pell1qrgaplem - Mathbox for Stefan O'Rear

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Theorem pell1qrgaplem 37437

Description: Lemma for pell1qrgap 37438. (Contributed by Stefan O'Rear, 18-Sep-2014.)

**Assertion**
Ref	Expression
pell1qrgaplem	$/\$ $/\$ $/\$ $/\$

**Proof of Theorem pell1qrgaplem**
Step	Hyp	Ref	Expression
1		nnrp 11842	. . . . . 6
2	1	ad2antrr 762	. . . . 5 $/\$ $/\$ $/\$ $/\$
3		1rp 11836	. . . . . 6
4	3	a1i 11	. . . . 5 $/\$ $/\$ $/\$ $/\$
5	2, 4	rpaddcld 11887	. . . 4 $/\$ $/\$ $/\$ $/\$
6	5	rpsqrtcld 14150	. . 3 $/\$ $/\$ $/\$ $/\$
7	6	rpred 11872	. 2 $/\$ $/\$ $/\$ $/\$
8	2	rpsqrtcld 14150	. . 3 $/\$ $/\$ $/\$ $/\$
9	8	rpred 11872	. 2 $/\$ $/\$ $/\$ $/\$
10		nn0re 11301	. . . 4
11	10	adantr 481	. . 3 $/\$
12	11	ad2antlr 763	. 2 $/\$ $/\$ $/\$ $/\$
13		nn0re 11301	. . . . 5
14	13	adantl 482	. . . 4 $/\$
15	14	ad2antlr 763	. . 3 $/\$ $/\$ $/\$ $/\$
16	9, 15	remulcld 10070	. 2 $/\$ $/\$ $/\$ $/\$
17	2	rpred 11872	. . . . . 6 $/\$ $/\$ $/\$ $/\$
18		1re 10039	. . . . . . . 8
19	18	a1i 11	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
20	15	resqcld 13035	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
21	19, 20	resubcld 10458	. . . . . 6 $/\$ $/\$ $/\$ $/\$
22	17, 21	remulcld 10070	. . . . 5 $/\$ $/\$ $/\$ $/\$
23		0red 10041	. . . . . 6 $/\$ $/\$ $/\$ $/\$
24	17, 23	remulcld 10070	. . . . 5 $/\$ $/\$ $/\$ $/\$
25	12	resqcld 13035	. . . . 5 $/\$ $/\$ $/\$ $/\$
26		sq1 12958	. . . . . . . . 9
27	26	a1i 11	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
28		nnge1 11046	. . . . . . . . . . 11
29	28	adantl 482	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
30		simplrl 800	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
31		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . 22
32	31	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 $/\$ $/\$ $/\$ $/\$ $/\$
33		sq0 12955	. . . . . . . . . . . . . . . . . . . . 21
34	32, 33	syl6eq 2672	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$ $/\$ $/\$ $/\$
35	34	oveq2d 6666	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$ $/\$ $/\$
36	2	rpcnd 11874	. . . . . . . . . . . . . . . . . . . . 21 $/\$ $/\$ $/\$ $/\$
37	36	adantr 481	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$ $/\$ $/\$ $/\$
38	37	mul01d 10235	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$ $/\$ $/\$
39	35, 38	eqtrd 2656	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $/\$
40	39	oveq2d 6666	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$
41		simplrr 801	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$
42	12	recnd 10068	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$ $/\$ $/\$
43	42	sqcld 13006	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$ $/\$
44	43	adantr 481	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $/\$
45	44	subid1d 10381	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$
46	40, 41, 45	3eqtr3d 2664	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$
47	26, 46	syl5req 2669	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$
48		nn0ge0 11318	. . . . . . . . . . . . . . . . . . 19
49	48	adantr 481	. . . . . . . . . . . . . . . . . 18 $/\$
50	49	ad2antlr 763	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$
51		0le1 10551	. . . . . . . . . . . . . . . . . 18
52	51	a1i 11	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$
53		sq11 12936	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$
54	12, 50, 19, 52, 53	syl22anc 1327	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
55	54	adantr 481	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$
56	47, 55	mpbid 222	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
57		simpr 477	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$
58	57	oveq2d 6666	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$
59	8	rpcnd 11874	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$
60	59	adantr 481	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$
61	60	mul01d 10235	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$
62	58, 61	eqtrd 2656	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
63	56, 62	oveq12d 6668	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
64		1p0e1 11133	. . . . . . . . . . . . 13
65	63, 64	syl6eq 2672	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
66	30, 65	breqtrd 4679	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
67	18	ltnri 10146	. . . . . . . . . . 11
68		pm2.24 121	. . . . . . . . . . 11
69	66, 67, 68	mpisyl 21	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
70		simplrr 801	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
71		elnn0 11294	. . . . . . . . . . 11 $\/$
72	70, 71	sylib 208	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $\/$
73	29, 69, 72	mpjaodan 827	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
74		nn0ge0 11318	. . . . . . . . . . . 12
75	74	adantl 482	. . . . . . . . . . 11 $/\$
76	75	ad2antlr 763	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
77	19, 15, 52, 76	le2sqd 13044	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
78	73, 77	mpbid 222	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
79	27, 78	eqbrtrrd 4677	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
80	19, 20	suble0d 10618	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
81	79, 80	mpbird 247	. . . . . 6 $/\$ $/\$ $/\$ $/\$
82	21, 23, 2	lemul2d 11916	. . . . . 6 $/\$ $/\$ $/\$ $/\$
83	81, 82	mpbid 222	. . . . 5 $/\$ $/\$ $/\$ $/\$
84	22, 24, 25, 83	leadd2dd 10642	. . . 4 $/\$ $/\$ $/\$ $/\$
85	5	rpcnd 11874	. . . . . 6 $/\$ $/\$ $/\$ $/\$
86	85	sqsqrtd 14178	. . . . 5 $/\$ $/\$ $/\$ $/\$
87		simprr 796	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
88	87	eqcomd 2628	. . . . . 6 $/\$ $/\$ $/\$ $/\$
89	88	oveq2d 6666	. . . . 5 $/\$ $/\$ $/\$ $/\$
90	15	recnd 10068	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
91	90	sqcld 13006	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
92	36, 91	mulcld 10060	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
93	36, 43, 92	addsub12d 10415	. . . . . 6 $/\$ $/\$ $/\$ $/\$
94	19	recnd 10068	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
95	36, 94, 91	subdid 10486	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
96	36	mulid1d 10057	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
97	96	oveq1d 6665	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
98	95, 97	eqtr2d 2657	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
99	98	oveq2d 6666	. . . . . 6 $/\$ $/\$ $/\$ $/\$
100	93, 99	eqtrd 2656	. . . . 5 $/\$ $/\$ $/\$ $/\$
101	86, 89, 100	3eqtrd 2660	. . . 4 $/\$ $/\$ $/\$ $/\$
102	36	mul01d 10235	. . . . . 6 $/\$ $/\$ $/\$ $/\$
103	102	oveq2d 6666	. . . . 5 $/\$ $/\$ $/\$ $/\$
104	43	addid1d 10236	. . . . 5 $/\$ $/\$ $/\$ $/\$
105	103, 104	eqtr2d 2657	. . . 4 $/\$ $/\$ $/\$ $/\$
106	84, 101, 105	3brtr4d 4685	. . 3 $/\$ $/\$ $/\$ $/\$
107	6	rpge0d 11876	. . . 4 $/\$ $/\$ $/\$ $/\$
108	7, 12, 107, 50	le2sqd 13044	. . 3 $/\$ $/\$ $/\$ $/\$
109	106, 108	mpbird 247	. 2 $/\$ $/\$ $/\$ $/\$
110	59	mulid1d 10057	. . 3 $/\$ $/\$ $/\$ $/\$
111	19, 15, 8	lemul2d 11916	. . . 4 $/\$ $/\$ $/\$ $/\$
112	73, 111	mpbid 222	. . 3 $/\$ $/\$ $/\$ $/\$
113	110, 112	eqbrtrrd 4677	. 2 $/\$ $/\$ $/\$ $/\$
114	7, 9, 12, 16, 109, 113	le2addd 10646	1 $/\$ $/\$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $\/$ wo 383 $/\$ wa 384

wceq 1483

wcel 1990 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

cn 11020

c2 11070

cn0 11292

crp 11832

cexp 12860

csqrt 13973

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: pell1qrgap 37438

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