irrapxlem5 - Mathbox for Stefan O'Rear

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Theorem irrapxlem5 37390

Description: Lemma for irrapx1 37392. Switching to real intervals and fraction syntax. (Contributed by Stefan O'Rear, 13-Sep-2014.)

**Assertion**
Ref	Expression
irrapxlem5	$/\$ $/\$ $/\$ denom

Distinct variable groups:

Proof of Theorem irrapxlem5 Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 477	. . . . . 6 $/\$
2	1	rpreccld 11882	. . . . 5 $/\$
3	2	rprege0d 11879	. . . 4 $/\$ $/\$
4		flge0nn0 12621	. . . 4 $/\$
5		nn0p1nn 11332	. . . 4
6	3, 4, 5	3syl 18	. . 3 $/\$
7		irrapxlem4 37389	. . 3 $/\$
8	6, 7	syldan 487	. 2 $/\$
9		simplrr 801	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
10		nnq 11801	. . . . . . 7
11	9, 10	syl 17	. . . . . 6 $/\$ $/\$ $/\$ $/\$
12		simplrl 800	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
13		nnq 11801	. . . . . . 7
14	12, 13	syl 17	. . . . . 6 $/\$ $/\$ $/\$ $/\$
15	12	nnne0d 11065	. . . . . 6 $/\$ $/\$ $/\$ $/\$
16		qdivcl 11809	. . . . . 6 $/\$ $/\$
17	11, 14, 15, 16	syl3anc 1326	. . . . 5 $/\$ $/\$ $/\$ $/\$
18	9	nnrpd 11870	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
19	12	nnrpd 11870	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
20	18, 19	rpdivcld 11889	. . . . . 6 $/\$ $/\$ $/\$ $/\$
21	20	rpgt0d 11875	. . . . 5 $/\$ $/\$ $/\$ $/\$
22	12	nnred 11035	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
23	12	nnnn0d 11351	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
24	23	nn0ge0d 11354	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
25	22, 24	absidd 14161	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
26	25	eqcomd 2628	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
27	26	oveq1d 6665	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
28	12	nncnd 11036	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
29		qre 11793	. . . . . . . . . . . . 13
30	17, 29	syl 17	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
31		rpre 11839	. . . . . . . . . . . . 13
32	31	ad3antrrr 766	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
33	30, 32	resubcld 10458	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
34	33	recnd 10068	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
35	28, 34	absmuld 14193	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
36	27, 35	eqtr4d 2659	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
37		qcn 11802	. . . . . . . . . . . 12
38	17, 37	syl 17	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
39		rpcn 11841	. . . . . . . . . . . 12
40	39	ad3antrrr 766	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
41	28, 38, 40	subdid 10486	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
42	9	nncnd 11036	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
43	42, 28, 15	divcan2d 10803	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
44	28, 40	mulcomd 10061	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
45	43, 44	oveq12d 6668	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
46	41, 45	eqtrd 2656	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
47	46	fveq2d 6195	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
48	32, 22	remulcld 10070	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
49	48	recnd 10068	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
50	42, 49	abssubd 14192	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
51	36, 47, 50	3eqtrd 2660	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
52	9	nnred 11035	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
53	48, 52	resubcld 10458	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
54	53	recnd 10068	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
55	54	abscld 14175	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
56		simpllr 799	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
57	56	rprecred 11883	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
58	56	rpreccld 11882	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
59	58	rpge0d 11876	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
60	57, 59, 4	syl2anc 693	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
61	60, 5	syl 17	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
62	61	nnrpd 11870	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
63	62, 19	ifcld 4131	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
64	63	rprecred 11883	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
65	56	rpred 11872	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
66	22, 65	remulcld 10070	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
67		simpr 477	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
68	58	rprecred 11883	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
69	61	nnred 11035	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
70	69, 22	ifcld 4131	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
71		fllep1 12602	. . . . . . . . . . . 12
72	57, 71	syl 17	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
73		max2 12018	. . . . . . . . . . . 12 $/\$
74	22, 69, 73	syl2anc 693	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
75	57, 69, 70, 72, 74	letrd 10194	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
76	58, 63	lerecd 11891	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
77	75, 76	mpbid 222	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
78	65	recnd 10068	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
79	56	rpne0d 11877	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
80	78, 79	recrecd 10798	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
81	78	mulid2d 10058	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
82	80, 81	eqtr4d 2659	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
83	12	nnge1d 11063	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
84		1red 10055	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
85	84, 22, 56	lemul1d 11915	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
86	83, 85	mpbid 222	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
87	82, 86	eqbrtrd 4675	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
88	64, 68, 66, 77, 87	letrd 10194	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
89	55, 64, 66, 67, 88	ltletrd 10197	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
90	51, 89	eqbrtrd 4675	. . . . . 6 $/\$ $/\$ $/\$ $/\$
91	34	abscld 14175	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
92	12	nngt0d 11064	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
93		ltmul2 10874	. . . . . . 7 $/\$ $/\$ $/\$
94	91, 65, 22, 92, 93	syl112anc 1330	. . . . . 6 $/\$ $/\$ $/\$ $/\$
95	90, 94	mpbird 247	. . . . 5 $/\$ $/\$ $/\$ $/\$
96	22, 22	remulcld 10070	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
97	22, 15	msqgt0d 10595	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
98	97	gt0ne0d 10592	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
99	96, 98	rereccld 10852	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
100		qdencl 15449	. . . . . . . . . . 11 denom
101	17, 100	syl 17	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ denom
102	101	nnred 11035	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ denom
103	102, 102	remulcld 10070	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ denom denom
104	101	nnne0d 11065	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ denom
105	102, 104	msqgt0d 10595	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ denom denom
106	105	gt0ne0d 10592	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ denom denom
107	103, 106	rereccld 10852	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ denom denom
108	22, 15	rereccld 10852	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
109		max1 12016	. . . . . . . . . . . 12 $/\$
110	22, 69, 109	syl2anc 693	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
111	19, 63	lerecd 11891	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
112	110, 111	mpbid 222	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
113	55, 64, 108, 67, 112	ltletrd 10197	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
114	28, 28, 28, 15, 15	divdiv1d 10832	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
115	28, 15	dividd 10799	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
116	115	oveq1d 6665	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
117	96	recnd 10068	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
118	28, 117, 98	divrecd 10804	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
119	114, 116, 118	3eqtr3rd 2665	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
120	113, 51, 119	3brtr4d 4685	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
121		ltmul2 10874	. . . . . . . . 9 $/\$ $/\$ $/\$
122	91, 99, 22, 92, 121	syl112anc 1330	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
123	120, 122	mpbird 247	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
124	9	nnzd 11481	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
125		divdenle 15457	. . . . . . . . . 10 $/\$ denom
126	124, 12, 125	syl2anc 693	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ denom
127	101	nnnn0d 11351	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ denom
128	127	nn0ge0d 11354	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ denom
129		le2msq 10923	. . . . . . . . . 10 denom $/\$ denom $/\$ $/\$ denom denom denom
130	102, 128, 22, 24, 129	syl22anc 1327	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ denom denom denom
131	126, 130	mpbid 222	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ denom denom
132		lerec 10906	. . . . . . . . 9 denom denom $/\$ denom denom $/\$ $/\$ denom denom denom denom
133	103, 105, 96, 97, 132	syl22anc 1327	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ denom denom denom denom
134	131, 133	mpbid 222	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ denom denom
135	91, 99, 107, 123, 134	ltletrd 10197	. . . . . 6 $/\$ $/\$ $/\$ $/\$ denom denom
136	101	nncnd 11036	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ denom
137		2nn0 11309	. . . . . . . 8
138		expneg 12868	. . . . . . . 8 denom $/\$ denom denom
139	136, 137, 138	sylancl 694	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ denom denom
140	136	sqvald 13005	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ denom denom denom
141	140	oveq2d 6666	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ denom denom denom
142	139, 141	eqtrd 2656	. . . . . 6 $/\$ $/\$ $/\$ $/\$ denom denom denom
143	135, 142	breqtrrd 4681	. . . . 5 $/\$ $/\$ $/\$ $/\$ denom
144		breq2 4657	. . . . . . 7
145		oveq1 6657	. . . . . . . . 9
146	145	fveq2d 6195	. . . . . . . 8
147	146	breq1d 4663	. . . . . . 7
148		fveq2 6191	. . . . . . . . 9 denom denom
149	148	oveq1d 6665	. . . . . . . 8 denom denom
150	146, 149	breq12d 4666	. . . . . . 7 denom denom
151	144, 147, 150	3anbi123d 1399	. . . . . 6 $/\$ $/\$ denom $/\$ $/\$ denom
152	151	rspcev 3309	. . . . 5 $/\$ $/\$ $/\$ denom $/\$ $/\$ denom
153	17, 21, 95, 143, 152	syl13anc 1328	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ denom
154	153	ex 450	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ denom
155	154	rexlimdvva 3038	. 2 $/\$ $/\$ $/\$ denom
156	8, 155	mpd 15	1 $/\$ $/\$ $/\$ denom

Colors of variables: wff setvar class

Syntax hints:

wi 4

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

wceq 1483

wcel 1990

wne 2794

wrex 2913

cif 4086 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

cq 11788

crp 11832

cfl 12591

cexp 12860

cabs 13974 denomcdenom 15442

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-rep 4771 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-int 4476 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-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-card 8765 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-xnn0 11364 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-ico 12181 df-fz 12327 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-dvds 14984 df-gcd 15217 df-numer 15443 df-denom 15444

This theorem is referenced by: irrapxlem6 37391

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