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Mirrors > Home > ILE Home > Th. List > resqrexlemnm | Unicode version |
Description: Lemma for resqrex 10954. The difference between two terms of the sequence. (Contributed by Mario Carneiro and Jim Kingdon, 31-Jul-2021.) |
Ref | Expression |
---|---|
resqrexlemex.seq | |
resqrexlemex.a | |
resqrexlemex.agt0 | |
resqrexlemnmsq.n | |
resqrexlemnmsq.m | |
resqrexlemnmsq.nm |
Ref | Expression |
---|---|
resqrexlemnm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resqrexlemex.seq | . . . . . . 7 | |
2 | resqrexlemex.a | . . . . . . 7 | |
3 | resqrexlemex.agt0 | . . . . . . 7 | |
4 | 1, 2, 3 | resqrexlemf 10935 | . . . . . 6 |
5 | resqrexlemnmsq.n | . . . . . 6 | |
6 | 4, 5 | ffvelrnd 5615 | . . . . 5 |
7 | 6 | rpred 9623 | . . . 4 |
8 | resqrexlemnmsq.m | . . . . . 6 | |
9 | 4, 8 | ffvelrnd 5615 | . . . . 5 |
10 | 9 | rpred 9623 | . . . 4 |
11 | 7, 10 | resubcld 8270 | . . 3 |
12 | 7 | resqcld 10603 | . . . . 5 |
13 | 10 | resqcld 10603 | . . . . 5 |
14 | 12, 13 | resubcld 8270 | . . . 4 |
15 | 2cn 8919 | . . . . . . 7 | |
16 | expm1t 10473 | . . . . . . 7 | |
17 | 15, 5, 16 | sylancr 411 | . . . . . 6 |
18 | 2nn 9009 | . . . . . . . . 9 | |
19 | 18 | a1i 9 | . . . . . . . 8 |
20 | 5 | nnnn0d 9158 | . . . . . . . 8 |
21 | 19, 20 | nnexpcld 10599 | . . . . . . 7 |
22 | 21 | nnrpd 9621 | . . . . . 6 |
23 | 17, 22 | eqeltrrd 2242 | . . . . 5 |
24 | 23 | rpred 9623 | . . . 4 |
25 | 14, 24 | remulcld 7920 | . . 3 |
26 | 1nn 8859 | . . . . . . . . 9 | |
27 | 26 | a1i 9 | . . . . . . . 8 |
28 | 4, 27 | ffvelrnd 5615 | . . . . . . 7 |
29 | 19 | nnzd 9303 | . . . . . . 7 |
30 | 28, 29 | rpexpcld 10601 | . . . . . 6 |
31 | 4re 8925 | . . . . . . . . 9 | |
32 | 4pos 8945 | . . . . . . . . 9 | |
33 | 31, 32 | elrpii 9583 | . . . . . . . 8 |
34 | 33 | a1i 9 | . . . . . . 7 |
35 | 5 | nnzd 9303 | . . . . . . . 8 |
36 | peano2zm 9220 | . . . . . . . 8 | |
37 | 35, 36 | syl 14 | . . . . . . 7 |
38 | 34, 37 | rpexpcld 10601 | . . . . . 6 |
39 | 30, 38 | rpdivcld 9641 | . . . . 5 |
40 | 39 | rpred 9623 | . . . 4 |
41 | 40, 24 | remulcld 7920 | . . 3 |
42 | 6, 9 | rpaddcld 9639 | . . . . . . 7 |
43 | 42, 23 | rpmulcld 9640 | . . . . . 6 |
44 | 43 | rpred 9623 | . . . . 5 |
45 | 2 | adantr 274 | . . . . . . . . 9 |
46 | 3 | adantr 274 | . . . . . . . . 9 |
47 | 5 | adantr 274 | . . . . . . . . 9 |
48 | 8 | adantr 274 | . . . . . . . . 9 |
49 | simpr 109 | . . . . . . . . 9 | |
50 | 1, 45, 46, 47, 48, 49 | resqrexlemdecn 10940 | . . . . . . . 8 |
51 | 10 | adantr 274 | . . . . . . . . 9 |
52 | 7 | adantr 274 | . . . . . . . . 9 |
53 | difrp 9619 | . . . . . . . . 9 | |
54 | 51, 52, 53 | syl2anc 409 | . . . . . . . 8 |
55 | 50, 54 | mpbid 146 | . . . . . . 7 |
56 | 55 | rpge0d 9627 | . . . . . 6 |
57 | 7 | recnd 7918 | . . . . . . . . 9 |
58 | 57 | subidd 8188 | . . . . . . . 8 |
59 | fveq2 5480 | . . . . . . . . 9 | |
60 | 59 | oveq2d 5852 | . . . . . . . 8 |
61 | 58, 60 | sylan9req 2218 | . . . . . . 7 |
62 | 0re 7890 | . . . . . . . 8 | |
63 | 62 | eqlei 7983 | . . . . . . 7 |
64 | 61, 63 | syl 14 | . . . . . 6 |
65 | resqrexlemnmsq.nm | . . . . . . 7 | |
66 | 8 | nnzd 9303 | . . . . . . . 8 |
67 | zleloe 9229 | . . . . . . . 8 | |
68 | 35, 66, 67 | syl2anc 409 | . . . . . . 7 |
69 | 65, 68 | mpbid 146 | . . . . . 6 |
70 | 56, 64, 69 | mpjaodan 788 | . . . . 5 |
71 | 1red 7905 | . . . . . 6 | |
72 | 21 | nnrecred 8895 | . . . . . . . . . . 11 |
73 | 72 | recnd 7918 | . . . . . . . . . 10 |
74 | 73 | addid1d 8038 | . . . . . . . . 9 |
75 | 0red 7891 | . . . . . . . . . 10 | |
76 | 1, 2, 3 | resqrexlemlo 10941 | . . . . . . . . . . 11 |
77 | 5, 76 | mpdan 418 | . . . . . . . . . 10 |
78 | 9 | rpgt0d 9626 | . . . . . . . . . 10 |
79 | 72, 75, 7, 10, 77, 78 | lt2addd 8456 | . . . . . . . . 9 |
80 | 74, 79 | eqbrtrrd 4000 | . . . . . . . 8 |
81 | 7, 10 | readdcld 7919 | . . . . . . . . 9 |
82 | 71, 81, 22 | ltdivmul2d 9676 | . . . . . . . 8 |
83 | 80, 82 | mpbid 146 | . . . . . . 7 |
84 | 17 | oveq2d 5852 | . . . . . . 7 |
85 | 83, 84 | breqtrd 4002 | . . . . . 6 |
86 | 71, 44, 85 | ltled 8008 | . . . . 5 |
87 | 11, 44, 70, 86 | lemulge11d 8823 | . . . 4 |
88 | 11 | recnd 7918 | . . . . . 6 |
89 | 81 | recnd 7918 | . . . . . 6 |
90 | 23 | rpcnd 9625 | . . . . . 6 |
91 | 88, 89, 90 | mulassd 7913 | . . . . 5 |
92 | 88, 89 | mulcomd 7911 | . . . . . . 7 |
93 | 10 | recnd 7918 | . . . . . . . 8 |
94 | subsq 10551 | . . . . . . . 8 | |
95 | 57, 93, 94 | syl2anc 409 | . . . . . . 7 |
96 | 92, 95 | eqtr4d 2200 | . . . . . 6 |
97 | 96 | oveq1d 5851 | . . . . 5 |
98 | 91, 97 | eqtr3d 2199 | . . . 4 |
99 | 87, 98 | breqtrd 4002 | . . 3 |
100 | 1, 2, 3, 5, 8, 65 | resqrexlemnmsq 10945 | . . . 4 |
101 | 14, 40, 23, 100 | ltmul1dd 9679 | . . 3 |
102 | 11, 25, 41, 99, 101 | lelttrd 8014 | . 2 |
103 | 40 | recnd 7918 | . . . . . 6 |
104 | 19 | nnrpd 9621 | . . . . . . . 8 |
105 | 104, 37 | rpexpcld 10601 | . . . . . . 7 |
106 | 105 | rpcnd 9625 | . . . . . 6 |
107 | 2cnd 8921 | . . . . . 6 | |
108 | 103, 106, 107 | mulassd 7913 | . . . . 5 |
109 | 30 | rpcnd 9625 | . . . . . . . 8 |
110 | 38 | rpcnd 9625 | . . . . . . . 8 |
111 | 38 | rpap0d 9629 | . . . . . . . 8 # |
112 | 109, 110, 106, 111 | div32apd 8701 | . . . . . . 7 |
113 | 4d2e2 9008 | . . . . . . . . . . . 12 | |
114 | 113 | oveq1i 5846 | . . . . . . . . . . 11 |
115 | 34 | rpcnd 9625 | . . . . . . . . . . . 12 |
116 | 104 | rpap0d 9629 | . . . . . . . . . . . 12 # |
117 | nnm1nn0 9146 | . . . . . . . . . . . . 13 | |
118 | 5, 117 | syl 14 | . . . . . . . . . . . 12 |
119 | 115, 107, 116, 118 | expdivapd 10591 | . . . . . . . . . . 11 |
120 | 114, 119 | eqtr3id 2211 | . . . . . . . . . 10 |
121 | 120 | oveq2d 5852 | . . . . . . . . 9 |
122 | 105 | rpap0d 9629 | . . . . . . . . . 10 # |
123 | 110, 106, 111, 122 | recdivapd 8694 | . . . . . . . . 9 |
124 | 121, 123 | eqtrd 2197 | . . . . . . . 8 |
125 | 124 | oveq2d 5852 | . . . . . . 7 |
126 | 112, 125 | eqtr4d 2200 | . . . . . 6 |
127 | 126 | oveq1d 5851 | . . . . 5 |
128 | 108, 127 | eqtr3d 2199 | . . . 4 |
129 | 106, 122 | recclapd 8668 | . . . . 5 |
130 | 109, 129, 107 | mul32d 8042 | . . . 4 |
131 | 128, 130 | eqtrd 2197 | . . 3 |
132 | 109, 107 | mulcld 7910 | . . . 4 |
133 | 132, 106, 122 | divrecapd 8680 | . . 3 |
134 | 131, 133 | eqtr4d 2200 | . 2 |
135 | 102, 134 | breqtrd 4002 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 698 wceq 1342 wcel 2135 csn 3570 class class class wbr 3976 cxp 4596 cfv 5182 (class class class)co 5836 cmpo 5838 cc 7742 cr 7743 cc0 7744 c1 7745 caddc 7747 cmul 7749 clt 7924 cle 7925 cmin 8060 cdiv 8559 cn 8848 c2 8899 c4 8901 cn0 9105 cz 9182 crp 9580 cseq 10370 cexp 10444 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-frec 6350 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-n0 9106 df-z 9183 df-uz 9458 df-rp 9581 df-seqfrec 10371 df-exp 10445 |
This theorem is referenced by: resqrexlemcvg 10947 |
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