Proof of Theorem zfz1isolemiso
Step | Hyp | Ref
| Expression |
1 | | zfz1isolemiso.g |
. . . . . 6
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2 | 1 | ad2antrr 488 |
. . . . 5
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3 | | simplr 528 |
. . . . 5
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4 | | simpr 110 |
. . . . 5
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5 | | isorel 5803 |
. . . . 5
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6 | 2, 3, 4, 5 | syl12anc 1236 |
. . . 4
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7 | | zfz1isolemiso.a |
. . . . . . . 8
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8 | 7 | adantr 276 |
. . . . . . 7
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9 | | elfzelz 10011 |
. . . . . . . . . . 11
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`"  
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10 | 7, 9 | syl 14 |
. . . . . . . . . 10
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11 | 10 | zred 9364 |
. . . . . . . . 9
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12 | 11 | adantr 276 |
. . . . . . . 8
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13 | | zfz1isolemiso.xf |
. . . . . . . . . . . . 13
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14 | | hashcl 10745 |
. . . . . . . . . . . . 13
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15 | 13, 14 | syl 14 |
. . . . . . . . . . . 12
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16 | 15 | nn0red 9219 |
. . . . . . . . . . 11
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17 | | peano2rem 8214 |
. . . . . . . . . . 11
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18 | 16, 17 | syl 14 |
. . . . . . . . . 10
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19 | 18 | adantr 276 |
. . . . . . . . 9
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20 | 16 | adantr 276 |
. . . . . . . . 9
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21 | | elfzle2 10014 |
. . . . . . . . . . 11
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22 | 21 | adantl 277 |
. . . . . . . . . 10
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23 | | zfz1isolemiso.mx |
. . . . . . . . . . . 12
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24 | | hashdifsn 10783 |
. . . . . . . . . . . 12
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25 | 13, 23, 24 | syl2anc 411 |
. . . . . . . . . . 11
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26 | 25 | adantr 276 |
. . . . . . . . . 10
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27 | 22, 26 | breqtrd 4026 |
. . . . . . . . 9
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28 | 20 | ltm1d 8878 |
. . . . . . . . 9
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29 | 12, 19, 20, 27, 28 | lelttrd 8072 |
. . . . . . . 8
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♯    |
30 | 12, 29 | gtned 8060 |
. . . . . . 7
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♯    |
31 | | fvunsng 5706 |
. . . . . . 7
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32 | 8, 30, 31 | syl2anc 411 |
. . . . . 6
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33 | 32 | adantr 276 |
. . . . 5
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34 | | zfz1isolemiso.b |
. . . . . . 7
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35 | 34 | ad2antrr 488 |
. . . . . 6
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36 | | elfzelz 10011 |
. . . . . . . . . 10
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  |
37 | 34, 36 | syl 14 |
. . . . . . . . 9
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38 | 37 | zred 9364 |
. . . . . . . 8
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39 | 38 | ad2antrr 488 |
. . . . . . 7
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  |
40 | 18 | ad2antrr 488 |
. . . . . . . 8
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41 | 16 | ad2antrr 488 |
. . . . . . . 8
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♯    |
42 | | elfzle2 10014 |
. . . . . . . . . 10
    ♯       ♯        |
43 | 42 | adantl 277 |
. . . . . . . . 9
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♯        |
44 | 25 | ad2antrr 488 |
. . . . . . . . 9
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♯      alt=")
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45 | 43, 44 | breqtrd 4026 |
. . . . . . . 8
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 ♯     |
46 | 16 | ltm1d 8878 |
. . . . . . . . 9
  ♯   ♯ |
47 | 46 | ad2antrr 488 |
. . . . . . . 8
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 ♯   ♯    |
48 | 39, 40, 41, 45, 47 | lelttrd 8072 |
. . . . . . 7
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♯    |
49 | 39, 48 | gtned 8060 |
. . . . . 6
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♯    |
50 | | fvunsng 5706 |
. . . . . 6
     ♯   ♯  
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51 | 35, 49, 50 | syl2anc 411 |
. . . . 5
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52 | 33, 51 | breq12d 4013 |
. . . 4
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53 | 6, 52 | bitr4d 191 |
. . 3
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54 | 29 | adantr 276 |
. . . . 5
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55 | | elsni 3609 |
. . . . . 6
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56 | 55 | adantl 277 |
. . . . 5
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57 | 54, 56 | breqtrrd 4028 |
. . . 4
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58 | | isof1o 5802 |
. . . . . . . . . . . . 13
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59 | 1, 58 | syl 14 |
. . . . . . . . . . . 12
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)"  |
60 | | f1of 5457 |
. . . . . . . . . . . 12
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`"               |
61 | 59, 60 | syl 14 |
. . . . . . . . . . 11
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)"          |
62 | 61 | ffvelcdmda 5647 |
. . . . . . . . . 10
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63 | 62 | eldifbd 3141 |
. . . . . . . . 9
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`"      |
64 | | elsn2g 3624 |
. . . . . . . . . . 11
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65 | 23, 64 | syl 14 |
. . . . . . . . . 10
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       |
66 | 65 | adantr 276 |
. . . . . . . . 9
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`"   
       |
67 | 63, 66 | mtbid 672 |
. . . . . . . 8
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68 | | breq1 4003 |
. . . . . . . . . 10
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69 | | zfz1isolemiso.m |
. . . . . . . . . . 11
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70 | 69 | adantr 276 |
. . . . . . . . . 10
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71 | 62 | eldifad 3140 |
. . . . . . . . . 10
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72 | 68, 70, 71 | rspcdva 2846 |
. . . . . . . . 9
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73 | | zfz1isolemiso.xz |
. . . . . . . . . . . 12
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74 | 73 | adantr 276 |
. . . . . . . . . . 11
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75 | 74, 71 | sseldd 3156 |
. . . . . . . . . 10
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76 | 73, 23 | sseldd 3156 |
. . . . . . . . . . 11
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77 | 76 | adantr 276 |
. . . . . . . . . 10
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78 | | zleloe 9289 |
. . . . . . . . . 10
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79 | 75, 77, 78 | syl2anc 411 |
. . . . . . . . 9
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80 | 72, 79 | mpbid 147 |
. . . . . . . 8
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81 | 67, 80 | ecased 1349 |
. . . . . . 7
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)" |
82 | 15 | nn0zd 9362 |
. . . . . . . . . . . . . . 15
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`"    |
83 | | peano2zm 9280 |
. . . . . . . . . . . . . . 15
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84 | 82, 83 | syl 14 |
. . . . . . . . . . . . . 14
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85 | | zltnle 9288 |
. . . . . . . . . . . . . 14
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 ♯      |
86 | 84, 82, 85 | syl2anc 411 |
. . . . . . . . . . . . 13
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`"   ♯      |
87 | 46, 86 | mpbid 147 |
. . . . . . . . . . . 12
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88 | 25 | breq2d 4012 |
. . . . . . . . . . . 12
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)"
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89 | 87, 88 | mtbird 673 |
. . . . . . . . . . 11
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90 | | elfzle2 10014 |
. . . . . . . . . . 11
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91 | 89, 90 | nsyl 628 |
. . . . . . . . . 10
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92 | | fdm 5367 |
. . . . . . . . . . 11
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93 | 61, 92 | syl 14 |
. . . . . . . . . 10
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)"    |
94 | 91, 93 | neleqtrrd 2276 |
. . . . . . . . 9
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)" |
95 | | fsnunfv 5713 |
. . . . . . . . 9
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96 | 15, 23, 94, 95 | syl3anc 1238 |
. . . . . . . 8
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97 | 96 | adantr 276 |
. . . . . . 7
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98 | 81, 32, 97 | 3brtr4d 4032 |
. . . . . 6
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99 | 98 | adantr 276 |
. . . . 5
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100 | 56 | fveq2d 5515 |
. . . . 5
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101 | 99, 100 | breqtrrd 4028 |
. . . 4
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102 | 57, 101 | 2thd 175 |
. . 3
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103 | 13, 23 | zfz1isolemsplit 10802 |
. . . . . 6
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104 | 34, 103 | eleqtrd 2256 |
. . . . 5
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105 | | elun 3276 |
. . . . 5
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106 | 104, 105 | sylib 122 |
. . . 4
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)"   ♯      |
107 | 106 | adantr 276 |
. . 3
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108 | 53, 102, 107 | mpjaodan 798 |
. 2
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109 | 38 | ad2antrr 488 |
. . . . 5
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110 | 11 | ad2antrr 488 |
. . . . 5
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111 | | elfzle2 10014 |
. . . . . . . 8
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♯    |
112 | 34, 111 | syl 14 |
. . . . . . 7
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♯   |
115 | 114 | ad2antlr 489 |
. . . . . 6
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)"   ♯    |
116 | 113, 115 | breqtrrd 4028 |
. . . . 5
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   ♯          
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)"  
  |
118 | 73 | ad2antrr 488 |
. . . . . . . 8
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)"     |
119 | 61 | ad2antrr 488 |
. . . . . . . . . 10
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)"          |
120 | | simpr 110 |
. . . . . . . . . 10
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)"      ♯         |
121 | 119, 120 | ffvelcdmd 5648 |
. . . . . . . . 9
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)"             |
122 | 121 | eldifad 3140 |
. . . . . . . 8
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)"                 |
125 | 76 | zred 9364 |
. . . . . . 7
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126 | 125 | ad2antrr 488 |
. . . . . 6
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)"     |
127 | | breq1 4003 |
. . . . . . 7
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)"  |
128 | 69 | ad2antrr 488 |
. . . . . . 7
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)" 
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)"      |
129 | 127, 128,
122 | rspcdva 2846 |
. . . . . 6
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)"         |
130 | 124, 126,
129 | lensymd 8069 |
. . . . 5
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)"  
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131 | 115 | fveq2d 5515 |
. . . . . . 7
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132 | 96 | ad2antrr 488 |
. . . . . . 7
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   ♯            ♯        ♯     |
133 | 131, 132 | eqtrd 2210 |
. . . . . 6
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134 | 34 | ad2antrr 488 |
. . . . . . 7
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)"   |
135 | 18 | ad2antrr 488 |
. . . . . . . . 9
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136 | 16 | ad2antrr 488 |
. . . . . . . . 9
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`"    |
137 | 42 | adantl 277 |
. . . . . . . . . 10
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)"   ♯        |
138 | 25 | ad2antrr 488 |
. . . . . . . . . 10
    ♯   
   ♯        ♯       ♯     |
139 | 137, 138 | breqtrd 4026 |
. . . . . . . . 9
    ♯   
   ♯         ♯     |
140 | 46 | ad2antrr 488 |
. . . . . . . . 9
    ♯   
   ♯         ♯   ♯    |
141 | 109, 135,
136, 139, 140 | lelttrd 8072 |
. . . . . . . 8
    ♯   
   ♯        ♯    |
142 | 109, 141 | gtned 8060 |
. . . . . . 7
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143 | 134, 142,
50 | syl2anc 411 |
. . . . . 6
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144 | 133, 143 | breq12d 4013 |
. . . . 5
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145 | 130, 144 | mtbird 673 |
. . . 4
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146 | 117, 145 | 2falsed 702 |
. . 3
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147 | 38 | ad2antrr 488 |
. . . . . 6
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148 | 147 | ltnrd 8059 |
. . . . 5
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149 | 114 | ad2antlr 489 |
. . . . . . 7
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150 | 55 | adantl 277 |
. . . . . . 7
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151 | 149, 150 | eqtr4d 2213 |
. . . . . 6
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)" 
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152 | 151 | breq1d 4010 |
. . . . 5
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   |
153 | 148, 152 | mtbird 673 |
. . . 4
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)" 
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  |
154 | 125 | ad2antrr 488 |
. . . . . 6
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)" 
 ♯      |
155 | 154 | ltnrd 8059 |
. . . . 5
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156 | 149 | fveq2d 5515 |
. . . . . . 7
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157 | 96 | ad2antrr 488 |
. . . . . . 7
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158 | 156, 157 | eqtrd 2210 |
. . . . . 6
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159 | 150 | fveq2d 5515 |
. . . . . . 7
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160 | 159, 157 | eqtrd 2210 |
. . . . . 6
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161 | 158, 160 | breq12d 4013 |
. . . . 5
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162 | 155, 161 | mtbird 673 |
. . . 4
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163 | 153, 162 | 2falsed 702 |
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164 | 106 | adantr 276 |
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165 | 146, 163,
164 | mpjaodan 798 |
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166 | 7, 103 | eleqtrd 2256 |
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167 | | elun 3276 |
. . 3
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168 | 166, 167 | sylib 122 |
. 2
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169 | 108, 165,
168 | mpjaodan 798 |
1
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