irrapxlem4 - Mathbox for Stefan O'Rear

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Theorem irrapxlem4 37389

Description: Lemma for irrapx1 37392. Eliminate ranges, use positivity of the input to force positivity of the output by increasing

as needed. (Contributed by Stefan O'Rear, 13-Sep-2014.)

**Assertion**
Ref	Expression
irrapxlem4	$/\$

Distinct variable groups:

Proof of Theorem irrapxlem4 Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfznn 12370	. . . 4
2	1	ad3antlr 767	. . 3 $/\$ $/\$ $/\$ $/\$
3		nn0z 11400	. . . . 5
4	3	ad2antlr 763	. . . 4 $/\$ $/\$ $/\$ $/\$
5		simpl 473	. . . . . . . . . . . . . 14 $/\$
6	5	ad3antrrr 766	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
7	6	rpred 11872	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
8	2	nnred 11035	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
9	7, 8	remulcld 10070	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
10		nn0re 11301	. . . . . . . . . . . 12
11	10	ad2antlr 763	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
12	9, 11	resubcld 10458	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
13	12	recnd 10068	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
14	13	abscld 14175	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
15	5	rpreccld 11882	. . . . . . . . . . . . 13 $/\$
16	15	rprege0d 11879	. . . . . . . . . . . 12 $/\$ $/\$
17		flge0nn0 12621	. . . . . . . . . . . 12 $/\$
18		nn0p1nn 11332	. . . . . . . . . . . 12
19	16, 17, 18	3syl 18	. . . . . . . . . . 11 $/\$
20	19	ad3antrrr 766	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
21		simpr 477	. . . . . . . . . . 11 $/\$
22	21	ad3antrrr 766	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
23	20, 22	ifcld 4131	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
24	23	nnrecred 11066	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
25		0red 10041	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
26	9, 25	resubcld 10458	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
27		simpr 477	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
28	20	nnrecred 11066	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
29	22	nnred 11035	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
30	6	rprecred 11883	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
31	30	flcld 12599	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
32	31	zred 11482	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
33		peano2re 10209	. . . . . . . . . . . . 13
34	32, 33	syl 17	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
35		max2 12018	. . . . . . . . . . . 12 $/\$
36	29, 34, 35	syl2anc 693	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
37	20	nngt0d 11064	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
38	23	nnred 11035	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
39	23	nngt0d 11064	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
40		lerec 10906	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
41	34, 37, 38, 39, 40	syl22anc 1327	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
42	36, 41	mpbid 222	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
43		fllep1 12602	. . . . . . . . . . . . 13
44	30, 43	syl 17	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
45	20	nncnd 11036	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
46	20	nnne0d 11065	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
47	45, 46	recrecd 10798	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
48	44, 47	breqtrrd 4681	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
49	34, 37	recgt0d 10958	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
50	6	rpgt0d 11875	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
51		lerec 10906	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
52	28, 49, 7, 50, 51	syl22anc 1327	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
53	48, 52	mpbird 247	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
54	24, 28, 7, 42, 53	letrd 10194	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
55	7	recnd 10068	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
56	55	mulid1d 10057	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
57	2	nnge1d 11063	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
58		1red 10055	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
59	58, 8, 6	lemul2d 11916	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
60	57, 59	mpbid 222	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
61	56, 60	eqbrtrrd 4677	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
62	9	recnd 10068	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
63	62	subid1d 10381	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
64	61, 63	breqtrrd 4681	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
65	24, 7, 26, 54, 64	letrd 10194	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
66	14, 24, 26, 27, 65	ltletrd 10197	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
67	12, 26	absltd 14168	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
68	66, 67	mpbid 222	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
69	68	simprd 479	. . . . 5 $/\$ $/\$ $/\$ $/\$
70	25, 11, 9	ltsub2d 10637	. . . . 5 $/\$ $/\$ $/\$ $/\$
71	69, 70	mpbird 247	. . . 4 $/\$ $/\$ $/\$ $/\$
72		elnnz 11387	. . . 4 $/\$
73	4, 71, 72	sylanbrc 698	. . 3 $/\$ $/\$ $/\$ $/\$
74	22, 2	ifcld 4131	. . . . 5 $/\$ $/\$ $/\$ $/\$
75	74	nnrecred 11066	. . . 4 $/\$ $/\$ $/\$ $/\$
76		elfzle2 12345	. . . . . . 7
77	76	ad3antlr 767	. . . . . 6 $/\$ $/\$ $/\$ $/\$
78		max1 12016	. . . . . . 7 $/\$
79	29, 34, 78	syl2anc 693	. . . . . 6 $/\$ $/\$ $/\$ $/\$
80		maxle 12022	. . . . . . 7 $/\$ $/\$ $/\$
81	8, 29, 38, 80	syl3anc 1326	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
82	77, 79, 81	mpbir2and 957	. . . . 5 $/\$ $/\$ $/\$ $/\$
83	29, 8	ifcld 4131	. . . . . 6 $/\$ $/\$ $/\$ $/\$
84	22	nngt0d 11064	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
85		max2 12018	. . . . . . . 8 $/\$
86	8, 29, 85	syl2anc 693	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
87	25, 29, 83, 84, 86	ltletrd 10197	. . . . . 6 $/\$ $/\$ $/\$ $/\$
88		lerec 10906	. . . . . 6 $/\$ $/\$ $/\$
89	83, 87, 38, 39, 88	syl22anc 1327	. . . . 5 $/\$ $/\$ $/\$ $/\$
90	82, 89	mpbid 222	. . . 4 $/\$ $/\$ $/\$ $/\$
91	14, 24, 75, 27, 90	ltletrd 10197	. . 3 $/\$ $/\$ $/\$ $/\$
92		oveq2 6658	. . . . . . 7
93	92	oveq1d 6665	. . . . . 6
94	93	fveq2d 6195	. . . . 5
95		breq1 4656	. . . . . . 7
96		id 22	. . . . . . 7
97	95, 96	ifbieq2d 4111	. . . . . 6
98	97	oveq2d 6666	. . . . 5
99	94, 98	breq12d 4666	. . . 4
100		oveq2 6658	. . . . . 6
101	100	fveq2d 6195	. . . . 5
102	101	breq1d 4663	. . . 4
103	99, 102	rspc2ev 3324	. . 3 $/\$ $/\$
104	2, 73, 91, 103	syl3anc 1326	. 2 $/\$ $/\$ $/\$ $/\$
105	19, 21	ifcld 4131	. . 3 $/\$
106		irrapxlem3 37388	. . 3 $/\$
107	5, 105, 106	syl2anc 693	. 2 $/\$
108	104, 107	r19.29vva 3081	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384

wcel 1990

wrex 2913

cif 4086 class class class wbr 4653

cfv 5888 (class class class)co 6650

cr 9935

cc0 9936

c1 9937

caddc 9939

cmul 9941

clt 10074

cle 10075

cmin 10266

cneg 10267

cdiv 10684

cn 11020

cn0 11292

cz 11377

crp 11832

cfz 12326

cfl 12591

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-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-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

This theorem is referenced by: irrapxlem5 37390

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