pellexlem2 - Mathbox for Stefan O'Rear

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Theorem pellexlem2 37394

Description: Lemma for pellex 37399. Arithmetical core of pellexlem3, norm upper bound. (Contributed by Stefan O'Rear, 14-Sep-2014.)

**Assertion**
Ref	Expression
pellexlem2	$/\$ $/\$ $/\$

**Proof of Theorem pellexlem2**
Step	Hyp	Ref	Expression
1		simpl3 1066	. . . . . . . . . 10 $/\$ $/\$ $/\$
2	1	nnred 11035	. . . . . . . . 9 $/\$ $/\$ $/\$
3	2	resqcld 13035	. . . . . . . 8 $/\$ $/\$ $/\$
4	2	sqge0d 13036	. . . . . . . 8 $/\$ $/\$ $/\$
5	3, 4	absidd 14161	. . . . . . 7 $/\$ $/\$ $/\$
6	5	eqcomd 2628	. . . . . 6 $/\$ $/\$ $/\$
7	6	oveq2d 6666	. . . . 5 $/\$ $/\$ $/\$
8		simpl2 1065	. . . . . . . . 9 $/\$ $/\$ $/\$
9	8	nncnd 11036	. . . . . . . 8 $/\$ $/\$ $/\$
10	9	sqcld 13006	. . . . . . 7 $/\$ $/\$ $/\$
11		simpl1 1064	. . . . . . . . 9 $/\$ $/\$ $/\$
12	11	nncnd 11036	. . . . . . . 8 $/\$ $/\$ $/\$
13	1	nncnd 11036	. . . . . . . . 9 $/\$ $/\$ $/\$
14	13	sqcld 13006	. . . . . . . 8 $/\$ $/\$ $/\$
15	12, 14	mulcld 10060	. . . . . . 7 $/\$ $/\$ $/\$
16	10, 15	subcld 10392	. . . . . 6 $/\$ $/\$ $/\$
17	1	nnne0d 11065	. . . . . . 7 $/\$ $/\$ $/\$
18		sqne0 12930	. . . . . . . 8
19	18	biimpar 502	. . . . . . 7 $/\$
20	13, 17, 19	syl2anc 693	. . . . . 6 $/\$ $/\$ $/\$
21	16, 14, 20	absdivd 14194	. . . . 5 $/\$ $/\$ $/\$
22	7, 21	eqtr4d 2659	. . . 4 $/\$ $/\$ $/\$
23	22	oveq2d 6666	. . 3 $/\$ $/\$ $/\$
24	16	abscld 14175	. . . . 5 $/\$ $/\$ $/\$
25	24	recnd 10068	. . . 4 $/\$ $/\$ $/\$
26	25, 14, 20	divcan2d 10803	. . 3 $/\$ $/\$ $/\$
27	10, 15, 14, 20	divsubdird 10840	. . . . . 6 $/\$ $/\$ $/\$
28	9, 13, 17	sqdivd 13021	. . . . . . . 8 $/\$ $/\$ $/\$
29	28	eqcomd 2628	. . . . . . 7 $/\$ $/\$ $/\$
30	11	nnred 11035	. . . . . . . . 9 $/\$ $/\$ $/\$
31	11	nnnn0d 11351	. . . . . . . . . 10 $/\$ $/\$ $/\$
32	31	nn0ge0d 11354	. . . . . . . . 9 $/\$ $/\$ $/\$
33		remsqsqrt 13997	. . . . . . . . 9 $/\$
34	30, 32, 33	syl2anc 693	. . . . . . . 8 $/\$ $/\$ $/\$
35	30, 32	resqrtcld 14156	. . . . . . . . . 10 $/\$ $/\$ $/\$
36	35	recnd 10068	. . . . . . . . 9 $/\$ $/\$ $/\$
37	36	sqvald 13005	. . . . . . . 8 $/\$ $/\$ $/\$
38	12, 14, 20	divcan4d 10807	. . . . . . . 8 $/\$ $/\$ $/\$
39	34, 37, 38	3eqtr4rd 2667	. . . . . . 7 $/\$ $/\$ $/\$
40	29, 39	oveq12d 6668	. . . . . 6 $/\$ $/\$ $/\$
41	9, 13, 17	divcld 10801	. . . . . . . 8 $/\$ $/\$ $/\$
42		subsq 12972	. . . . . . . 8 $/\$
43	41, 36, 42	syl2anc 693	. . . . . . 7 $/\$ $/\$ $/\$
44	41, 36	addcld 10059	. . . . . . . 8 $/\$ $/\$ $/\$
45	8	nnred 11035	. . . . . . . . . . 11 $/\$ $/\$ $/\$
46	45, 1	nndivred 11069	. . . . . . . . . 10 $/\$ $/\$ $/\$
47	46, 35	resubcld 10458	. . . . . . . . 9 $/\$ $/\$ $/\$
48	47	recnd 10068	. . . . . . . 8 $/\$ $/\$ $/\$
49	44, 48	mulcomd 10061	. . . . . . 7 $/\$ $/\$ $/\$
50	43, 49	eqtrd 2656	. . . . . 6 $/\$ $/\$ $/\$
51	27, 40, 50	3eqtrd 2660	. . . . 5 $/\$ $/\$ $/\$
52	51	fveq2d 6195	. . . 4 $/\$ $/\$ $/\$
53	52	oveq2d 6666	. . 3 $/\$ $/\$ $/\$
54	23, 26, 53	3eqtr3d 2664	. 2 $/\$ $/\$ $/\$
55	48, 44	absmuld 14193	. . . 4 $/\$ $/\$ $/\$
56	55	oveq2d 6666	. . 3 $/\$ $/\$ $/\$
57	48	abscld 14175	. . . . . 6 $/\$ $/\$ $/\$
58	44	abscld 14175	. . . . . 6 $/\$ $/\$ $/\$
59	57, 58	remulcld 10070	. . . . 5 $/\$ $/\$ $/\$
60	3, 59	remulcld 10070	. . . 4 $/\$ $/\$ $/\$
61		2nn0 11309	. . . . . . . . 9
62	61	nn0negzi 11416	. . . . . . . 8
63	62	a1i 11	. . . . . . 7 $/\$ $/\$ $/\$
64	2, 17, 63	reexpclzd 13034	. . . . . 6 $/\$ $/\$ $/\$
65	64, 58	remulcld 10070	. . . . 5 $/\$ $/\$ $/\$
66	3, 65	remulcld 10070	. . . 4 $/\$ $/\$ $/\$
67		1red 10055	. . . . 5 $/\$ $/\$ $/\$
68		2re 11090	. . . . . . 7
69	68	a1i 11	. . . . . 6 $/\$ $/\$ $/\$
70	69, 35	remulcld 10070	. . . . 5 $/\$ $/\$ $/\$
71	67, 70	readdcld 10069	. . . 4 $/\$ $/\$ $/\$
72		simpr 477	. . . . . 6 $/\$ $/\$ $/\$
73	8	nngt0d 11064	. . . . . . . . . . 11 $/\$ $/\$ $/\$
74	1	nngt0d 11064	. . . . . . . . . . 11 $/\$ $/\$ $/\$
75	45, 2, 73, 74	divgt0d 10959	. . . . . . . . . 10 $/\$ $/\$ $/\$
76	11	nngt0d 11064	. . . . . . . . . . 11 $/\$ $/\$ $/\$
77		sqrtgt0 13999	. . . . . . . . . . 11 $/\$
78	30, 76, 77	syl2anc 693	. . . . . . . . . 10 $/\$ $/\$ $/\$
79	46, 35, 75, 78	addgt0d 10602	. . . . . . . . 9 $/\$ $/\$ $/\$
80	79	gt0ne0d 10592	. . . . . . . 8 $/\$ $/\$ $/\$
81		absgt0 14064	. . . . . . . . 9
82	81	biimpa 501	. . . . . . . 8 $/\$
83	44, 80, 82	syl2anc 693	. . . . . . 7 $/\$ $/\$ $/\$
84		ltmul1 10873	. . . . . . 7 $/\$ $/\$ $/\$
85	57, 64, 58, 83, 84	syl112anc 1330	. . . . . 6 $/\$ $/\$ $/\$
86	72, 85	mpbid 222	. . . . 5 $/\$ $/\$ $/\$
87	2, 17	sqgt0d 13037	. . . . . 6 $/\$ $/\$ $/\$
88		ltmul2 10874	. . . . . 6 $/\$ $/\$ $/\$
89	59, 65, 3, 87, 88	syl112anc 1330	. . . . 5 $/\$ $/\$ $/\$
90	86, 89	mpbid 222	. . . 4 $/\$ $/\$ $/\$
91	13, 17, 63	expclzd 13013	. . . . . . 7 $/\$ $/\$ $/\$
92	58	recnd 10068	. . . . . . 7 $/\$ $/\$ $/\$
93		mulass 10024	. . . . . . . 8 $/\$ $/\$
94	93	eqcomd 2628	. . . . . . 7 $/\$ $/\$
95	14, 91, 92, 94	syl3anc 1326	. . . . . 6 $/\$ $/\$ $/\$
96		expneg 12868	. . . . . . . . . 10 $/\$
97	13, 61, 96	sylancl 694	. . . . . . . . 9 $/\$ $/\$ $/\$
98	97	oveq2d 6666	. . . . . . . 8 $/\$ $/\$ $/\$
99	14, 20	recidd 10796	. . . . . . . 8 $/\$ $/\$ $/\$
100	98, 99	eqtrd 2656	. . . . . . 7 $/\$ $/\$ $/\$
101	100	oveq1d 6665	. . . . . 6 $/\$ $/\$ $/\$
102	92	mulid2d 10058	. . . . . 6 $/\$ $/\$ $/\$
103	95, 101, 102	3eqtrd 2660	. . . . 5 $/\$ $/\$ $/\$
104	41, 36	addcomd 10238	. . . . . . . 8 $/\$ $/\$ $/\$
105		ppncan 10323	. . . . . . . . . 10 $/\$ $/\$
106	105	eqcomd 2628	. . . . . . . . 9 $/\$ $/\$
107	36, 36, 41, 106	syl3anc 1326	. . . . . . . 8 $/\$ $/\$ $/\$
108	36, 36	addcld 10059	. . . . . . . . . 10 $/\$ $/\$ $/\$
109	108, 48	addcomd 10238	. . . . . . . . 9 $/\$ $/\$ $/\$
110		2times 11145	. . . . . . . . . . . 12
111	110	eqcomd 2628	. . . . . . . . . . 11
112	36, 111	syl 17	. . . . . . . . . 10 $/\$ $/\$ $/\$
113	112	oveq2d 6666	. . . . . . . . 9 $/\$ $/\$ $/\$
114	109, 113	eqtrd 2656	. . . . . . . 8 $/\$ $/\$ $/\$
115	104, 107, 114	3eqtrd 2660	. . . . . . 7 $/\$ $/\$ $/\$
116	115	fveq2d 6195	. . . . . 6 $/\$ $/\$ $/\$
117	47, 70	readdcld 10069	. . . . . . . . 9 $/\$ $/\$ $/\$
118	117	recnd 10068	. . . . . . . 8 $/\$ $/\$ $/\$
119	118	abscld 14175	. . . . . . 7 $/\$ $/\$ $/\$
120	70	recnd 10068	. . . . . . . . 9 $/\$ $/\$ $/\$
121	120	abscld 14175	. . . . . . . 8 $/\$ $/\$ $/\$
122	57, 121	readdcld 10069	. . . . . . 7 $/\$ $/\$ $/\$
123	48, 120	abstrid 14195	. . . . . . 7 $/\$ $/\$ $/\$
124		0le2 11111	. . . . . . . . . . . 12
125	124	a1i 11	. . . . . . . . . . 11 $/\$ $/\$ $/\$
126	30, 32	sqrtge0d 14159	. . . . . . . . . . 11 $/\$ $/\$ $/\$
127	69, 35, 125, 126	mulge0d 10604	. . . . . . . . . 10 $/\$ $/\$ $/\$
128	70, 127	absidd 14161	. . . . . . . . 9 $/\$ $/\$ $/\$
129	128	oveq2d 6666	. . . . . . . 8 $/\$ $/\$ $/\$
130	1	nnsqcld 13029	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
131	130	nnge1d 11063	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
132		0lt1 10550	. . . . . . . . . . . . . . . 16
133	132	a1i 11	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
134		lerec 10906	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
135	67, 133, 3, 87, 134	syl22anc 1327	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
136	131, 135	mpbid 222	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
137		1div1e1 10717	. . . . . . . . . . . . 13
138	136, 137	syl6breq 4694	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
139	97, 138	eqbrtrd 4675	. . . . . . . . . . 11 $/\$ $/\$ $/\$
140	57, 64, 67, 72, 139	ltletrd 10197	. . . . . . . . . 10 $/\$ $/\$ $/\$
141	57, 67, 140	ltled 10185	. . . . . . . . 9 $/\$ $/\$ $/\$
142	57, 67, 70, 141	leadd1dd 10641	. . . . . . . 8 $/\$ $/\$ $/\$
143	129, 142	eqbrtrd 4675	. . . . . . 7 $/\$ $/\$ $/\$
144	119, 122, 71, 123, 143	letrd 10194	. . . . . 6 $/\$ $/\$ $/\$
145	116, 144	eqbrtrd 4675	. . . . 5 $/\$ $/\$ $/\$
146	103, 145	eqbrtrd 4675	. . . 4 $/\$ $/\$ $/\$
147	60, 66, 71, 90, 146	ltletrd 10197	. . 3 $/\$ $/\$ $/\$
148	56, 147	eqbrtrd 4675	. 2 $/\$ $/\$ $/\$
149	54, 148	eqbrtrd 4675	1 $/\$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wne 2794 class class class wbr 4653

cfv 5888 (class class class)co 6650

cc 9934

cr 9935

cc0 9936

c1 9937

caddc 9939

cmul 9941

clt 10074

cle 10075

cmin 10266

cneg 10267

cdiv 10684

cn 11020

c2 11070

cn0 11292

cz 11377

cexp 12860

csqrt 13973

cabs 13974

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-2nd 7169 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-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-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976

This theorem is referenced by: pellexlem3 37395

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