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Mirrors > Home > ILE Home > Th. List > resqrexlemcalc1 | Unicode version |
Description: Lemma for resqrex 10766. Some of the calculations involved in showing that the sequence converges. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.) |
Ref | Expression |
---|---|
resqrexlemex.seq | |
resqrexlemex.a | |
resqrexlemex.agt0 |
Ref | Expression |
---|---|
resqrexlemcalc1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resqrexlemex.seq | . . . . . . . 8 | |
2 | resqrexlemex.a | . . . . . . . 8 | |
3 | resqrexlemex.agt0 | . . . . . . . 8 | |
4 | 1, 2, 3 | resqrexlemfp1 10749 | . . . . . . 7 |
5 | 4 | oveq1d 5757 | . . . . . 6 |
6 | 1, 2, 3 | resqrexlemf 10747 | . . . . . . . . . . 11 |
7 | 6 | ffvelrnda 5523 | . . . . . . . . . 10 |
8 | 7 | rpred 9451 | . . . . . . . . 9 |
9 | 2 | adantr 274 | . . . . . . . . . 10 |
10 | 9, 7 | rerpdivcld 9483 | . . . . . . . . 9 |
11 | 8, 10 | readdcld 7763 | . . . . . . . 8 |
12 | 11 | recnd 7762 | . . . . . . 7 |
13 | 2cnd 8761 | . . . . . . 7 | |
14 | 2ap0 8781 | . . . . . . . 8 # | |
15 | 14 | a1i 9 | . . . . . . 7 # |
16 | 12, 13, 15 | sqdivapd 10405 | . . . . . 6 |
17 | 5, 16 | eqtrd 2150 | . . . . 5 |
18 | sq2 10356 | . . . . . 6 | |
19 | 18 | oveq2i 5753 | . . . . 5 |
20 | 17, 19 | syl6eq 2166 | . . . 4 |
21 | 9 | recnd 7762 | . . . . . 6 |
22 | 4cn 8766 | . . . . . . 7 | |
23 | 22 | a1i 9 | . . . . . 6 |
24 | 4re 8765 | . . . . . . . 8 | |
25 | 24 | a1i 9 | . . . . . . 7 |
26 | 4pos 8785 | . . . . . . . 8 | |
27 | 26 | a1i 9 | . . . . . . 7 |
28 | 25, 27 | gt0ap0d 8359 | . . . . . 6 # |
29 | 21, 23, 28 | divcanap3d 8523 | . . . . 5 |
30 | 29 | eqcomd 2123 | . . . 4 |
31 | 20, 30 | oveq12d 5760 | . . 3 |
32 | 12 | sqcld 10390 | . . . 4 |
33 | 23, 21 | mulcld 7754 | . . . 4 |
34 | 32, 33, 23, 28 | divsubdirapd 8558 | . . 3 |
35 | 31, 34 | eqtr4d 2153 | . 2 |
36 | 8 | recnd 7762 | . . . . . . . . 9 |
37 | 36 | sqcld 10390 | . . . . . . . 8 |
38 | 13, 21 | mulcld 7754 | . . . . . . . 8 |
39 | 37, 38, 33 | addsubassd 8061 | . . . . . . 7 |
40 | 2cn 8759 | . . . . . . . . . . . 12 | |
41 | 22, 40 | negsubdi2i 8016 | . . . . . . . . . . 11 |
42 | 2p2e4 8815 | . . . . . . . . . . . . . 14 | |
43 | 42 | oveq1i 5752 | . . . . . . . . . . . . 13 |
44 | 40, 40 | pncan3oi 7946 | . . . . . . . . . . . . 13 |
45 | 43, 44 | eqtr3i 2140 | . . . . . . . . . . . 12 |
46 | 45 | negeqi 7924 | . . . . . . . . . . 11 |
47 | 41, 46 | eqtr3i 2140 | . . . . . . . . . 10 |
48 | 47 | oveq1i 5752 | . . . . . . . . 9 |
49 | 13, 23, 21 | subdird 8145 | . . . . . . . . 9 |
50 | 13, 21 | mulneg1d 8141 | . . . . . . . . 9 |
51 | 48, 49, 50 | 3eqtr3a 2174 | . . . . . . . 8 |
52 | 51 | oveq2d 5758 | . . . . . . 7 |
53 | 37, 38 | negsubd 8047 | . . . . . . 7 |
54 | 39, 52, 53 | 3eqtrd 2154 | . . . . . 6 |
55 | 54 | oveq1d 5757 | . . . . 5 |
56 | 10 | recnd 7762 | . . . . . . . . 9 |
57 | binom2 10371 | . . . . . . . . 9 | |
58 | 36, 56, 57 | syl2anc 408 | . . . . . . . 8 |
59 | 7 | rpap0d 9457 | . . . . . . . . . . . 12 # |
60 | 21, 36, 59 | divcanap2d 8520 | . . . . . . . . . . 11 |
61 | 60 | oveq2d 5758 | . . . . . . . . . 10 |
62 | 61 | oveq2d 5758 | . . . . . . . . 9 |
63 | 62 | oveq1d 5757 | . . . . . . . 8 |
64 | 58, 63 | eqtrd 2150 | . . . . . . 7 |
65 | 64 | oveq1d 5757 | . . . . . 6 |
66 | 37, 38 | addcld 7753 | . . . . . . 7 |
67 | 56 | sqcld 10390 | . . . . . . 7 |
68 | 66, 67, 33 | addsubd 8062 | . . . . . 6 |
69 | 65, 68 | eqtrd 2150 | . . . . 5 |
70 | 37, 38 | subcld 8041 | . . . . . . 7 |
71 | 70, 67 | addcld 7753 | . . . . . 6 |
72 | 2z 9050 | . . . . . . . . 9 | |
73 | 72 | a1i 9 | . . . . . . . 8 |
74 | 7, 73 | rpexpcld 10416 | . . . . . . 7 |
75 | 74 | rpap0d 9457 | . . . . . 6 # |
76 | 71, 37, 75 | divcanap4d 8524 | . . . . 5 |
77 | 55, 69, 76 | 3eqtr4d 2160 | . . . 4 |
78 | 37, 38, 37 | subdird 8145 | . . . . . . . 8 |
79 | 37 | sqvald 10389 | . . . . . . . . 9 |
80 | 13, 21, 37 | mul32d 7883 | . . . . . . . . . 10 |
81 | 13, 37, 21 | mulassd 7757 | . . . . . . . . . 10 |
82 | 80, 81 | eqtr2d 2151 | . . . . . . . . 9 |
83 | 79, 82 | oveq12d 5760 | . . . . . . . 8 |
84 | 78, 83 | eqtr4d 2153 | . . . . . . 7 |
85 | 21, 36, 59 | sqdivapd 10405 | . . . . . . . . 9 |
86 | 85 | oveq1d 5757 | . . . . . . . 8 |
87 | 21 | sqcld 10390 | . . . . . . . . 9 |
88 | 87, 37, 75 | divcanap1d 8519 | . . . . . . . 8 |
89 | 86, 88 | eqtrd 2150 | . . . . . . 7 |
90 | 84, 89 | oveq12d 5760 | . . . . . 6 |
91 | 70, 67, 37 | adddird 7759 | . . . . . 6 |
92 | binom2sub 10373 | . . . . . . 7 | |
93 | 37, 21, 92 | syl2anc 408 | . . . . . 6 |
94 | 90, 91, 93 | 3eqtr4d 2160 | . . . . 5 |
95 | 94 | oveq1d 5757 | . . . 4 |
96 | 77, 95 | eqtrd 2150 | . . 3 |
97 | 96 | oveq1d 5757 | . 2 |
98 | 37, 21 | subcld 8041 | . . . . 5 |
99 | 98 | sqcld 10390 | . . . 4 |
100 | 99, 37, 23, 75, 28 | divdivap1d 8550 | . . 3 |
101 | 37, 23 | mulcomd 7755 | . . . 4 |
102 | 101 | oveq2d 5758 | . . 3 |
103 | 100, 102 | eqtrd 2150 | . 2 |
104 | 35, 97, 103 | 3eqtrd 2154 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wcel 1465 csn 3497 class class class wbr 3899 cxp 4507 cfv 5093 (class class class)co 5742 cmpo 5744 cc 7586 cr 7587 cc0 7588 c1 7589 caddc 7591 cmul 7593 clt 7768 cle 7769 cmin 7901 cneg 7902 # cap 8311 cdiv 8400 cn 8688 c2 8739 c4 8741 cz 9022 crp 9409 cseq 10186 cexp 10260 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-ilim 4261 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-frec 6256 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8305 df-ap 8312 df-div 8401 df-inn 8689 df-2 8747 df-3 8748 df-4 8749 df-n0 8946 df-z 9023 df-uz 9295 df-rp 9410 df-seqfrec 10187 df-exp 10261 |
This theorem is referenced by: resqrexlemcalc2 10755 |
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