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Mirrors > Home > MPE Home > Th. List > 2lgslem3c | Structured version Visualization version Unicode version |
Description: Lemma for 2lgslem3c1 25127. (Contributed by AV, 16-Jul-2021.) |
Ref | Expression |
---|---|
2lgslem2.n |
Ref | Expression |
---|---|
2lgslem3c |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2lgslem2.n | . . 3 | |
2 | oveq1 6657 | . . . . 5 | |
3 | 2 | oveq1d 6665 | . . . 4 |
4 | oveq1 6657 | . . . . 5 | |
5 | 4 | fveq2d 6195 | . . . 4 |
6 | 3, 5 | oveq12d 6668 | . . 3 |
7 | 1, 6 | syl5eq 2668 | . 2 |
8 | 8nn0 11315 | . . . . . . . . . . 11 | |
9 | 8 | a1i 11 | . . . . . . . . . 10 |
10 | id 22 | . . . . . . . . . 10 | |
11 | 9, 10 | nn0mulcld 11356 | . . . . . . . . 9 |
12 | 11 | nn0cnd 11353 | . . . . . . . 8 |
13 | 5cn 11100 | . . . . . . . . 9 | |
14 | 13 | a1i 11 | . . . . . . . 8 |
15 | 1cnd 10056 | . . . . . . . 8 | |
16 | 12, 14, 15 | addsubassd 10412 | . . . . . . 7 |
17 | 4t2e8 11181 | . . . . . . . . . . . 12 | |
18 | 17 | eqcomi 2631 | . . . . . . . . . . 11 |
19 | 18 | a1i 11 | . . . . . . . . . 10 |
20 | 19 | oveq1d 6665 | . . . . . . . . 9 |
21 | 4cn 11098 | . . . . . . . . . . 11 | |
22 | 21 | a1i 11 | . . . . . . . . . 10 |
23 | 2cn 11091 | . . . . . . . . . . 11 | |
24 | 23 | a1i 11 | . . . . . . . . . 10 |
25 | nn0cn 11302 | . . . . . . . . . 10 | |
26 | 22, 24, 25 | mul32d 10246 | . . . . . . . . 9 |
27 | 20, 26 | eqtrd 2656 | . . . . . . . 8 |
28 | df-5 11082 | . . . . . . . . . . 11 | |
29 | 28 | oveq1i 6660 | . . . . . . . . . 10 |
30 | pncan1 10454 | . . . . . . . . . . 11 | |
31 | 21, 30 | ax-mp 5 | . . . . . . . . . 10 |
32 | 29, 31 | eqtri 2644 | . . . . . . . . 9 |
33 | 32 | a1i 11 | . . . . . . . 8 |
34 | 27, 33 | oveq12d 6668 | . . . . . . 7 |
35 | 16, 34 | eqtrd 2656 | . . . . . 6 |
36 | 35 | oveq1d 6665 | . . . . 5 |
37 | 4nn0 11311 | . . . . . . . . . 10 | |
38 | 37 | a1i 11 | . . . . . . . . 9 |
39 | 38, 10 | nn0mulcld 11356 | . . . . . . . 8 |
40 | 39 | nn0cnd 11353 | . . . . . . 7 |
41 | 40, 24 | mulcld 10060 | . . . . . 6 |
42 | 2rp 11837 | . . . . . . . 8 | |
43 | 42 | a1i 11 | . . . . . . 7 |
44 | 43 | rpcnne0d 11881 | . . . . . 6 |
45 | divdir 10710 | . . . . . 6 | |
46 | 41, 22, 44, 45 | syl3anc 1326 | . . . . 5 |
47 | 2ne0 11113 | . . . . . . . 8 | |
48 | 47 | a1i 11 | . . . . . . 7 |
49 | 40, 24, 48 | divcan4d 10807 | . . . . . 6 |
50 | 4d2e2 11184 | . . . . . . 7 | |
51 | 50 | a1i 11 | . . . . . 6 |
52 | 49, 51 | oveq12d 6668 | . . . . 5 |
53 | 36, 46, 52 | 3eqtrd 2660 | . . . 4 |
54 | 4ne0 11117 | . . . . . . . . . 10 | |
55 | 21, 54 | pm3.2i 471 | . . . . . . . . 9 |
56 | 55 | a1i 11 | . . . . . . . 8 |
57 | divdir 10710 | . . . . . . . 8 | |
58 | 12, 14, 56, 57 | syl3anc 1326 | . . . . . . 7 |
59 | 8cn 11106 | . . . . . . . . . . 11 | |
60 | 59 | a1i 11 | . . . . . . . . . 10 |
61 | div23 10704 | . . . . . . . . . 10 | |
62 | 60, 25, 56, 61 | syl3anc 1326 | . . . . . . . . 9 |
63 | 18 | oveq1i 6660 | . . . . . . . . . . . 12 |
64 | 23, 21, 54 | divcan3i 10771 | . . . . . . . . . . . 12 |
65 | 63, 64 | eqtri 2644 | . . . . . . . . . . 11 |
66 | 65 | a1i 11 | . . . . . . . . . 10 |
67 | 66 | oveq1d 6665 | . . . . . . . . 9 |
68 | 62, 67 | eqtrd 2656 | . . . . . . . 8 |
69 | 68 | oveq1d 6665 | . . . . . . 7 |
70 | 58, 69 | eqtrd 2656 | . . . . . 6 |
71 | 70 | fveq2d 6195 | . . . . 5 |
72 | 1lt4 11199 | . . . . . 6 | |
73 | 2nn0 11309 | . . . . . . . . . . . 12 | |
74 | 73 | a1i 11 | . . . . . . . . . . 11 |
75 | 74, 10 | nn0mulcld 11356 | . . . . . . . . . 10 |
76 | 75 | nn0zd 11480 | . . . . . . . . 9 |
77 | 76 | peano2zd 11485 | . . . . . . . 8 |
78 | 1nn0 11308 | . . . . . . . . 9 | |
79 | 78 | a1i 11 | . . . . . . . 8 |
80 | 4nn 11187 | . . . . . . . . 9 | |
81 | 80 | a1i 11 | . . . . . . . 8 |
82 | adddivflid 12619 | . . . . . . . 8 | |
83 | 77, 79, 81, 82 | syl3anc 1326 | . . . . . . 7 |
84 | 24, 25 | mulcld 10060 | . . . . . . . . . . 11 |
85 | 54 | a1i 11 | . . . . . . . . . . . 12 |
86 | 22, 85 | reccld 10794 | . . . . . . . . . . 11 |
87 | 84, 15, 86 | addassd 10062 | . . . . . . . . . 10 |
88 | 28 | oveq1i 6660 | . . . . . . . . . . . . . 14 |
89 | ax-1cn 9994 | . . . . . . . . . . . . . . 15 | |
90 | 21, 89, 21, 54 | divdiri 10782 | . . . . . . . . . . . . . 14 |
91 | 21, 54 | dividi 10758 | . . . . . . . . . . . . . . 15 |
92 | 91 | oveq1i 6660 | . . . . . . . . . . . . . 14 |
93 | 88, 90, 92 | 3eqtri 2648 | . . . . . . . . . . . . 13 |
94 | 93 | a1i 11 | . . . . . . . . . . . 12 |
95 | 94 | eqcomd 2628 | . . . . . . . . . . 11 |
96 | 95 | oveq2d 6666 | . . . . . . . . . 10 |
97 | 87, 96 | eqtrd 2656 | . . . . . . . . 9 |
98 | 97 | fveq2d 6195 | . . . . . . . 8 |
99 | 98 | eqeq1d 2624 | . . . . . . 7 |
100 | 83, 99 | bitrd 268 | . . . . . 6 |
101 | 72, 100 | mpbii 223 | . . . . 5 |
102 | 71, 101 | eqtrd 2656 | . . . 4 |
103 | 53, 102 | oveq12d 6668 | . . 3 |
104 | 75 | nn0cnd 11353 | . . . 4 |
105 | 40, 24, 104, 15 | addsub4d 10439 | . . 3 |
106 | 2t2e4 11177 | . . . . . . . . . 10 | |
107 | 106 | eqcomi 2631 | . . . . . . . . 9 |
108 | 107 | a1i 11 | . . . . . . . 8 |
109 | 108 | oveq1d 6665 | . . . . . . 7 |
110 | 24, 24, 25 | mulassd 10063 | . . . . . . 7 |
111 | 109, 110 | eqtrd 2656 | . . . . . 6 |
112 | 111 | oveq1d 6665 | . . . . 5 |
113 | 2txmxeqx 11149 | . . . . . 6 | |
114 | 104, 113 | syl 17 | . . . . 5 |
115 | 112, 114 | eqtrd 2656 | . . . 4 |
116 | 2m1e1 11135 | . . . . 5 | |
117 | 116 | a1i 11 | . . . 4 |
118 | 115, 117 | oveq12d 6668 | . . 3 |
119 | 103, 105, 118 | 3eqtrd 2660 | . 2 |
120 | 7, 119 | sylan9eqr 2678 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wne 2794 class class class wbr 4653 cfv 5888 (class class class)co 6650 cc 9934 cc0 9936 c1 9937 caddc 9939 cmul 9941 clt 10074 cmin 10266 cdiv 10684 cn 11020 c2 11070 c4 11072 c5 11073 c8 11076 cn0 11292 cz 11377 crp 11832 cfl 12591 |
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 ax-pre-sup 10014 |
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-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 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-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 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-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 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-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fl 12593 |
This theorem is referenced by: 2lgslem3c1 25127 |
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