aaliou3lem8 - Metamath Proof Explorer

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Theorem aaliou3lem8 24100

Description: Lemma for aaliou3 24106. (Contributed by Stefan O'Rear, 20-Nov-2014.)

**Assertion**
Ref	Expression
aaliou3lem8	$/\$

Distinct variable groups:

Proof of Theorem aaliou3lem8 Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2rp 11837	. . . . . 6
2		rpdivcl 11856	. . . . . 6 $/\$
3	1, 2	mpan 706	. . . . 5
4	3	rpred 11872	. . . 4
5		2re 11090	. . . . 5
6		1lt2 11194	. . . . 5
7		expnbnd 12993	. . . . 5 $/\$ $/\$
8	5, 6, 7	mp3an23 1416	. . . 4
9	4, 8	syl 17	. . 3
10	9	adantl 482	. 2 $/\$
11		simprl 794	. . . 4 $/\$ $/\$ $/\$
12		simpll 790	. . . 4 $/\$ $/\$ $/\$
13		nnaddm1cl 11434	. . . 4 $/\$
14	11, 12, 13	syl2anc 693	. . 3 $/\$ $/\$ $/\$
15		simplr 792	. . . . . . . . . 10 $/\$ $/\$ $/\$
16		rerpdivcl 11861	. . . . . . . . . 10 $/\$
17	5, 15, 16	sylancr 695	. . . . . . . . 9 $/\$ $/\$ $/\$
18	11	nnnn0d 11351	. . . . . . . . . 10 $/\$ $/\$ $/\$
19		reexpcl 12877	. . . . . . . . . 10 $/\$
20	5, 18, 19	sylancr 695	. . . . . . . . 9 $/\$ $/\$ $/\$
21	11, 12	nnaddcld 11067	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$
22		nnm1nn0 11334	. . . . . . . . . . . . . . . 16
23	21, 22	syl 17	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
24		peano2nn0 11333	. . . . . . . . . . . . . . 15
25	23, 24	syl 17	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
26		faccl 13070	. . . . . . . . . . . . . 14
27	25, 26	syl 17	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
28	27	nnzd 11481	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
29		faccl 13070	. . . . . . . . . . . . . . 15
30	23, 29	syl 17	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
31	30	nnzd 11481	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
32	12	nnzd 11481	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
33	31, 32	zmulcld 11488	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
34	28, 33	zsubcld 11487	. . . . . . . . . . 11 $/\$ $/\$ $/\$
35		rpexpcl 12879	. . . . . . . . . . 11 $/\$
36	1, 34, 35	sylancr 695	. . . . . . . . . 10 $/\$ $/\$ $/\$
37	36	rpred 11872	. . . . . . . . 9 $/\$ $/\$ $/\$
38		simprr 796	. . . . . . . . . 10 $/\$ $/\$ $/\$
39	17, 20, 38	ltled 10185	. . . . . . . . 9 $/\$ $/\$ $/\$
40	5	a1i 11	. . . . . . . . . 10 $/\$ $/\$ $/\$
41		1le2 11241	. . . . . . . . . . 11
42	41	a1i 11	. . . . . . . . . 10 $/\$ $/\$ $/\$
43	11	nnred 11035	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
44	30	nnred 11035	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
45	18	nn0ge0d 11354	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
46	30	nnge1d 11063	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
47	43, 44, 45, 46	lemulge12d 10962	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
48		facp1 13065	. . . . . . . . . . . . . . 15
49	23, 48	syl 17	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
50	49	oveq1d 6665	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
51	30	nncnd 11036	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
52	25	nn0cnd 11353	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
53	12	nncnd 11036	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
54	51, 52, 53	subdid 10486	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
55	21	nncnd 11036	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$
56		1cnd 10056	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$
57	55, 56	npcand 10396	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$
58	57	oveq1d 6665	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
59	11	nncnd 11036	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$
60	59, 53	pncand 10393	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
61	58, 60	eqtrd 2656	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
62	61	oveq2d 6666	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
63	50, 54, 62	3eqtr2d 2662	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
64	47, 63	breqtrrd 4681	. . . . . . . . . . 11 $/\$ $/\$ $/\$
65	11	nnzd 11481	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
66		eluz 11701	. . . . . . . . . . . 12 $/\$
67	65, 34, 66	syl2anc 693	. . . . . . . . . . 11 $/\$ $/\$ $/\$
68	64, 67	mpbird 247	. . . . . . . . . 10 $/\$ $/\$ $/\$
69	40, 42, 68	leexp2ad 13041	. . . . . . . . 9 $/\$ $/\$ $/\$
70	17, 20, 37, 39, 69	letrd 10194	. . . . . . . 8 $/\$ $/\$ $/\$
71		rpcnne0 11850	. . . . . . . . . . 11 $/\$
72	1, 71	mp1i 13	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
73		expsub 12908	. . . . . . . . . 10 $/\$ $/\$ $/\$
74	72, 28, 33, 73	syl12anc 1324	. . . . . . . . 9 $/\$ $/\$ $/\$
75		2cn 11091	. . . . . . . . . . . 12
76	75	a1i 11	. . . . . . . . . . 11 $/\$ $/\$ $/\$
77	12	nnnn0d 11351	. . . . . . . . . . 11 $/\$ $/\$ $/\$
78	30	nnnn0d 11351	. . . . . . . . . . 11 $/\$ $/\$ $/\$
79	76, 77, 78	expmuld 13011	. . . . . . . . . 10 $/\$ $/\$ $/\$
80	79	oveq2d 6666	. . . . . . . . 9 $/\$ $/\$ $/\$
81		rpexpcl 12879	. . . . . . . . . . . 12 $/\$
82	1, 28, 81	sylancr 695	. . . . . . . . . . 11 $/\$ $/\$ $/\$
83	82	rpcnd 11874	. . . . . . . . . 10 $/\$ $/\$ $/\$
84		rpexpcl 12879	. . . . . . . . . . . . 13 $/\$
85	1, 31, 84	sylancr 695	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
86	85, 32	rpexpcld 13032	. . . . . . . . . . 11 $/\$ $/\$ $/\$
87	86	rpcnd 11874	. . . . . . . . . 10 $/\$ $/\$ $/\$
88	86	rpne0d 11877	. . . . . . . . . 10 $/\$ $/\$ $/\$
89	83, 87, 88	divrecd 10804	. . . . . . . . 9 $/\$ $/\$ $/\$
90	74, 80, 89	3eqtrrd 2661	. . . . . . . 8 $/\$ $/\$ $/\$
91	70, 90	breqtrrd 4681	. . . . . . 7 $/\$ $/\$ $/\$
92	86	rpreccld 11882	. . . . . . . . . 10 $/\$ $/\$ $/\$
93	82, 92	rpmulcld 11888	. . . . . . . . 9 $/\$ $/\$ $/\$
94	93	rpred 11872	. . . . . . . 8 $/\$ $/\$ $/\$
95	40, 94, 15	ledivmuld 11925	. . . . . . 7 $/\$ $/\$ $/\$
96	91, 95	mpbid 222	. . . . . 6 $/\$ $/\$ $/\$
97	15	rpcnd 11874	. . . . . . 7 $/\$ $/\$ $/\$
98	92	rpcnd 11874	. . . . . . 7 $/\$ $/\$ $/\$
99	97, 83, 98	mul12d 10245	. . . . . 6 $/\$ $/\$ $/\$
100	96, 99	breqtrd 4679	. . . . 5 $/\$ $/\$ $/\$
101	15, 92	rpmulcld 11888	. . . . . . 7 $/\$ $/\$ $/\$
102	101	rpred 11872	. . . . . 6 $/\$ $/\$ $/\$
103	40, 102, 82	ledivmuld 11925	. . . . 5 $/\$ $/\$ $/\$
104	100, 103	mpbird 247	. . . 4 $/\$ $/\$ $/\$
105	27	nnnn0d 11351	. . . . . . 7 $/\$ $/\$ $/\$
106		expneg 12868	. . . . . . 7 $/\$
107	75, 105, 106	sylancr 695	. . . . . 6 $/\$ $/\$ $/\$
108	107	oveq2d 6666	. . . . 5 $/\$ $/\$ $/\$
109	82	rpne0d 11877	. . . . . 6 $/\$ $/\$ $/\$
110	76, 83, 109	divrecd 10804	. . . . 5 $/\$ $/\$ $/\$
111	108, 110	eqtr4d 2659	. . . 4 $/\$ $/\$ $/\$
112	97, 87, 88	divrecd 10804	. . . 4 $/\$ $/\$ $/\$
113	104, 111, 112	3brtr4d 4685	. . 3 $/\$ $/\$ $/\$
114		oveq1 6657	. . . . . . . . 9
115	114	fveq2d 6195	. . . . . . . 8
116	115	negeqd 10275	. . . . . . 7
117	116	oveq2d 6666	. . . . . 6
118	117	oveq2d 6666	. . . . 5
119		fveq2 6191	. . . . . . . 8
120	119	oveq2d 6666	. . . . . . 7
121	120	oveq1d 6665	. . . . . 6
122	121	oveq2d 6666	. . . . 5
123	118, 122	breq12d 4666	. . . 4
124	123	rspcev 3309	. . 3 $/\$
125	14, 113, 124	syl2anc 693	. 2 $/\$ $/\$ $/\$
126	10, 125	rexlimddv 3035	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384

wceq 1483

wcel 1990

wne 2794

wrex 2913 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

cuz 11687

crp 11832

cexp 12860

cfa 13060

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-inf 8349 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-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fl 12593 df-seq 12802 df-exp 12861 df-fac 13061

This theorem is referenced by: aaliou3lem9 24105

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