ 4 _ 1 4 ) e. RR ) |
|
32 | 30 31 | ax-mp | |- ( 1 . _ 4 _ 1 4 ) e. RR |
33 | 23 32 | remulcli | |- ( ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) x. ( 1 . _ 4 _ 1 4 ) ) e. RR |
34 | 6re | |- 6 e. RR |
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35 | 1re | |- 1 e. RR |
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36 | 5 35 | pm3.2i | |- ( 5 e. RR /\ 1 e. RR ) |
37 | dp2cl | |- ( ( 5 e. RR /\ 1 e. RR ) -> _ 5 1 e. RR ) |
|
38 | 36 37 | ax-mp | |- _ 5 1 e. RR |
39 | 34 38 | pm3.2i | |- ( 6 e. RR /\ _ 5 1 e. RR ) |
40 | dp2cl | |- ( ( 6 e. RR /\ _ 5 1 e. RR ) -> _ 6 _ 5 1 e. RR ) |
|
41 | 39 40 | ax-mp | |- _ 6 _ 5 1 e. RR |
42 | dpcl | |- ( ( 1 e. NN0 /\ _ 6 _ 5 1 e. RR ) -> ( 1 . _ 6 _ 5 1 ) e. RR ) |
|
43 | 1 41 42 | mp2an | |- ( 1 . _ 6 _ 5 1 ) e. RR |
44 | 33 43 | pm3.2i | |- ( ( ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) x. ( 1 . _ 4 _ 1 4 ) ) e. RR /\ ( 1 . _ 6 _ 5 1 ) e. RR ) |
45 | 22 | sqge0i | |- 0 <_ ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) |
46 | rpgt0 | |- ( ( 1 . _ 4 _ 1 4 ) e. RR+ -> 0 < ( 1 . _ 4 _ 1 4 ) ) |
|
47 | 30 46 | ax-mp | |- 0 < ( 1 . _ 4 _ 1 4 ) |
48 | 2 32 47 | ltleii | |- 0 <_ ( 1 . _ 4 _ 1 4 ) |
49 | 23 32 | mulge0i | |- ( ( 0 <_ ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) /\ 0 <_ ( 1 . _ 4 _ 1 4 ) ) -> 0 <_ ( ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) x. ( 1 . _ 4 _ 1 4 ) ) ) |
50 | 45 48 49 | mp2an | |- 0 <_ ( ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) x. ( 1 . _ 4 _ 1 4 ) ) |
51 | 0nn0 | |- 0 e. NN0 |
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52 | 7nn0 | |- 7 e. NN0 |
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53 | 9nn0 | |- 9 e. NN0 |
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54 | 5nn0 | |- 5 e. NN0 |
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55 | 5nn | |- 5 e. NN |
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56 | nnrp | |- ( 5 e. NN -> 5 e. RR+ ) |
|
57 | 55 56 | ax-mp | |- 5 e. RR+ |
58 | 54 57 | rpdp2cl | |- _ 5 5 e. RR+ |
59 | 53 58 | rpdp2cl | |- _ 9 _ 5 5 e. RR+ |
60 | 53 59 | rpdp2cl | |- _ 9 _ 9 _ 5 5 e. RR+ |
61 | 52 60 | rpdp2cl | |- _ 7 _ 9 _ 9 _ 5 5 e. RR+ |
62 | 51 61 | rpdp2cl | |- _ 0 _ 7 _ 9 _ 9 _ 5 5 e. RR+ |
63 | 8nn | |- 8 e. NN |
|
64 | 63 | rpdp2cl2 | |- _ 8 0 e. RR+ |
65 | 51 64 | rpdp2cl | |- _ 0 _ 8 0 e. RR+ |
66 | 9lt10 | |- 9 < ; 1 0 |
|
67 | 5lt10 | |- 5 < ; 1 0 |
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68 | 54 57 67 67 | dp2lt10 | |- _ 5 5 < ; 1 0 |
69 | 53 58 66 68 | dp2lt10 | |- _ 9 _ 5 5 < ; 1 0 |
70 | 53 59 66 69 | dp2lt10 | |- _ 9 _ 9 _ 5 5 < ; 1 0 |
71 | 7p1e8 | |- ( 7 + 1 ) = 8 |
|
72 | 52 60 70 71 | dp2ltsuc | |- _ 7 _ 9 _ 9 _ 5 5 < 8 |
73 | 8nn0 | |- 8 e. NN0 |
|
74 | 73 | dp20u | |- _ 8 0 = 8 |
75 | 72 74 | breqtrri | |- _ 7 _ 9 _ 9 _ 5 5 < _ 8 0 |
76 | 51 61 64 75 | dp2lt | |- _ 0 _ 7 _ 9 _ 9 _ 5 5 < _ 0 _ 8 0 |
77 | 1 62 65 76 | dplt | |- ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) < ( 1 . _ 0 _ 8 0 ) |
78 | 1 62 | rpdpcl | |- ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) e. RR+ |
79 | rpge0 | |- ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) e. RR+ -> 0 <_ ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ) |
|
80 | 78 79 | ax-mp | |- 0 <_ ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) |
81 | 1 65 | rpdpcl | |- ( 1 . _ 0 _ 8 0 ) e. RR+ |
82 | rpge0 | |- ( ( 1 . _ 0 _ 8 0 ) e. RR+ -> 0 <_ ( 1 . _ 0 _ 8 0 ) ) |
|
83 | 81 82 | ax-mp | |- 0 <_ ( 1 . _ 0 _ 8 0 ) |
84 | 8re | |- 8 e. RR |
|
85 | 84 2 | pm3.2i | |- ( 8 e. RR /\ 0 e. RR ) |
86 | dp2cl | |- ( ( 8 e. RR /\ 0 e. RR ) -> _ 8 0 e. RR ) |
|
87 | 85 86 | ax-mp | |- _ 8 0 e. RR |
88 | 2 87 | pm3.2i | |- ( 0 e. RR /\ _ 8 0 e. RR ) |
89 | dp2cl | |- ( ( 0 e. RR /\ _ 8 0 e. RR ) -> _ 0 _ 8 0 e. RR ) |
|
90 | 88 89 | ax-mp | |- _ 0 _ 8 0 e. RR |
91 | dpcl | |- ( ( 1 e. NN0 /\ _ 0 _ 8 0 e. RR ) -> ( 1 . _ 0 _ 8 0 ) e. RR ) |
|
92 | 1 90 91 | mp2an | |- ( 1 . _ 0 _ 8 0 ) e. RR |
93 | 22 92 | lt2sqi | |- ( ( 0 <_ ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) /\ 0 <_ ( 1 . _ 0 _ 8 0 ) ) -> ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) < ( 1 . _ 0 _ 8 0 ) <-> ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) < ( ( 1 . _ 0 _ 8 0 ) ^ 2 ) ) ) |
94 | 80 83 93 | mp2an | |- ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) < ( 1 . _ 0 _ 8 0 ) <-> ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) < ( ( 1 . _ 0 _ 8 0 ) ^ 2 ) ) |
95 | 77 94 | mpbi | |- ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) < ( ( 1 . _ 0 _ 8 0 ) ^ 2 ) |
96 | 92 | recni | |- ( 1 . _ 0 _ 8 0 ) e. CC |
97 | 96 | sqvali | |- ( ( 1 . _ 0 _ 8 0 ) ^ 2 ) = ( ( 1 . _ 0 _ 8 0 ) x. ( 1 . _ 0 _ 8 0 ) ) |
98 | 6nn0 | |- 6 e. NN0 |
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99 | 1 98 | deccl | |- ; 1 6 e. NN0 |
100 | 98 24 | deccl | |- ; 6 4 e. NN0 |
101 | 4lt10 | |- 4 < ; 1 0 |
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102 | 10pos | |- 0 < ; 1 0 |
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103 | 99 51 | deccl | |- ; ; 1 6 0 e. NN0 |
104 | eqid | |- ; ; ; 1 6 0 0 = ; ; ; 1 6 0 0 |
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105 | eqid | |- ; 6 4 = ; 6 4 |
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106 | eqid | |- ; ; 1 6 0 = ; ; 1 6 0 |
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107 | 98 | dec0h | |- 6 = ; 0 6 |
108 | 99 | nn0cni | |- ; 1 6 e. CC |
109 | 108 | addid1i | |- ( ; 1 6 + 0 ) = ; 1 6 |
110 | 6cn | |- 6 e. CC |
|
111 | 110 | addid2i | |- ( 0 + 6 ) = 6 |
112 | 99 51 51 98 106 107 109 111 | decadd | |- ( ; ; 1 6 0 + 6 ) = ; ; 1 6 6 |
113 | 4cn | |- 4 e. CC |
|
114 | 113 | addid2i | |- ( 0 + 4 ) = 4 |
115 | 103 51 98 24 104 105 112 114 | decadd | |- ( ; ; ; 1 6 0 0 + ; 6 4 ) = ; ; ; 1 6 6 4 |
116 | 1t1e1 | |- ( 1 x. 1 ) = 1 |
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117 | 1 | dp0u | |- ( 1 . 0 ) = 1 |
118 | 117 117 | oveq12i | |- ( ( 1 . 0 ) x. ( 1 . 0 ) ) = ( 1 x. 1 ) |
119 | 51 | dp20u | |- _ 0 0 = 0 |
120 | 119 | oveq2i | |- ( 1 . _ 0 0 ) = ( 1 . 0 ) |
121 | 120 117 | eqtri | |- ( 1 . _ 0 0 ) = 1 |
122 | 116 118 121 | 3eqtr4i | |- ( ( 1 . 0 ) x. ( 1 . 0 ) ) = ( 1 . _ 0 0 ) |
123 | 8t8e64 | |- ( 8 x. 8 ) = ; 6 4 |
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124 | 73 | dp0u | |- ( 8 . 0 ) = 8 |
125 | 124 124 | oveq12i | |- ( ( 8 . 0 ) x. ( 8 . 0 ) ) = ( 8 x. 8 ) |
126 | 119 | oveq2i | |- ( ; 6 4 . _ 0 0 ) = ( ; 6 4 . 0 ) |
127 | 100 | dp0u | |- ( ; 6 4 . 0 ) = ; 6 4 |
128 | 126 127 | eqtri | |- ( ; 6 4 . _ 0 0 ) = ; 6 4 |
129 | 123 125 128 | 3eqtr4i | |- ( ( 8 . 0 ) x. ( 8 . 0 ) ) = ( ; 6 4 . _ 0 0 ) |
130 | 10nn0 | |- ; 1 0 e. NN0 |
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131 | 130 51 | deccl | |- ; ; 1 0 0 e. NN0 |
132 | eqid | |- ; ; ; 1 0 0 1 = ; ; ; 1 0 0 1 |
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133 | eqid | |- ; ; 1 6 6 = ; ; 1 6 6 |
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134 | eqid | |- ; ; 1 0 0 = ; ; 1 0 0 |
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135 | eqid | |- ; 1 6 = ; 1 6 |
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136 | dec10p | |- ( ; 1 0 + 1 ) = ; 1 1 |
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137 | 130 51 1 98 134 135 136 111 | decadd | |- ( ; ; 1 0 0 + ; 1 6 ) = ; ; 1 1 6 |
138 | ax-1cn | |- 1 e. CC |
|
139 | 138 110 | addcomi | |- ( 1 + 6 ) = ( 6 + 1 ) |
140 | 6p1e7 | |- ( 6 + 1 ) = 7 |
|
141 | 139 140 | eqtri | |- ( 1 + 6 ) = 7 |
142 | 131 1 99 98 132 133 137 141 | decadd | |- ( ; ; ; 1 0 0 1 + ; ; 1 6 6 ) = ; ; ; 1 1 6 7 |
143 | eqid | |- ; 1 7 = ; 1 7 |
|
144 | 141 | oveq1i | |- ( ( 1 + 6 ) + 1 ) = ( 7 + 1 ) |
145 | 144 71 | eqtri | |- ( ( 1 + 6 ) + 1 ) = 8 |
146 | 7p4e11 | |- ( 7 + 4 ) = ; 1 1 |
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147 | 1 52 98 24 143 105 145 1 146 | decaddc | |- ( ; 1 7 + ; 6 4 ) = ; 8 1 |
148 | 119 | oveq2i | |- ( ; 1 6 . _ 0 0 ) = ( ; 1 6 . 0 ) |
149 | 99 | dp0u | |- ( ; 1 6 . 0 ) = ; 1 6 |
150 | 148 149 | eqtri | |- ( ; 1 6 . _ 0 0 ) = ; 1 6 |
151 | 121 150 | oveq12i | |- ( ( 1 . _ 0 0 ) + ( ; 1 6 . _ 0 0 ) ) = ( 1 + ; 1 6 ) |
152 | 1 | dec0h | |- 1 = ; 0 1 |
153 | 138 | addid2i | |- ( 0 + 1 ) = 1 |
154 | 51 1 1 98 152 135 153 141 | decadd | |- ( 1 + ; 1 6 ) = ; 1 7 |
155 | 151 154 | eqtri | |- ( ( 1 . _ 0 0 ) + ( ; 1 6 . _ 0 0 ) ) = ; 1 7 |
156 | 155 128 | oveq12i | |- ( ( ( 1 . _ 0 0 ) + ( ; 1 6 . _ 0 0 ) ) + ( ; 6 4 . _ 0 0 ) ) = ( ; 1 7 + ; 6 4 ) |
157 | 117 124 | oveq12i | |- ( ( 1 . 0 ) + ( 8 . 0 ) ) = ( 1 + 8 ) |
158 | 8cn | |- 8 e. CC |
|
159 | 138 158 | addcomi | |- ( 1 + 8 ) = ( 8 + 1 ) |
160 | 8p1e9 | |- ( 8 + 1 ) = 9 |
|
161 | 157 159 160 | 3eqtri | |- ( ( 1 . 0 ) + ( 8 . 0 ) ) = 9 |
162 | 161 161 | oveq12i | |- ( ( ( 1 . 0 ) + ( 8 . 0 ) ) x. ( ( 1 . 0 ) + ( 8 . 0 ) ) ) = ( 9 x. 9 ) |
163 | 9t9e81 | |- ( 9 x. 9 ) = ; 8 1 |
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164 | 162 163 | eqtri | |- ( ( ( 1 . 0 ) + ( 8 . 0 ) ) x. ( ( 1 . 0 ) + ( 8 . 0 ) ) ) = ; 8 1 |
165 | 147 156 164 | 3eqtr4ri | |- ( ( ( 1 . 0 ) + ( 8 . 0 ) ) x. ( ( 1 . 0 ) + ( 8 . 0 ) ) ) = ( ( ( 1 . _ 0 0 ) + ( ; 1 6 . _ 0 0 ) ) + ( ; 6 4 . _ 0 0 ) ) |
166 | 1 51 73 51 1 73 51 51 51 51 1 99 51 51 100 51 51 1 98 98 24 1 1 98 52 101 102 102 115 122 129 142 165 | dpmul4 | |- ( ( 1 . _ 0 _ 8 0 ) x. ( 1 . _ 0 _ 8 0 ) ) < ( 1 . _ 1 _ 6 7 ) |
167 | 97 166 | eqbrtri | |- ( ( 1 . _ 0 _ 8 0 ) ^ 2 ) < ( 1 . _ 1 _ 6 7 ) |
168 | 92 | resqcli | |- ( ( 1 . _ 0 _ 8 0 ) ^ 2 ) e. RR |
169 | 34 3 | pm3.2i | |- ( 6 e. RR /\ 7 e. RR ) |
170 | dp2cl | |- ( ( 6 e. RR /\ 7 e. RR ) -> _ 6 7 e. RR ) |
|
171 | 169 170 | ax-mp | |- _ 6 7 e. RR |
172 | 35 171 | pm3.2i | |- ( 1 e. RR /\ _ 6 7 e. RR ) |
173 | dp2cl | |- ( ( 1 e. RR /\ _ 6 7 e. RR ) -> _ 1 _ 6 7 e. RR ) |
|
174 | 172 173 | ax-mp | |- _ 1 _ 6 7 e. RR |
175 | dpcl | |- ( ( 1 e. NN0 /\ _ 1 _ 6 7 e. RR ) -> ( 1 . _ 1 _ 6 7 ) e. RR ) |
|
176 | 1 174 175 | mp2an | |- ( 1 . _ 1 _ 6 7 ) e. RR |
177 | 23 168 176 | lttri | |- ( ( ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) < ( ( 1 . _ 0 _ 8 0 ) ^ 2 ) /\ ( ( 1 . _ 0 _ 8 0 ) ^ 2 ) < ( 1 . _ 1 _ 6 7 ) ) -> ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) < ( 1 . _ 1 _ 6 7 ) ) |
178 | 95 167 177 | mp2an | |- ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) < ( 1 . _ 1 _ 6 7 ) |
179 | 23 176 32 47 | ltmul1ii | |- ( ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) < ( 1 . _ 1 _ 6 7 ) <-> ( ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) x. ( 1 . _ 4 _ 1 4 ) ) < ( ( 1 . _ 1 _ 6 7 ) x. ( 1 . _ 4 _ 1 4 ) ) ) |
180 | 178 179 | mpbi | |- ( ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) x. ( 1 . _ 4 _ 1 4 ) ) < ( ( 1 . _ 1 _ 6 7 ) x. ( 1 . _ 4 _ 1 4 ) ) |
181 | 2nn0 | |- 2 e. NN0 |
|
182 | 3nn0 | |- 3 e. NN0 |
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183 | 1lt10 | |- 1 < ; 1 0 |
|
184 | 3lt10 | |- 3 < ; 1 0 |
|
185 | 8lt10 | |- 8 < ; 1 0 |
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186 | 130 53 | deccl | |- ; ; 1 0 9 e. NN0 |
187 | eqid | |- ; ; ; 1 0 9 2 = ; ; ; 1 0 9 2 |
|
188 | 53 | dec0h | |- 9 = ; 0 9 |
189 | 186 | nn0cni | |- ; ; 1 0 9 e. CC |
190 | 189 | addid1i | |- ( ; ; 1 0 9 + 0 ) = ; ; 1 0 9 |
191 | dec10p | |- ( ; 1 0 + 0 ) = ; 1 0 |
|
192 | 138 | addid1i | |- ( 1 + 0 ) = 1 |
193 | 1 51 51 1 191 152 192 153 | decadd | |- ( ( ; 1 0 + 0 ) + 1 ) = ; 1 1 |
194 | 9p1e10 | |- ( 9 + 1 ) = ; 1 0 |
|
195 | 130 53 51 1 190 152 193 51 194 | decaddc | |- ( ( ; ; 1 0 9 + 0 ) + 1 ) = ; ; 1 1 0 |
196 | 9cn | |- 9 e. CC |
|
197 | 2cn | |- 2 e. CC |
|
198 | 196 197 | addcomi | |- ( 9 + 2 ) = ( 2 + 9 ) |
199 | 9p2e11 | |- ( 9 + 2 ) = ; 1 1 |
|
200 | 198 199 | eqtr3i | |- ( 2 + 9 ) = ; 1 1 |
201 | 186 181 51 53 187 188 195 1 200 | decaddc | |- ( ; ; ; 1 0 9 2 + 9 ) = ; ; ; 1 1 0 1 |
202 | 113 138 | mulcomi | |- ( 4 x. 1 ) = ( 1 x. 4 ) |
203 | 113 | mulid1i | |- ( 4 x. 1 ) = 4 |
204 | 202 203 | eqtr3i | |- ( 1 x. 4 ) = 4 |
205 | 24 | dec0h | |- 4 = ; 0 4 |
206 | 203 202 205 | 3eqtr3i | |- ( 1 x. 4 ) = ; 0 4 |
207 | 138 113 | addcli | |- ( 1 + 4 ) e. CC |
208 | 207 | addid1i | |- ( ( 1 + 4 ) + 0 ) = ( 1 + 4 ) |
209 | 113 138 | addcomi | |- ( 4 + 1 ) = ( 1 + 4 ) |
210 | 4p1e5 | |- ( 4 + 1 ) = 5 |
|
211 | 208 209 210 | 3eqtr2i | |- ( ( 1 + 4 ) + 0 ) = 5 |
212 | 54 | dec0h | |- 5 = ; 0 5 |
213 | 211 212 | eqtri | |- ( ( 1 + 4 ) + 0 ) = ; 0 5 |
214 | 1 1 1 24 51 51 54 24 116 204 116 206 213 192 | dpmul | |- ( ( 1 . 1 ) x. ( 1 . 4 ) ) = ( 1 . _ 5 4 ) |
215 | 110 | mulid1i | |- ( 6 x. 1 ) = 6 |
216 | 6t4e24 | |- ( 6 x. 4 ) = ; 2 4 |
|
217 | 7cn | |- 7 e. CC |
|
218 | 217 | mulid1i | |- ( 7 x. 1 ) = 7 |
219 | 7t4e28 | |- ( 7 x. 4 ) = ; 2 8 |
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220 | 181 24 | deccl | |- ; 2 4 e. NN0 |
221 | 220 | nn0cni | |- ; 2 4 e. CC |
222 | 221 217 | addcomi | |- ( ; 2 4 + 7 ) = ( 7 + ; 2 4 ) |
223 | eqid | |- ; 2 4 = ; 2 4 |
|
224 | 2p1e3 | |- ( 2 + 1 ) = 3 |
|
225 | 217 113 146 | addcomli | |- ( 4 + 7 ) = ; 1 1 |
226 | 181 24 52 223 224 1 225 | decaddci | |- ( ; 2 4 + 7 ) = ; 3 1 |
227 | 222 226 | eqtr3i | |- ( 7 + ; 2 4 ) = ; 3 1 |
228 | 227 | oveq1i | |- ( ( 7 + ; 2 4 ) + 2 ) = ( ; 3 1 + 2 ) |
229 | eqid | |- ; 3 1 = ; 3 1 |
|
230 | 197 138 224 | addcomli | |- ( 1 + 2 ) = 3 |
231 | 182 1 181 229 230 | decaddi | |- ( ; 3 1 + 2 ) = ; 3 3 |
232 | 228 231 | eqtri | |- ( ( 7 + ; 2 4 ) + 2 ) = ; 3 3 |
233 | 6p3e9 | |- ( 6 + 3 ) = 9 |
|
234 | 98 52 1 24 181 182 182 73 215 216 218 219 232 233 | dpmul | |- ( ( 6 . 7 ) x. ( 1 . 4 ) ) = ( 9 . _ 3 8 ) |
235 | 1 54 | deccl | |- ; 1 5 e. NN0 |
236 | 235 24 | deccl | |- ; ; 1 5 4 e. NN0 |
237 | 51 1 | deccl | |- ; 0 1 e. NN0 |
238 | 237 1 | deccl | |- ; ; 0 1 1 e. NN0 |
239 | eqid | |- ; ; ; 1 5 4 1 = ; ; ; 1 5 4 1 |
|
240 | 152 | deceq1i | |- ; 1 1 = ; ; 0 1 1 |
241 | 240 | deceq1i | |- ; ; 1 1 0 = ; ; ; 0 1 1 0 |
242 | eqid | |- ; ; 1 5 4 = ; ; 1 5 4 |
|
243 | eqid | |- ; ; 0 1 1 = ; ; 0 1 1 |
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244 | 152 | oveq2i | |- ( ; 1 5 + 1 ) = ( ; 1 5 + ; 0 1 ) |
245 | eqid | |- ; 1 5 = ; 1 5 |
|
246 | 5p1e6 | |- ( 5 + 1 ) = 6 |
|
247 | 1 54 1 245 246 | decaddi | |- ( ; 1 5 + 1 ) = ; 1 6 |
248 | 244 247 | eqtr3i | |- ( ; 1 5 + ; 0 1 ) = ; 1 6 |
249 | 235 24 237 1 242 243 248 210 | decadd | |- ( ; ; 1 5 4 + ; ; 0 1 1 ) = ; ; 1 6 5 |
250 | 236 1 238 51 239 241 249 192 | decadd | |- ( ; ; ; 1 5 4 1 + ; ; 1 1 0 ) = ; ; ; 1 6 5 1 |
251 | 7t2e14 | |- ( 7 x. 2 ) = ; 1 4 |
|
252 | 8t7e56 | |- ( 8 x. 7 ) = ; 5 6 |
|
253 | 158 217 252 | mulcomli | |- ( 7 x. 8 ) = ; 5 6 |
254 | 8t2e16 | |- ( 8 x. 2 ) = ; 1 6 |
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255 | eqid | |- ; 5 6 = ; 5 6 |
|
256 | 5cn | |- 5 e. CC |
|
257 | 256 138 246 | addcomli | |- ( 1 + 5 ) = 6 |
258 | 257 | oveq1i | |- ( ( 1 + 5 ) + 1 ) = ( 6 + 1 ) |
259 | 258 140 | eqtri | |- ( ( 1 + 5 ) + 1 ) = 7 |
260 | 6p6e12 | |- ( 6 + 6 ) = ; 1 2 |
|
261 | 1 98 54 98 135 255 259 181 260 | decaddc | |- ( ; 1 6 + ; 5 6 ) = ; 7 2 |
262 | 261 | oveq1i | |- ( ( ; 1 6 + ; 5 6 ) + 6 ) = ( ; 7 2 + 6 ) |
263 | eqid | |- ; 7 2 = ; 7 2 |
|
264 | 6p2e8 | |- ( 6 + 2 ) = 8 |
|
265 | 110 197 264 | addcomli | |- ( 2 + 6 ) = 8 |
266 | 52 181 98 263 265 | decaddi | |- ( ; 7 2 + 6 ) = ; 7 8 |
267 | 262 266 | eqtri | |- ( ( ; 1 6 + ; 5 6 ) + 6 ) = ; 7 8 |
268 | eqid | |- ; 1 4 = ; 1 4 |
|
269 | 1p1e2 | |- ( 1 + 1 ) = 2 |
|
270 | 1 24 52 268 269 1 225 | decaddci | |- ( ; 1 4 + 7 ) = ; 2 1 |
271 | 52 73 181 73 98 52 73 24 251 253 254 123 267 270 | dpmul | |- ( ( 7 . 8 ) x. ( 2 . 8 ) ) = ( ; 2 1 . _ 8 4 ) |
272 | eqid | |- ; 1 1 = ; 1 1 |
|
273 | eqid | |- ; 6 7 = ; 6 7 |
|
274 | 217 138 71 | addcomli | |- ( 1 + 7 ) = 8 |
275 | 1 1 98 52 272 273 141 274 | decadd | |- ( ; 1 1 + ; 6 7 ) = ; 7 8 |
276 | 1 1 98 52 52 73 275 | dpadd | |- ( ( 1 . 1 ) + ( 6 . 7 ) ) = ( 7 . 8 ) |
277 | 4p4e8 | |- ( 4 + 4 ) = 8 |
|
278 | 1 24 1 24 268 268 269 277 | decadd | |- ( ; 1 4 + ; 1 4 ) = ; 2 8 |
279 | 1 24 1 24 181 73 278 | dpadd | |- ( ( 1 . 4 ) + ( 1 . 4 ) ) = ( 2 . 8 ) |
280 | 276 279 | oveq12i | |- ( ( ( 1 . 1 ) + ( 6 . 7 ) ) x. ( ( 1 . 4 ) + ( 1 . 4 ) ) ) = ( ( 7 . 8 ) x. ( 2 . 8 ) ) |
281 | 1 181 | deccl | |- ; 1 2 e. NN0 |
282 | eqid | |- ; ; 1 0 9 = ; ; 1 0 9 |
|
283 | 130 | nn0cni | |- ; 1 0 e. CC |
284 | 283 138 136 | addcomli | |- ( 1 + ; 1 0 ) = ; 1 1 |
285 | 1 1 269 284 | decsuc | |- ( ( 1 + ; 1 0 ) + 1 ) = ; 1 2 |
286 | 9p5e14 | |- ( 9 + 5 ) = ; 1 4 |
|
287 | 196 256 286 | addcomli | |- ( 5 + 9 ) = ; 1 4 |
288 | 1 54 130 53 245 282 285 24 287 | decaddc | |- ( ; 1 5 + ; ; 1 0 9 ) = ; ; 1 2 4 |
289 | 4p2e6 | |- ( 4 + 2 ) = 6 |
|
290 | 235 24 186 181 242 187 288 289 | decadd | |- ( ; ; 1 5 4 + ; ; ; 1 0 9 2 ) = ; ; ; 1 2 4 6 |
291 | 1 54 24 130 53 281 181 24 98 290 | dpadd3 | |- ( ( 1 . _ 5 4 ) + ( ; 1 0 . _ 9 2 ) ) = ( ; 1 2 . _ 4 6 ) |
292 | 291 | oveq1i | |- ( ( ( 1 . _ 5 4 ) + ( ; 1 0 . _ 9 2 ) ) + ( 9 . _ 3 8 ) ) = ( ( ; 1 2 . _ 4 6 ) + ( 9 . _ 3 8 ) ) |
293 | 181 1 | deccl | |- ; 2 1 e. NN0 |
294 | 281 24 | deccl | |- ; ; 1 2 4 e. NN0 |
295 | 53 182 | deccl | |- ; 9 3 e. NN0 |
296 | eqid | |- ; ; ; 1 2 4 6 = ; ; ; 1 2 4 6 |
|
297 | eqid | |- ; ; 9 3 8 = ; ; 9 3 8 |
|
298 | eqid | |- ; ; 1 2 4 = ; ; 1 2 4 |
|
299 | eqid | |- ; 9 3 = ; 9 3 |
|
300 | eqid | |- ; 1 2 = ; 1 2 |
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301 | 1 181 53 300 269 1 200 | decaddci | |- ( ; 1 2 + 9 ) = ; 2 1 |
302 | 4p3e7 | |- ( 4 + 3 ) = 7 |
|
303 | 281 24 53 182 298 299 301 302 | decadd | |- ( ; ; 1 2 4 + ; 9 3 ) = ; ; 2 1 7 |
304 | 293 52 71 303 | decsuc | |- ( ( ; ; 1 2 4 + ; 9 3 ) + 1 ) = ; ; 2 1 8 |
305 | 8p6e14 | |- ( 8 + 6 ) = ; 1 4 |
|
306 | 158 110 305 | addcomli | |- ( 6 + 8 ) = ; 1 4 |
307 | 294 98 295 73 296 297 304 24 306 | decaddc | |- ( ; ; ; 1 2 4 6 + ; ; 9 3 8 ) = ; ; ; 2 1 8 4 |
308 | 281 24 98 53 182 293 73 73 24 307 | dpadd3 | |- ( ( ; 1 2 . _ 4 6 ) + ( 9 . _ 3 8 ) ) = ( ; 2 1 . _ 8 4 ) |
309 | 292 308 | eqtri | |- ( ( ( 1 . _ 5 4 ) + ( ; 1 0 . _ 9 2 ) ) + ( 9 . _ 3 8 ) ) = ( ; 2 1 . _ 8 4 ) |
310 | 271 280 309 | 3eqtr4i | |- ( ( ( 1 . 1 ) + ( 6 . 7 ) ) x. ( ( 1 . 4 ) + ( 1 . 4 ) ) ) = ( ( ( 1 . _ 5 4 ) + ( ; 1 0 . _ 9 2 ) ) + ( 9 . _ 3 8 ) ) |
311 | 1 1 98 52 1 1 54 24 24 24 1 130 53 181 53 182 73 1 1 51 1 1 98 54 1 183 184 185 201 214 234 250 310 | dpmul4 | |- ( ( 1 . _ 1 _ 6 7 ) x. ( 1 . _ 4 _ 1 4 ) ) < ( 1 . _ 6 _ 5 1 ) |
312 | 176 32 | remulcli | |- ( ( 1 . _ 1 _ 6 7 ) x. ( 1 . _ 4 _ 1 4 ) ) e. RR |
313 | 33 312 43 | lttri | |- ( ( ( ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) x. ( 1 . _ 4 _ 1 4 ) ) < ( ( 1 . _ 1 _ 6 7 ) x. ( 1 . _ 4 _ 1 4 ) ) /\ ( ( 1 . _ 1 _ 6 7 ) x. ( 1 . _ 4 _ 1 4 ) ) < ( 1 . _ 6 _ 5 1 ) ) -> ( ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) x. ( 1 . _ 4 _ 1 4 ) ) < ( 1 . _ 6 _ 5 1 ) ) |
314 | 180 311 313 | mp2an | |- ( ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) x. ( 1 . _ 4 _ 1 4 ) ) < ( 1 . _ 6 _ 5 1 ) |
315 | 50 314 | pm3.2i | |- ( 0 <_ ( ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) x. ( 1 . _ 4 _ 1 4 ) ) /\ ( ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) x. ( 1 . _ 4 _ 1 4 ) ) < ( 1 . _ 6 _ 5 1 ) ) |
316 | 44 315 | pm3.2i | |- ( ( ( ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) x. ( 1 . _ 4 _ 1 4 ) ) e. RR /\ ( 1 . _ 6 _ 5 1 ) e. RR ) /\ ( 0 <_ ( ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) x. ( 1 . _ 4 _ 1 4 ) ) /\ ( ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) x. ( 1 . _ 4 _ 1 4 ) ) < ( 1 . _ 6 _ 5 1 ) ) ) |
317 | 4re | |- 4 e. RR |
|
318 | 2re | |- 2 e. RR |
|
319 | 3re | |- 3 e. RR |
|
320 | 34 319 | pm3.2i | |- ( 6 e. RR /\ 3 e. RR ) |
321 | dp2cl | |- ( ( 6 e. RR /\ 3 e. RR ) -> _ 6 3 e. RR ) |
|
322 | 320 321 | ax-mp | |- _ 6 3 e. RR |
323 | 318 322 | pm3.2i | |- ( 2 e. RR /\ _ 6 3 e. RR ) |
324 | dp2cl | |- ( ( 2 e. RR /\ _ 6 3 e. RR ) -> _ 2 _ 6 3 e. RR ) |
|
325 | 323 324 | ax-mp | |- _ 2 _ 6 3 e. RR |
326 | 317 325 | pm3.2i | |- ( 4 e. RR /\ _ 2 _ 6 3 e. RR ) |
327 | dp2cl | |- ( ( 4 e. RR /\ _ 2 _ 6 3 e. RR ) -> _ 4 _ 2 _ 6 3 e. RR ) |
|
328 | 326 327 | ax-mp | |- _ 4 _ 2 _ 6 3 e. RR |
329 | dpcl | |- ( ( 1 e. NN0 /\ _ 4 _ 2 _ 6 3 e. RR ) -> ( 1 . _ 4 _ 2 _ 6 3 ) e. RR ) |
|
330 | 1 328 329 | mp2an | |- ( 1 . _ 4 _ 2 _ 6 3 ) e. RR |
331 | 84 319 | pm3.2i | |- ( 8 e. RR /\ 3 e. RR ) |
332 | dp2cl | |- ( ( 8 e. RR /\ 3 e. RR ) -> _ 8 3 e. RR ) |
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333 | 331 332 | ax-mp | |- _ 8 3 e. RR |
334 | 84 333 | pm3.2i | |- ( 8 e. RR /\ _ 8 3 e. RR ) |
335 | dp2cl | |- ( ( 8 e. RR /\ _ 8 3 e. RR ) -> _ 8 _ 8 3 e. RR ) |
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336 | 334 335 | ax-mp | |- _ 8 _ 8 3 e. RR |
337 | 319 336 | pm3.2i | |- ( 3 e. RR /\ _ 8 _ 8 3 e. RR ) |
338 | dp2cl | |- ( ( 3 e. RR /\ _ 8 _ 8 3 e. RR ) -> _ 3 _ 8 _ 8 3 e. RR ) |
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339 | 337 338 | ax-mp | |- _ 3 _ 8 _ 8 3 e. RR |
340 | 2 339 | pm3.2i | |- ( 0 e. RR /\ _ 3 _ 8 _ 8 3 e. RR ) |
341 | dp2cl | |- ( ( 0 e. RR /\ _ 3 _ 8 _ 8 3 e. RR ) -> _ 0 _ 3 _ 8 _ 8 3 e. RR ) |
|
342 | 340 341 | ax-mp | |- _ 0 _ 3 _ 8 _ 8 3 e. RR |
343 | dpcl | |- ( ( 1 e. NN0 /\ _ 0 _ 3 _ 8 _ 8 3 e. RR ) -> ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) e. RR ) |
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344 | 1 342 343 | mp2an | |- ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) e. RR |
345 | 330 344 | remulcli | |- ( ( 1 . _ 4 _ 2 _ 6 3 ) x. ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) ) e. RR |
346 | 317 333 | pm3.2i | |- ( 4 e. RR /\ _ 8 3 e. RR ) |
347 | dp2cl | |- ( ( 4 e. RR /\ _ 8 3 e. RR ) -> _ 4 _ 8 3 e. RR ) |
|
348 | 346 347 | ax-mp | |- _ 4 _ 8 3 e. RR |
349 | dpcl | |- ( ( 1 e. NN0 /\ _ 4 _ 8 3 e. RR ) -> ( 1 . _ 4 _ 8 3 ) e. RR ) |
|
350 | 1 348 349 | mp2an | |- ( 1 . _ 4 _ 8 3 ) e. RR |
351 | 345 350 | pm3.2i | |- ( ( ( 1 . _ 4 _ 2 _ 6 3 ) x. ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) ) e. RR /\ ( 1 . _ 4 _ 8 3 ) e. RR ) |
352 | 3rp | |- 3 e. RR+ |
|
353 | 98 352 | rpdp2cl | |- _ 6 3 e. RR+ |
354 | 181 353 | rpdp2cl | |- _ 2 _ 6 3 e. RR+ |
355 | 24 354 | rpdp2cl | |- _ 4 _ 2 _ 6 3 e. RR+ |
356 | 1 355 | rpdpcl | |- ( 1 . _ 4 _ 2 _ 6 3 ) e. RR+ |
357 | rpge0 | |- ( ( 1 . _ 4 _ 2 _ 6 3 ) e. RR+ -> 0 <_ ( 1 . _ 4 _ 2 _ 6 3 ) ) |
|
358 | 356 357 | ax-mp | |- 0 <_ ( 1 . _ 4 _ 2 _ 6 3 ) |
359 | 73 352 | rpdp2cl | |- _ 8 3 e. RR+ |
360 | 73 359 | rpdp2cl | |- _ 8 _ 8 3 e. RR+ |
361 | 182 360 | rpdp2cl | |- _ 3 _ 8 _ 8 3 e. RR+ |
362 | 51 361 | rpdp2cl | |- _ 0 _ 3 _ 8 _ 8 3 e. RR+ |
363 | 1 362 | rpdpcl | |- ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) e. RR+ |
364 | rpge0 | |- ( ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) e. RR+ -> 0 <_ ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) ) |
|
365 | 363 364 | ax-mp | |- 0 <_ ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) |
366 | 330 344 | mulge0i | |- ( ( 0 <_ ( 1 . _ 4 _ 2 _ 6 3 ) /\ 0 <_ ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) ) -> 0 <_ ( ( 1 . _ 4 _ 2 _ 6 3 ) x. ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) ) ) |
367 | 358 365 366 | mp2an | |- 0 <_ ( ( 1 . _ 4 _ 2 _ 6 3 ) x. ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) ) |
368 | 318 3 | pm3.2i | |- ( 2 e. RR /\ 7 e. RR ) |
369 | dp2cl | |- ( ( 2 e. RR /\ 7 e. RR ) -> _ 2 7 e. RR ) |
|
370 | 368 369 | ax-mp | |- _ 2 7 e. RR |
371 | 317 370 | pm3.2i | |- ( 4 e. RR /\ _ 2 7 e. RR ) |
372 | dp2cl | |- ( ( 4 e. RR /\ _ 2 7 e. RR ) -> _ 4 _ 2 7 e. RR ) |
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373 | 371 372 | ax-mp | |- _ 4 _ 2 7 e. RR |
374 | dpcl | |- ( ( 1 e. NN0 /\ _ 4 _ 2 7 e. RR ) -> ( 1 . _ 4 _ 2 7 ) e. RR ) |
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375 | 1 373 374 | mp2an | |- ( 1 . _ 4 _ 2 7 ) e. RR |
376 | 330 375 | pm3.2i | |- ( ( 1 . _ 4 _ 2 _ 6 3 ) e. RR /\ ( 1 . _ 4 _ 2 7 ) e. RR ) |
377 | 7nn | |- 7 e. NN |
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378 | nnrp | |- ( 7 e. NN -> 7 e. RR+ ) |
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379 | 377 378 | ax-mp | |- 7 e. RR+ |
380 | 181 379 | rpdp2cl | |- _ 2 7 e. RR+ |
381 | 24 380 | rpdp2cl | |- _ 4 _ 2 7 e. RR+ |
382 | 98 352 184 140 | dp2ltsuc | |- _ 6 3 < 7 |
383 | 181 353 379 382 | dp2lt | |- _ 2 _ 6 3 < _ 2 7 |
384 | 24 354 380 383 | dp2lt | |- _ 4 _ 2 _ 6 3 < _ 4 _ 2 7 |
385 | 1 355 381 384 | dplt | |- ( 1 . _ 4 _ 2 _ 6 3 ) < ( 1 . _ 4 _ 2 7 ) |
386 | 358 385 | pm3.2i | |- ( 0 <_ ( 1 . _ 4 _ 2 _ 6 3 ) /\ ( 1 . _ 4 _ 2 _ 6 3 ) < ( 1 . _ 4 _ 2 7 ) ) |
387 | 376 386 | pm3.2i | |- ( ( ( 1 . _ 4 _ 2 _ 6 3 ) e. RR /\ ( 1 . _ 4 _ 2 7 ) e. RR ) /\ ( 0 <_ ( 1 . _ 4 _ 2 _ 6 3 ) /\ ( 1 . _ 4 _ 2 _ 6 3 ) < ( 1 . _ 4 _ 2 7 ) ) ) |
388 | 319 4 | pm3.2i | |- ( 3 e. RR /\ 9 e. RR ) |
389 | dp2cl | |- ( ( 3 e. RR /\ 9 e. RR ) -> _ 3 9 e. RR ) |
|
390 | 388 389 | ax-mp | |- _ 3 9 e. RR |
391 | 2 390 | pm3.2i | |- ( 0 e. RR /\ _ 3 9 e. RR ) |
392 | dp2cl | |- ( ( 0 e. RR /\ _ 3 9 e. RR ) -> _ 0 _ 3 9 e. RR ) |
|
393 | 391 392 | ax-mp | |- _ 0 _ 3 9 e. RR |
394 | dpcl | |- ( ( 1 e. NN0 /\ _ 0 _ 3 9 e. RR ) -> ( 1 . _ 0 _ 3 9 ) e. RR ) |
|
395 | 1 393 394 | mp2an | |- ( 1 . _ 0 _ 3 9 ) e. RR |
396 | 344 395 | pm3.2i | |- ( ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) e. RR /\ ( 1 . _ 0 _ 3 9 ) e. RR ) |
397 | 9nn | |- 9 e. NN |
|
398 | nnrp | |- ( 9 e. NN -> 9 e. RR+ ) |
|
399 | 397 398 | ax-mp | |- 9 e. RR+ |
400 | 182 399 | rpdp2cl | |- _ 3 9 e. RR+ |
401 | 51 400 | rpdp2cl | |- _ 0 _ 3 9 e. RR+ |
402 | 73 352 185 184 | dp2lt10 | |- _ 8 3 < ; 1 0 |
403 | 73 359 402 160 | dp2ltsuc | |- _ 8 _ 8 3 < 9 |
404 | 182 360 399 403 | dp2lt | |- _ 3 _ 8 _ 8 3 < _ 3 9 |
405 | 51 361 400 404 | dp2lt | |- _ 0 _ 3 _ 8 _ 8 3 < _ 0 _ 3 9 |
406 | 1 362 401 405 | dplt | |- ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) < ( 1 . _ 0 _ 3 9 ) |
407 | 365 406 | pm3.2i | |- ( 0 <_ ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) /\ ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) < ( 1 . _ 0 _ 3 9 ) ) |
408 | 396 407 | pm3.2i | |- ( ( ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) e. RR /\ ( 1 . _ 0 _ 3 9 ) e. RR ) /\ ( 0 <_ ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) /\ ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) < ( 1 . _ 0 _ 3 9 ) ) ) |
409 | ltmul12a | |- ( ( ( ( ( 1 . _ 4 _ 2 _ 6 3 ) e. RR /\ ( 1 . _ 4 _ 2 7 ) e. RR ) /\ ( 0 <_ ( 1 . _ 4 _ 2 _ 6 3 ) /\ ( 1 . _ 4 _ 2 _ 6 3 ) < ( 1 . _ 4 _ 2 7 ) ) ) /\ ( ( ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) e. RR /\ ( 1 . _ 0 _ 3 9 ) e. RR ) /\ ( 0 <_ ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) /\ ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) < ( 1 . _ 0 _ 3 9 ) ) ) ) -> ( ( 1 . _ 4 _ 2 _ 6 3 ) x. ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) ) < ( ( 1 . _ 4 _ 2 7 ) x. ( 1 . _ 0 _ 3 9 ) ) ) |
|
410 | 387 408 409 | mp2an | |- ( ( 1 . _ 4 _ 2 _ 6 3 ) x. ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) ) < ( ( 1 . _ 4 _ 2 7 ) x. ( 1 . _ 0 _ 3 9 ) ) |
411 | 6lt10 | |- 6 < ; 1 0 |
|
412 | 73 1 | deccl | |- ; 8 1 e. NN0 |
413 | eqid | |- ; ; 8 1 6 = ; ; 8 1 6 |
|
414 | eqid | |- ; 1 0 = ; 1 0 |
|
415 | eqid | |- ; 8 1 = ; 8 1 |
|
416 | 73 1 269 415 | decsuc | |- ( ; 8 1 + 1 ) = ; 8 2 |
417 | 73 | dec0h | |- 8 = ; 0 8 |
418 | 417 | deceq1i | |- ; 8 2 = ; ; 0 8 2 |
419 | 416 418 | eqtri | |- ( ; 8 1 + 1 ) = ; ; 0 8 2 |
420 | 110 | addid1i | |- ( 6 + 0 ) = 6 |
421 | 412 98 1 51 413 414 419 420 | decadd | |- ( ; ; 8 1 6 + ; 1 0 ) = ; ; ; 0 8 2 6 |
422 | 138 | mul01i | |- ( 1 x. 0 ) = 0 |
423 | 113 | mul01i | |- ( 4 x. 0 ) = 0 |
424 | 51 | dec0h | |- 0 = ; 0 0 |
425 | 423 424 | eqtri | |- ( 4 x. 0 ) = ; 0 0 |
426 | 113 | addid1i | |- ( 4 + 0 ) = 4 |
427 | 426 | oveq1i | |- ( ( 4 + 0 ) + 0 ) = ( 4 + 0 ) |
428 | 427 426 205 | 3eqtri | |- ( ( 4 + 0 ) + 0 ) = ; 0 4 |
429 | 1 24 1 51 51 51 24 51 116 422 203 425 428 192 | dpmul | |- ( ( 1 . 4 ) x. ( 1 . 0 ) ) = ( 1 . _ 4 0 ) |
430 | 3cn | |- 3 e. CC |
|
431 | 3t2e6 | |- ( 3 x. 2 ) = 6 |
|
432 | 430 197 431 | mulcomli | |- ( 2 x. 3 ) = 6 |
433 | 9t2e18 | |- ( 9 x. 2 ) = ; 1 8 |
|
434 | 196 197 433 | mulcomli | |- ( 2 x. 9 ) = ; 1 8 |
435 | 7t3e21 | |- ( 7 x. 3 ) = ; 2 1 |
|
436 | 9t7e63 | |- ( 9 x. 7 ) = ; 6 3 |
|
437 | 196 217 436 | mulcomli | |- ( 7 x. 9 ) = ; 6 3 |
438 | eqid | |- ; 2 1 = ; 2 1 |
|
439 | eqid | |- ; 1 8 = ; 1 8 |
|
440 | 159 160 | eqtri | |- ( 1 + 8 ) = 9 |
441 | 181 1 1 73 438 439 224 440 | decadd | |- ( ; 2 1 + ; 1 8 ) = ; 3 9 |
442 | 441 | oveq1i | |- ( ( ; 2 1 + ; 1 8 ) + 6 ) = ( ; 3 9 + 6 ) |
443 | eqid | |- ; 3 9 = ; 3 9 |
|
444 | 3p1e4 | |- ( 3 + 1 ) = 4 |
|
445 | 9p6e15 | |- ( 9 + 6 ) = ; 1 5 |
|
446 | 182 53 98 443 444 54 445 | decaddci | |- ( ; 3 9 + 6 ) = ; 4 5 |
447 | 442 446 | eqtri | |- ( ( ; 2 1 + ; 1 8 ) + 6 ) = ; 4 5 |
448 | 6p4e10 | |- ( 6 + 4 ) = ; 1 0 |
|
449 | 181 52 182 53 98 24 54 182 432 434 435 437 447 448 | dpmul | |- ( ( 2 . 7 ) x. ( 3 . 9 ) ) = ( ; 1 0 . _ 5 3 ) |
450 | 1 24 | deccl | |- ; 1 4 e. NN0 |
451 | 450 51 | deccl | |- ; ; 1 4 0 e. NN0 |
452 | 417 73 | eqeltrri | |- ; 0 8 e. NN0 |
453 | eqid | |- ; ; ; 1 4 0 1 = ; ; ; 1 4 0 1 |
|
454 | eqid | |- ; ; 0 8 2 = ; ; 0 8 2 |
|
455 | eqid | |- ; ; 1 4 0 = ; ; 1 4 0 |
|
456 | 417 158 | eqeltrri | |- ; 0 8 e. CC |
457 | 0cn | |- 0 e. CC |
|
458 | 417 | oveq1i | |- ( 8 + 0 ) = ( ; 0 8 + 0 ) |
459 | 158 | addid1i | |- ( 8 + 0 ) = 8 |
460 | 458 459 | eqtr3i | |- ( ; 0 8 + 0 ) = 8 |
461 | 456 457 460 | addcomli | |- ( 0 + ; 0 8 ) = 8 |
462 | 450 51 452 455 461 | decaddi | |- ( ; ; 1 4 0 + ; 0 8 ) = ; ; 1 4 8 |
463 | 451 1 452 181 453 454 462 230 | decadd | |- ( ; ; ; 1 4 0 1 + ; ; 0 8 2 ) = ; ; ; 1 4 8 3 |
464 | 4t4e16 | |- ( 4 x. 4 ) = ; 1 6 |
|
465 | 9t4e36 | |- ( 9 x. 4 ) = ; 3 6 |
|
466 | 196 113 465 | mulcomli | |- ( 4 x. 9 ) = ; 3 6 |
467 | 196 | mulid1i | |- ( 9 x. 1 ) = 9 |
468 | 467 188 | eqtri | |- ( 9 x. 1 ) = ; 0 9 |
469 | 196 138 468 | mulcomli | |- ( 1 x. 9 ) = ; 0 9 |
470 | 182 98 | deccl | |- ; 3 6 e. NN0 |
471 | 470 | nn0cni | |- ; 3 6 e. CC |
472 | eqid | |- ; 3 6 = ; 3 6 |
|
473 | 182 98 24 472 444 51 448 | decaddci | |- ( ; 3 6 + 4 ) = ; 4 0 |
474 | 471 113 473 | addcomli | |- ( 4 + ; 3 6 ) = ; 4 0 |
475 | 474 | oveq1i | |- ( ( 4 + ; 3 6 ) + 0 ) = ( ; 4 0 + 0 ) |
476 | 24 51 | deccl | |- ; 4 0 e. NN0 |
477 | 476 | nn0cni | |- ; 4 0 e. CC |
478 | 477 | addid1i | |- ( ; 4 0 + 0 ) = ; 4 0 |
479 | 475 478 | eqtri | |- ( ( 4 + ; 3 6 ) + 0 ) = ; 4 0 |
480 | 1 98 24 135 269 51 448 | decaddci | |- ( ; 1 6 + 4 ) = ; 2 0 |
481 | 24 1 24 53 51 24 51 53 464 466 204 469 479 480 | dpmul | |- ( ( 4 . 1 ) x. ( 4 . 9 ) ) = ( ; 2 0 . _ 0 9 ) |
482 | eqid | |- ; 2 7 = ; 2 7 |
|
483 | 230 | oveq1i | |- ( ( 1 + 2 ) + 1 ) = ( 3 + 1 ) |
484 | 483 444 | eqtri | |- ( ( 1 + 2 ) + 1 ) = 4 |
485 | 1 24 181 52 268 482 484 1 225 | decaddc | |- ( ; 1 4 + ; 2 7 ) = ; 4 1 |
486 | 1 24 181 52 24 1 485 | dpadd | |- ( ( 1 . 4 ) + ( 2 . 7 ) ) = ( 4 . 1 ) |
487 | 430 138 444 | addcomli | |- ( 1 + 3 ) = 4 |
488 | 196 | addid2i | |- ( 0 + 9 ) = 9 |
489 | 1 51 182 53 414 443 487 488 | decadd | |- ( ; 1 0 + ; 3 9 ) = ; 4 9 |
490 | 1 51 182 53 24 53 489 | dpadd | |- ( ( 1 . 0 ) + ( 3 . 9 ) ) = ( 4 . 9 ) |
491 | 486 490 | oveq12i | |- ( ( ( 1 . 4 ) + ( 2 . 7 ) ) x. ( ( 1 . 0 ) + ( 3 . 9 ) ) ) = ( ( 4 . 1 ) x. ( 4 . 9 ) ) |
492 | 1 24 73 1 268 415 440 210 | decadd | |- ( ; 1 4 + ; 8 1 ) = ; 9 5 |
493 | 450 51 412 98 455 413 492 111 | decadd | |- ( ; ; 1 4 0 + ; ; 8 1 6 ) = ; ; 9 5 6 |
494 | 1 24 51 73 1 53 98 54 98 493 | dpadd3 | |- ( ( 1 . _ 4 0 ) + ( 8 . _ 1 6 ) ) = ( 9 . _ 5 6 ) |
495 | 494 | oveq1i | |- ( ( ( 1 . _ 4 0 ) + ( 8 . _ 1 6 ) ) + ( ; 1 0 . _ 5 3 ) ) = ( ( 9 . _ 5 6 ) + ( ; 1 0 . _ 5 3 ) ) |
496 | 181 51 | deccl | |- ; 2 0 e. NN0 |
497 | 53 54 | deccl | |- ; 9 5 e. NN0 |
498 | 130 54 | deccl | |- ; ; 1 0 5 e. NN0 |
499 | eqid | |- ; ; 9 5 6 = ; ; 9 5 6 |
|
500 | eqid | |- ; ; ; 1 0 5 3 = ; ; ; 1 0 5 3 |
|
501 | eqid | |- ; 9 5 = ; 9 5 |
|
502 | eqid | |- ; ; 1 0 5 = ; ; 1 0 5 |
|
503 | dec10p | |- ( ; 1 0 + 9 ) = ; 1 9 |
|
504 | 283 196 503 | addcomli | |- ( 9 + ; 1 0 ) = ; 1 9 |
505 | 504 | oveq1i | |- ( ( 9 + ; 1 0 ) + 1 ) = ( ; 1 9 + 1 ) |
506 | eqid | |- ; 1 9 = ; 1 9 |
|
507 | 1 53 1 506 269 51 194 | decaddci | |- ( ; 1 9 + 1 ) = ; 2 0 |
508 | 505 507 | eqtri | |- ( ( 9 + ; 1 0 ) + 1 ) = ; 2 0 |
509 | 5p5e10 | |- ( 5 + 5 ) = ; 1 0 |
|
510 | 53 54 130 54 501 502 508 51 509 | decaddc | |- ( ; 9 5 + ; ; 1 0 5 ) = ; ; 2 0 0 |
511 | 497 98 498 182 499 500 510 233 | decadd | |- ( ; ; 9 5 6 + ; ; ; 1 0 5 3 ) = ; ; ; 2 0 0 9 |
512 | 53 54 98 130 54 496 182 51 53 511 | dpadd3 | |- ( ( 9 . _ 5 6 ) + ( ; 1 0 . _ 5 3 ) ) = ( ; 2 0 . _ 0 9 ) |
513 | 495 512 | eqtri | |- ( ( ( 1 . _ 4 0 ) + ( 8 . _ 1 6 ) ) + ( ; 1 0 . _ 5 3 ) ) = ( ; 2 0 . _ 0 9 ) |
514 | 481 491 513 | 3eqtr4i | |- ( ( ( 1 . 4 ) + ( 2 . 7 ) ) x. ( ( 1 . 0 ) + ( 3 . 9 ) ) ) = ( ( ( 1 . _ 4 0 ) + ( 8 . _ 1 6 ) ) + ( ; 1 0 . _ 5 3 ) ) |
515 | 1 24 181 52 1 182 24 51 51 53 1 73 1 98 130 54 182 51 73 181 98 1 24 73 182 411 67 184 421 429 449 463 514 | dpmul4 | |- ( ( 1 . _ 4 _ 2 7 ) x. ( 1 . _ 0 _ 3 9 ) ) < ( 1 . _ 4 _ 8 3 ) |
516 | 375 395 | remulcli | |- ( ( 1 . _ 4 _ 2 7 ) x. ( 1 . _ 0 _ 3 9 ) ) e. RR |
517 | 345 516 350 | lttri | |- ( ( ( ( 1 . _ 4 _ 2 _ 6 3 ) x. ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) ) < ( ( 1 . _ 4 _ 2 7 ) x. ( 1 . _ 0 _ 3 9 ) ) /\ ( ( 1 . _ 4 _ 2 7 ) x. ( 1 . _ 0 _ 3 9 ) ) < ( 1 . _ 4 _ 8 3 ) ) -> ( ( 1 . _ 4 _ 2 _ 6 3 ) x. ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) ) < ( 1 . _ 4 _ 8 3 ) ) |
518 | 410 515 517 | mp2an | |- ( ( 1 . _ 4 _ 2 _ 6 3 ) x. ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) ) < ( 1 . _ 4 _ 8 3 ) |
519 | 367 518 | pm3.2i | |- ( 0 <_ ( ( 1 . _ 4 _ 2 _ 6 3 ) x. ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) ) /\ ( ( 1 . _ 4 _ 2 _ 6 3 ) x. ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) ) < ( 1 . _ 4 _ 8 3 ) ) |
520 | 351 519 | pm3.2i | |- ( ( ( ( 1 . _ 4 _ 2 _ 6 3 ) x. ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) ) e. RR /\ ( 1 . _ 4 _ 8 3 ) e. RR ) /\ ( 0 <_ ( ( 1 . _ 4 _ 2 _ 6 3 ) x. ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) ) /\ ( ( 1 . _ 4 _ 2 _ 6 3 ) x. ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) ) < ( 1 . _ 4 _ 8 3 ) ) ) |
521 | ltmul12a | |- ( ( ( ( ( ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) x. ( 1 . _ 4 _ 1 4 ) ) e. RR /\ ( 1 . _ 6 _ 5 1 ) e. RR ) /\ ( 0 <_ ( ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) x. ( 1 . _ 4 _ 1 4 ) ) /\ ( ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) x. ( 1 . _ 4 _ 1 4 ) ) < ( 1 . _ 6 _ 5 1 ) ) ) /\ ( ( ( ( 1 . _ 4 _ 2 _ 6 3 ) x. ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) ) e. RR /\ ( 1 . _ 4 _ 8 3 ) e. RR ) /\ ( 0 <_ ( ( 1 . _ 4 _ 2 _ 6 3 ) x. ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) ) /\ ( ( 1 . _ 4 _ 2 _ 6 3 ) x. ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) ) < ( 1 . _ 4 _ 8 3 ) ) ) ) -> ( ( ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) x. ( 1 . _ 4 _ 1 4 ) ) x. ( ( 1 . _ 4 _ 2 _ 6 3 ) x. ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) ) ) < ( ( 1 . _ 6 _ 5 1 ) x. ( 1 . _ 4 _ 8 3 ) ) ) |
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522 | 316 520 521 | mp2an | |- ( ( ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) x. ( 1 . _ 4 _ 1 4 ) ) x. ( ( 1 . _ 4 _ 2 _ 6 3 ) x. ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) ) ) < ( ( 1 . _ 6 _ 5 1 ) x. ( 1 . _ 4 _ 8 3 ) ) |
523 | 24 181 | deccl | |- ; 4 2 e. NN0 |
524 | 496 24 | deccl | |- ; ; 2 0 4 e. NN0 |
525 | eqid | |- ; ; ; 2 0 4 2 = ; ; ; 2 0 4 2 |
|
526 | eqid | |- ; 4 2 = ; 4 2 |
|
527 | eqid | |- ; ; 2 0 4 = ; ; 2 0 4 |
|
528 | 496 24 24 527 277 | decaddi | |- ( ; ; 2 0 4 + 4 ) = ; ; 2 0 8 |
529 | 2p2e4 | |- ( 2 + 2 ) = 4 |
|
530 | 524 181 24 181 525 526 528 529 | decadd | |- ( ; ; ; 2 0 4 2 + ; 4 2 ) = ; ; ; 2 0 8 4 |
531 | 448 | oveq1i | |- ( ( 6 + 4 ) + 2 ) = ( ; 1 0 + 2 ) |
532 | dec10p | |- ( ; 1 0 + 2 ) = ; 1 2 |
|
533 | 531 532 | eqtri | |- ( ( 6 + 4 ) + 2 ) = ; 1 2 |
534 | 1 98 1 24 181 1 181 24 116 204 215 216 533 269 | dpmul | |- ( ( 1 . 6 ) x. ( 1 . 4 ) ) = ( 2 . _ 2 4 ) |
535 | 8t5e40 | |- ( 8 x. 5 ) = ; 4 0 |
|
536 | 158 256 535 | mulcomli | |- ( 5 x. 8 ) = ; 4 0 |
537 | 5t3e15 | |- ( 5 x. 3 ) = ; 1 5 |
|
538 | 158 | mulid2i | |- ( 1 x. 8 ) = 8 |
539 | 430 | mulid2i | |- ( 1 x. 3 ) = 3 |
540 | 182 | dec0h | |- 3 = ; 0 3 |
541 | 539 540 | eqtri | |- ( 1 x. 3 ) = ; 0 3 |
542 | 235 | nn0cni | |- ; 1 5 e. CC |
543 | 8p5e13 | |- ( 8 + 5 ) = ; 1 3 |
|
544 | 158 256 543 | addcomli | |- ( 5 + 8 ) = ; 1 3 |
545 | 1 54 73 245 269 182 544 | decaddci | |- ( ; 1 5 + 8 ) = ; 2 3 |
546 | 542 158 545 | addcomli | |- ( 8 + ; 1 5 ) = ; 2 3 |
547 | 546 | oveq1i | |- ( ( 8 + ; 1 5 ) + 0 ) = ( ; 2 3 + 0 ) |
548 | 181 182 | deccl | |- ; 2 3 e. NN0 |
549 | 548 | nn0cni | |- ; 2 3 e. CC |
550 | 549 | addid1i | |- ( ; 2 3 + 0 ) = ; 2 3 |
551 | 547 550 | eqtri | |- ( ( 8 + ; 1 5 ) + 0 ) = ; 2 3 |
552 | eqid | |- ; 4 0 = ; 4 0 |
|
553 | 197 | addid2i | |- ( 0 + 2 ) = 2 |
554 | 24 51 181 552 553 | decaddi | |- ( ; 4 0 + 2 ) = ; 4 2 |
555 | 54 1 73 182 51 181 182 182 536 537 538 541 551 554 | dpmul | |- ( ( 5 . 1 ) x. ( 8 . 3 ) ) = ( ; 4 2 . _ 3 3 ) |
556 | 181 181 | deccl | |- ; 2 2 e. NN0 |
557 | 556 24 | deccl | |- ; ; 2 2 4 e. NN0 |
558 | eqid | |- ; ; ; 2 2 4 1 = ; ; ; 2 2 4 1 |
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559 | eqid | |- ; ; 2 0 8 = ; ; 2 0 8 |
|
560 | eqid | |- ; ; 2 2 4 = ; ; 2 2 4 |
|
561 | eqid | |- ; 2 0 = ; 2 0 |
|
562 | eqid | |- ; 2 2 = ; 2 2 |
|
563 | 181 181 181 562 529 | decaddi | |- ( ; 2 2 + 2 ) = ; 2 4 |
564 | 556 24 181 51 560 561 563 426 | decadd | |- ( ; ; 2 2 4 + ; 2 0 ) = ; ; 2 4 4 |
565 | 557 1 496 73 558 559 564 440 | decadd | |- ( ; ; ; 2 2 4 1 + ; ; 2 0 8 ) = ; ; ; 2 4 4 9 |
566 | 556 98 | deccl | |- ; ; 2 2 6 e. NN0 |
567 | 523 182 | deccl | |- ; ; 4 2 3 e. NN0 |
568 | eqid | |- ; ; ; 2 2 6 6 = ; ; ; 2 2 6 6 |
|
569 | eqid | |- ; ; ; 4 2 3 3 = ; ; ; 4 2 3 3 |
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570 | eqid | |- ; ; 2 2 6 = ; ; 2 2 6 |
|
571 | eqid | |- ; ; 4 2 3 = ; ; 4 2 3 |
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572 | 113 197 289 | addcomli | |- ( 2 + 4 ) = 6 |
573 | 181 181 24 181 562 526 572 529 | decadd | |- ( ; 2 2 + ; 4 2 ) = ; 6 4 |
574 | 556 98 523 182 570 571 573 233 | decadd | |- ( ; ; 2 2 6 + ; ; 4 2 3 ) = ; ; 6 4 9 |
575 | 566 98 567 182 568 569 574 233 | decadd | |- ( ; ; ; 2 2 6 6 + ; ; ; 4 2 3 3 ) = ; ; ; 6 4 9 9 |
576 | 556 98 98 523 182 100 182 53 53 575 | dpadd3 | |- ( ( ; 2 2 . _ 6 6 ) + ( ; 4 2 . _ 3 3 ) ) = ( ; 6 4 . _ 9 9 ) |
577 | 496 | nn0cni | |- ; 2 0 e. CC |
578 | 181 51 181 561 553 | decaddi | |- ( ; 2 0 + 2 ) = ; 2 2 |
579 | 577 197 578 | addcomli | |- ( 2 + ; 2 0 ) = ; 2 2 |
580 | 181 181 496 24 562 527 579 572 | decadd | |- ( ; 2 2 + ; ; 2 0 4 ) = ; ; 2 2 6 |
581 | 556 24 524 181 560 525 580 289 | decadd | |- ( ; ; 2 2 4 + ; ; ; 2 0 4 2 ) = ; ; ; 2 2 6 6 |
582 | 181 181 24 496 24 556 181 98 98 581 | dpadd3 | |- ( ( 2 . _ 2 4 ) + ( ; 2 0 . _ 4 2 ) ) = ( ; 2 2 . _ 6 6 ) |
583 | 582 | oveq1i | |- ( ( ( 2 . _ 2 4 ) + ( ; 2 0 . _ 4 2 ) ) + ( ; 4 2 . _ 3 3 ) ) = ( ( ; 2 2 . _ 6 6 ) + ( ; 4 2 . _ 3 3 ) ) |
584 | eqid | |- ; 5 1 = ; 5 1 |
|
585 | 1 98 54 1 135 584 257 140 | decadd | |- ( ; 1 6 + ; 5 1 ) = ; 6 7 |
586 | 1 98 54 1 98 52 585 | dpadd | |- ( ( 1 . 6 ) + ( 5 . 1 ) ) = ( 6 . 7 ) |
587 | eqid | |- ; 8 3 = ; 8 3 |
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588 | 1 24 73 182 268 587 440 302 | decadd | |- ( ; 1 4 + ; 8 3 ) = ; 9 7 |
589 | 1 24 73 182 53 52 588 | dpadd | |- ( ( 1 . 4 ) + ( 8 . 3 ) ) = ( 9 . 7 ) |
590 | 586 589 | oveq12i | |- ( ( ( 1 . 6 ) + ( 5 . 1 ) ) x. ( ( 1 . 4 ) + ( 8 . 3 ) ) ) = ( ( 6 . 7 ) x. ( 9 . 7 ) ) |
591 | 9t6e54 | |- ( 9 x. 6 ) = ; 5 4 |
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592 | 196 110 591 | mulcomli | |- ( 6 x. 9 ) = ; 5 4 |
593 | 7t6e42 | |- ( 7 x. 6 ) = ; 4 2 |
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594 | 217 110 593 | mulcomli | |- ( 6 x. 7 ) = ; 4 2 |
595 | 7t7e49 | |- ( 7 x. 7 ) = ; 4 9 |
|
596 | eqid | |- ; 6 3 = ; 6 3 |
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597 | 3p2e5 | |- ( 3 + 2 ) = 5 |
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598 | 98 182 24 181 596 526 448 597 | decadd | |- ( ; 6 3 + ; 4 2 ) = ; ; 1 0 5 |
599 | 598 | oveq1i | |- ( ( ; 6 3 + ; 4 2 ) + 4 ) = ( ; ; 1 0 5 + 4 ) |
600 | 5p4e9 | |- ( 5 + 4 ) = 9 |
|
601 | 130 54 24 502 600 | decaddi | |- ( ; ; 1 0 5 + 4 ) = ; ; 1 0 9 |
602 | 599 601 | eqtri | |- ( ( ; 6 3 + ; 4 2 ) + 4 ) = ; ; 1 0 9 |
603 | eqid | |- ; 5 4 = ; 5 4 |
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604 | 54 24 1 51 603 414 246 426 | decadd | |- ( ; 5 4 + ; 1 0 ) = ; 6 4 |
605 | 98 52 53 52 24 130 53 53 592 594 437 595 602 604 | dpmul | |- ( ( 6 . 7 ) x. ( 9 . 7 ) ) = ( ; 6 4 . _ 9 9 ) |
606 | 590 605 | eqtri | |- ( ( ( 1 . 6 ) + ( 5 . 1 ) ) x. ( ( 1 . 4 ) + ( 8 . 3 ) ) ) = ( ; 6 4 . _ 9 9 ) |
607 | 576 583 606 | 3eqtr4ri | |- ( ( ( 1 . 6 ) + ( 5 . 1 ) ) x. ( ( 1 . 4 ) + ( 8 . 3 ) ) ) = ( ( ( 2 . _ 2 4 ) + ( ; 2 0 . _ 4 2 ) ) + ( ; 4 2 . _ 3 3 ) ) |
608 | 1 98 54 1 1 73 181 24 24 182 181 496 24 181 523 182 182 181 51 73 24 181 24 24 53 101 184 184 530 534 555 565 607 | dpmul4 | |- ( ( 1 . _ 6 _ 5 1 ) x. ( 1 . _ 4 _ 8 3 ) ) < ( 2 . _ 4 _ 4 9 ) |
609 | 33 345 | remulcli | |- ( ( ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) x. ( 1 . _ 4 _ 1 4 ) ) x. ( ( 1 . _ 4 _ 2 _ 6 3 ) x. ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) ) ) e. RR |
610 | 43 350 | remulcli | |- ( ( 1 . _ 6 _ 5 1 ) x. ( 1 . _ 4 _ 8 3 ) ) e. RR |
611 | 24 399 | rpdp2cl | |- _ 4 9 e. RR+ |
612 | 24 611 | rpdp2cl | |- _ 4 _ 4 9 e. RR+ |
613 | 181 612 | rpdpcl | |- ( 2 . _ 4 _ 4 9 ) e. RR+ |
614 | rpre | |- ( ( 2 . _ 4 _ 4 9 ) e. RR+ -> ( 2 . _ 4 _ 4 9 ) e. RR ) |
|
615 | 613 614 | ax-mp | |- ( 2 . _ 4 _ 4 9 ) e. RR |
616 | 609 610 615 | lttri | |- ( ( ( ( ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) x. ( 1 . _ 4 _ 1 4 ) ) x. ( ( 1 . _ 4 _ 2 _ 6 3 ) x. ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) ) ) < ( ( 1 . _ 6 _ 5 1 ) x. ( 1 . _ 4 _ 8 3 ) ) /\ ( ( 1 . _ 6 _ 5 1 ) x. ( 1 . _ 4 _ 8 3 ) ) < ( 2 . _ 4 _ 4 9 ) ) -> ( ( ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) x. ( 1 . _ 4 _ 1 4 ) ) x. ( ( 1 . _ 4 _ 2 _ 6 3 ) x. ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) ) ) < ( 2 . _ 4 _ 4 9 ) ) |
617 | 522 608 616 | mp2an | |- ( ( ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) x. ( 1 . _ 4 _ 1 4 ) ) x. ( ( 1 . _ 4 _ 2 _ 6 3 ) x. ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) ) ) < ( 2 . _ 4 _ 4 9 ) |
618 | 3pos | |- 0 < 3 |
|
619 | 609 615 319 | ltmul2i | |- ( 0 < 3 -> ( ( ( ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) x. ( 1 . _ 4 _ 1 4 ) ) x. ( ( 1 . _ 4 _ 2 _ 6 3 ) x. ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) ) ) < ( 2 . _ 4 _ 4 9 ) <-> ( 3 x. ( ( ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) x. ( 1 . _ 4 _ 1 4 ) ) x. ( ( 1 . _ 4 _ 2 _ 6 3 ) x. ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) ) ) ) < ( 3 x. ( 2 . _ 4 _ 4 9 ) ) ) ) |
620 | 618 619 | ax-mp | |- ( ( ( ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) x. ( 1 . _ 4 _ 1 4 ) ) x. ( ( 1 . _ 4 _ 2 _ 6 3 ) x. ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) ) ) < ( 2 . _ 4 _ 4 9 ) <-> ( 3 x. ( ( ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) x. ( 1 . _ 4 _ 1 4 ) ) x. ( ( 1 . _ 4 _ 2 _ 6 3 ) x. ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) ) ) ) < ( 3 x. ( 2 . _ 4 _ 4 9 ) ) ) |
621 | 617 620 | mpbi | |- ( 3 x. ( ( ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) x. ( 1 . _ 4 _ 1 4 ) ) x. ( ( 1 . _ 4 _ 2 _ 6 3 ) x. ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) ) ) ) < ( 3 x. ( 2 . _ 4 _ 4 9 ) ) |
622 | 119 | dp2eq2i | |- _ 0 _ 0 0 = _ 0 0 |
623 | 622 119 | eqtri | |- _ 0 _ 0 0 = 0 |
624 | 623 | oveq2i | |- ( 3 . _ 0 _ 0 0 ) = ( 3 . 0 ) |
625 | 182 | dp0u | |- ( 3 . 0 ) = 3 |
626 | 624 625 | eqtr2i | |- 3 = ( 3 . _ 0 _ 0 0 ) |
627 | 626 | oveq1i | |- ( 3 x. ( 2 . _ 4 _ 4 9 ) ) = ( ( 3 . _ 0 _ 0 0 ) x. ( 2 . _ 4 _ 4 9 ) ) |
628 | 450 52 | deccl | |- ; ; 1 4 7 e. NN0 |
629 | 628 51 | deccl | |- ; ; ; 1 4 7 0 e. NN0 |
630 | 629 | nn0cni | |- ; ; ; 1 4 7 0 e. CC |
631 | 630 | addid1i | |- ( ; ; ; 1 4 7 0 + 0 ) = ; ; ; 1 4 7 0 |
632 | 4t3e12 | |- ( 4 x. 3 ) = ; 1 2 |
|
633 | 113 430 632 | mulcomli | |- ( 3 x. 4 ) = ; 1 2 |
634 | 197 | mul02i | |- ( 0 x. 2 ) = 0 |
635 | 113 457 425 | mulcomli | |- ( 0 x. 4 ) = ; 0 0 |
636 | 51 51 1 181 424 300 153 553 | decadd | |- ( 0 + ; 1 2 ) = ; 1 2 |
637 | 636 | oveq1i | |- ( ( 0 + ; 1 2 ) + 0 ) = ( ; 1 2 + 0 ) |
638 | 281 | nn0cni | |- ; 1 2 e. CC |
639 | 638 | addid1i | |- ( ; 1 2 + 0 ) = ; 1 2 |
640 | 637 639 | eqtri | |- ( ( 0 + ; 1 2 ) + 0 ) = ; 1 2 |
641 | 182 51 181 24 51 1 181 51 431 633 634 635 640 140 | dpmul | |- ( ( 3 . 0 ) x. ( 2 . 4 ) ) = ( 7 . _ 2 0 ) |
642 | 51 | dp0u | |- ( 0 . 0 ) = 0 |
643 | 642 | oveq1i | |- ( ( 0 . 0 ) x. ( 4 . 9 ) ) = ( 0 x. ( 4 . 9 ) ) |
644 | dpcl | |- ( ( 4 e. NN0 /\ 9 e. RR ) -> ( 4 . 9 ) e. RR ) |
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645 | 24 4 644 | mp2an | |- ( 4 . 9 ) e. RR |
646 | 645 | recni | |- ( 4 . 9 ) e. CC |
647 | 646 | mul02i | |- ( 0 x. ( 4 . 9 ) ) = 0 |
648 | 643 647 | eqtri | |- ( ( 0 . 0 ) x. ( 4 . 9 ) ) = 0 |
649 | 119 | oveq2i | |- ( 0 . _ 0 0 ) = ( 0 . 0 ) |
650 | 649 642 | eqtri | |- ( 0 . _ 0 0 ) = 0 |
651 | 648 650 | eqtr4i | |- ( ( 0 . 0 ) x. ( 4 . 9 ) ) = ( 0 . _ 0 0 ) |
652 | 52 181 | deccl | |- ; 7 2 e. NN0 |
653 | 652 51 | deccl | |- ; ; 7 2 0 e. NN0 |
654 | eqid | |- ; ; ; 7 2 0 1 = ; ; ; 7 2 0 1 |
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655 | eqid | |- ; ; 1 4 7 = ; ; 1 4 7 |
|
656 | eqid | |- ; ; 7 2 0 = ; ; 7 2 0 |
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657 | 52 181 224 263 | decsuc | |- ( ; 7 2 + 1 ) = ; 7 3 |
658 | 652 51 1 24 656 268 657 114 | decadd | |- ( ; ; 7 2 0 + ; 1 4 ) = ; ; 7 3 4 |
659 | 653 1 450 52 654 655 658 274 | decadd | |- ( ; ; ; 7 2 0 1 + ; ; 1 4 7 ) = ; ; ; 7 3 4 8 |
660 | 642 | oveq2i | |- ( ( 3 . 0 ) + ( 0 . 0 ) ) = ( ( 3 . 0 ) + 0 ) |
661 | 625 430 | eqeltri | |- ( 3 . 0 ) e. CC |
662 | 661 | addid1i | |- ( ( 3 . 0 ) + 0 ) = ( 3 . 0 ) |
663 | 660 662 | eqtri | |- ( ( 3 . 0 ) + ( 0 . 0 ) ) = ( 3 . 0 ) |
664 | eqid | |- ; 4 9 = ; 4 9 |
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665 | 572 | oveq1i | |- ( ( 2 + 4 ) + 1 ) = ( 6 + 1 ) |
666 | 665 140 | eqtri | |- ( ( 2 + 4 ) + 1 ) = 7 |
667 | 9p4e13 | |- ( 9 + 4 ) = ; 1 3 |
|
668 | 196 113 667 | addcomli | |- ( 4 + 9 ) = ; 1 3 |
669 | 181 24 24 53 223 664 666 182 668 | decaddc | |- ( ; 2 4 + ; 4 9 ) = ; 7 3 |
670 | 181 24 24 53 52 182 669 | dpadd | |- ( ( 2 . 4 ) + ( 4 . 9 ) ) = ( 7 . 3 ) |
671 | 663 670 | oveq12i | |- ( ( ( 3 . 0 ) + ( 0 . 0 ) ) x. ( ( 2 . 4 ) + ( 4 . 9 ) ) ) = ( ( 3 . 0 ) x. ( 7 . 3 ) ) |
672 | 217 430 435 | mulcomli | |- ( 3 x. 7 ) = ; 2 1 |
673 | 3t3e9 | |- ( 3 x. 3 ) = 9 |
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674 | 217 | mul01i | |- ( 7 x. 0 ) = 0 |
675 | 217 457 674 | mulcomli | |- ( 0 x. 7 ) = 0 |
676 | 430 | mul01i | |- ( 3 x. 0 ) = 0 |
677 | 676 424 | eqtri | |- ( 3 x. 0 ) = ; 0 0 |
678 | 430 457 677 | mulcomli | |- ( 0 x. 3 ) = ; 0 0 |
679 | 196 | addid1i | |- ( 9 + 0 ) = 9 |
680 | 679 | oveq1i | |- ( ( 9 + 0 ) + 0 ) = ( 9 + 0 ) |
681 | 680 679 188 | 3eqtri | |- ( ( 9 + 0 ) + 0 ) = ; 0 9 |
682 | 196 457 | addcomi | |- ( 9 + 0 ) = ( 0 + 9 ) |
683 | 682 | oveq1i | |- ( ( 9 + 0 ) + 0 ) = ( ( 0 + 9 ) + 0 ) |
684 | 683 | eqeq1i | |- ( ( ( 9 + 0 ) + 0 ) = ; 0 9 <-> ( ( 0 + 9 ) + 0 ) = ; 0 9 ) |
685 | 681 684 | mpbi | |- ( ( 0 + 9 ) + 0 ) = ; 0 9 |
686 | 181 1 51 438 192 | decaddi | |- ( ; 2 1 + 0 ) = ; 2 1 |
687 | 182 51 52 182 51 51 53 51 672 673 675 678 685 686 | dpmul | |- ( ( 3 . 0 ) x. ( 7 . 3 ) ) = ( ; 2 1 . _ 9 0 ) |
688 | 671 687 | eqtri | |- ( ( ( 3 . 0 ) + ( 0 . 0 ) ) x. ( ( 2 . 4 ) + ( 4 . 9 ) ) ) = ( ; 2 1 . _ 9 0 ) |
689 | 650 | oveq2i | |- ( ( ( 7 . _ 2 0 ) + ( ; 1 4 . _ 7 0 ) ) + ( 0 . _ 0 0 ) ) = ( ( ( 7 . _ 2 0 ) + ( ; 1 4 . _ 7 0 ) ) + 0 ) |
690 | 318 2 | pm3.2i | |- ( 2 e. RR /\ 0 e. RR ) |
691 | dp2cl | |- ( ( 2 e. RR /\ 0 e. RR ) -> _ 2 0 e. RR ) |
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692 | 690 691 | ax-mp | |- _ 2 0 e. RR |
693 | dpcl | |- ( ( 7 e. NN0 /\ _ 2 0 e. RR ) -> ( 7 . _ 2 0 ) e. RR ) |
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694 | 52 692 693 | mp2an | |- ( 7 . _ 2 0 ) e. RR |
695 | 694 | recni | |- ( 7 . _ 2 0 ) e. CC |
696 | 3 2 | pm3.2i | |- ( 7 e. RR /\ 0 e. RR ) |
697 | dp2cl | |- ( ( 7 e. RR /\ 0 e. RR ) -> _ 7 0 e. RR ) |
|
698 | 696 697 | ax-mp | |- _ 7 0 e. RR |
699 | dpcl | |- ( ( ; 1 4 e. NN0 /\ _ 7 0 e. RR ) -> ( ; 1 4 . _ 7 0 ) e. RR ) |
|
700 | 450 698 699 | mp2an | |- ( ; 1 4 . _ 7 0 ) e. RR |
701 | 700 | recni | |- ( ; 1 4 . _ 7 0 ) e. CC |
702 | 695 701 | addcli | |- ( ( 7 . _ 2 0 ) + ( ; 1 4 . _ 7 0 ) ) e. CC |
703 | 702 | addid1i | |- ( ( ( 7 . _ 2 0 ) + ( ; 1 4 . _ 7 0 ) ) + 0 ) = ( ( 7 . _ 2 0 ) + ( ; 1 4 . _ 7 0 ) ) |
704 | eqid | |- ; ; ; 1 4 7 0 = ; ; ; 1 4 7 0 |
|
705 | 450 | nn0cni | |- ; 1 4 e. CC |
706 | 705 217 270 | addcomli | |- ( 7 + ; 1 4 ) = ; 2 1 |
707 | 7p2e9 | |- ( 7 + 2 ) = 9 |
|
708 | 217 197 707 | addcomli | |- ( 2 + 7 ) = 9 |
709 | 52 181 450 52 263 655 706 708 | decadd | |- ( ; 7 2 + ; ; 1 4 7 ) = ; ; 2 1 9 |
710 | 00id | |- ( 0 + 0 ) = 0 |
|
711 | 652 51 628 51 656 704 709 710 | decadd | |- ( ; ; 7 2 0 + ; ; ; 1 4 7 0 ) = ; ; ; 2 1 9 0 |
712 | 52 181 51 450 52 293 51 53 51 711 | dpadd3 | |- ( ( 7 . _ 2 0 ) + ( ; 1 4 . _ 7 0 ) ) = ( ; 2 1 . _ 9 0 ) |
713 | 689 703 712 | 3eqtri | |- ( ( ( 7 . _ 2 0 ) + ( ; 1 4 . _ 7 0 ) ) + ( 0 . _ 0 0 ) ) = ( ; 2 1 . _ 9 0 ) |
714 | 688 713 | eqtr4i | |- ( ( ( 3 . 0 ) + ( 0 . 0 ) ) x. ( ( 2 . 4 ) + ( 4 . 9 ) ) ) = ( ( ( 7 . _ 2 0 ) + ( ; 1 4 . _ 7 0 ) ) + ( 0 . _ 0 0 ) ) |
715 | 182 51 51 51 181 24 181 51 24 53 52 450 52 51 51 51 51 1 24 52 51 52 182 24 73 102 102 102 631 641 651 659 714 | dpmul4 | |- ( ( 3 . _ 0 _ 0 0 ) x. ( 2 . _ 4 _ 4 9 ) ) < ( 7 . _ 3 _ 4 8 ) |
716 | 627 715 | eqbrtri | |- ( 3 x. ( 2 . _ 4 _ 4 9 ) ) < ( 7 . _ 3 _ 4 8 ) |
717 | 319 609 | remulcli | |- ( 3 x. ( ( ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) x. ( 1 . _ 4 _ 1 4 ) ) x. ( ( 1 . _ 4 _ 2 _ 6 3 ) x. ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) ) ) ) e. RR |
718 | 319 615 | remulcli | |- ( 3 x. ( 2 . _ 4 _ 4 9 ) ) e. RR |
719 | nnrp | |- ( 8 e. NN -> 8 e. RR+ ) |
|
720 | 63 719 | ax-mp | |- 8 e. RR+ |
721 | 24 720 | rpdp2cl | |- _ 4 8 e. RR+ |
722 | 182 721 | rpdp2cl | |- _ 3 _ 4 8 e. RR+ |
723 | 52 722 | rpdpcl | |- ( 7 . _ 3 _ 4 8 ) e. RR+ |
724 | rpre | |- ( ( 7 . _ 3 _ 4 8 ) e. RR+ -> ( 7 . _ 3 _ 4 8 ) e. RR ) |
|
725 | 723 724 | ax-mp | |- ( 7 . _ 3 _ 4 8 ) e. RR |
726 | 717 718 725 | lttri | |- ( ( ( 3 x. ( ( ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) x. ( 1 . _ 4 _ 1 4 ) ) x. ( ( 1 . _ 4 _ 2 _ 6 3 ) x. ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) ) ) ) < ( 3 x. ( 2 . _ 4 _ 4 9 ) ) /\ ( 3 x. ( 2 . _ 4 _ 4 9 ) ) < ( 7 . _ 3 _ 4 8 ) ) -> ( 3 x. ( ( ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) x. ( 1 . _ 4 _ 1 4 ) ) x. ( ( 1 . _ 4 _ 2 _ 6 3 ) x. ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) ) ) ) < ( 7 . _ 3 _ 4 8 ) ) |
727 | 621 716 726 | mp2an | |- ( 3 x. ( ( ( ( 1 . _ 0 _ 7 _ 9 _ 9 _ 5 5 ) ^ 2 ) x. ( 1 . _ 4 _ 1 4 ) ) x. ( ( 1 . _ 4 _ 2 _ 6 3 ) x. ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) ) ) ) < ( 7 . _ 3 _ 4 8 ) |