expmulnbnd - Metamath Proof Explorer

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Theorem expmulnbnd 12996

Description: Exponentiation with a mantissa greater than 1 is not bounded by any linear function. (Contributed by Mario Carneiro, 31-Mar-2015.)

**Assertion**
Ref	Expression
expmulnbnd	$/\$ $/\$

Distinct variable groups:

Proof of Theorem expmulnbnd Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2re 11090	. . . . 5
2		simp1 1061	. . . . 5 $/\$ $/\$
3		remulcl 10021	. . . . 5 $/\$
4	1, 2, 3	sylancr 695	. . . 4 $/\$ $/\$
5		simp3 1063	. . . . 5 $/\$ $/\$
6		1re 10039	. . . . . 6
7		simp2 1062	. . . . . 6 $/\$ $/\$
8		difrp 11868	. . . . . 6 $/\$
9	6, 7, 8	sylancr 695	. . . . 5 $/\$ $/\$
10	5, 9	mpbid 222	. . . 4 $/\$ $/\$
11	4, 10	rerpdivcld 11903	. . 3 $/\$ $/\$
12		expnbnd 12993	. . 3 $/\$ $/\$
13	11, 7, 5, 12	syl3anc 1326	. 2 $/\$ $/\$
14		2nn0 11309	. . . 4
15		nnnn0 11299	. . . . 5
16	15	ad2antrl 764	. . . 4 $/\$ $/\$ $/\$ $/\$
17		nn0mulcl 11329	. . . 4 $/\$
18	14, 16, 17	sylancr 695	. . 3 $/\$ $/\$ $/\$ $/\$
19	2	ad2antrr 762	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
20		2nn 11185	. . . . . . . . 9
21		simprl 794	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
22		nnmulcl 11043	. . . . . . . . 9 $/\$
23	20, 21, 22	sylancr 695	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
24		eluznn 11758	. . . . . . . 8 $/\$
25	23, 24	sylan 488	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
26	25	nnred 11035	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
27	19, 26	remulcld 10070	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
28		0re 10040	. . . . . . . 8
29		ifcl 4130	. . . . . . . 8 $/\$
30	19, 28, 29	sylancl 694	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
31		remulcl 10021	. . . . . . 7 $/\$
32	1, 30, 31	sylancr 695	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
33		simplrl 800	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
34	33	nnred 11035	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
35	26, 34	resubcld 10458	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
36	32, 35	remulcld 10070	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
37	7	ad2antrr 762	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
38	25	nnnn0d 11351	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
39		reexpcl 12877	. . . . . 6 $/\$
40	37, 38, 39	syl2anc 693	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
41		remulcl 10021	. . . . . . . 8 $/\$
42	1, 35, 41	sylancr 695	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
43	38	nn0ge0d 11354	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
44		max1 12016	. . . . . . . 8 $/\$
45	28, 19, 44	sylancr 695	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
46		remulcl 10021	. . . . . . . . . . . 12 $/\$
47	1, 34, 46	sylancr 695	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
48		eluzle 11700	. . . . . . . . . . . 12
49	48	adantl 482	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
50	47, 26, 26, 49	leadd2dd 10642	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
51	26	recnd 10068	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
52	51	2timesd 11275	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
53	50, 52	breqtrrd 4681	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
54		remulcl 10021	. . . . . . . . . . 11 $/\$
55	1, 26, 54	sylancr 695	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
56		leaddsub 10504	. . . . . . . . . 10 $/\$ $/\$
57	26, 47, 55, 56	syl3anc 1326	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
58	53, 57	mpbid 222	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
59		2cnd 11093	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
60	34	recnd 10068	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
61	59, 51, 60	subdid 10486	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
62	58, 61	breqtrrd 4681	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
63		max2 12018	. . . . . . . 8 $/\$
64	28, 19, 63	sylancr 695	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
65	26, 42, 19, 30, 43, 45, 62, 64	lemul12bd 10967	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
66	19	recnd 10068	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
67	66, 51	mulcomd 10061	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
68	30	recnd 10068	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
69	35	recnd 10068	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
70	59, 68, 69	mul32d 10246	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
71	65, 67, 70	3brtr4d 4685	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
72	10	ad2antrr 762	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
73	72	rpred 11872	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
74	73, 35	remulcld 10070	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
75	33	nnnn0d 11351	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
76		reexpcl 12877	. . . . . . . 8 $/\$
77	37, 75, 76	syl2anc 693	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
78	74, 77	remulcld 10070	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
79		simplrr 801	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
80	1, 19, 3	sylancr 695	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
81	80, 77, 72	ltdivmuld 11923	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
82	79, 81	mpbid 222	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
83	5	ad2antrr 762	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
84		posdif 10521	. . . . . . . . . . . 12 $/\$
85	6, 37, 84	sylancr 695	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
86	83, 85	mpbid 222	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
87	33	nnzd 11481	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
88	28	a1i 11	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
89	6	a1i 11	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
90		0lt1 10550	. . . . . . . . . . . . 13
91	90	a1i 11	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
92	88, 89, 37, 91, 83	lttrd 10198	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
93		expgt0 12893	. . . . . . . . . . 11 $/\$ $/\$
94	37, 87, 92, 93	syl3anc 1326	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
95	73, 77, 86, 94	mulgt0d 10192	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
96		oveq2 6658	. . . . . . . . . . 11
97	96	breq1d 4663	. . . . . . . . . 10
98		2t0e0 11183	. . . . . . . . . . . 12
99		oveq2 6658	. . . . . . . . . . . 12
100	98, 99	syl5eqr 2670	. . . . . . . . . . 11
101	100	breq1d 4663	. . . . . . . . . 10
102	97, 101	ifboth 4124	. . . . . . . . 9 $/\$
103	82, 95, 102	syl2anc 693	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
104	73, 77	remulcld 10070	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
105		simpr 477	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
106	60	2timesd 11275	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
107	106	fveq2d 6195	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
108	105, 107	eleqtrd 2703	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
109		eluzsub 11717	. . . . . . . . . . . 12 $/\$ $/\$
110	87, 87, 108, 109	syl3anc 1326	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
111		eluznn 11758	. . . . . . . . . . 11 $/\$
112	33, 110, 111	syl2anc 693	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
113	112	nngt0d 11064	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
114		ltmul1 10873	. . . . . . . . 9 $/\$ $/\$ $/\$
115	32, 104, 35, 113, 114	syl112anc 1330	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
116	103, 115	mpbid 222	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
117	73	recnd 10068	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
118	77	recnd 10068	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
119	117, 118, 69	mul32d 10246	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
120	116, 119	breqtrd 4679	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
121		peano2re 10209	. . . . . . . . . 10
122	74, 121	syl 17	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
123	112	nnnn0d 11351	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
124		reexpcl 12877	. . . . . . . . . 10 $/\$
125	37, 123, 124	syl2anc 693	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
126	74	ltp1d 10954	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
127	88, 37, 92	ltled 10185	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
128		bernneq2 12991	. . . . . . . . . 10 $/\$ $/\$
129	37, 123, 127, 128	syl3anc 1326	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
130	74, 122, 125, 126, 129	ltletrd 10197	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
131	37	recnd 10068	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
132	92	gt0ne0d 10592	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
133		eluzelz 11697	. . . . . . . . . 10
134	133	adantl 482	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
135		expsub 12908	. . . . . . . . 9 $/\$ $/\$ $/\$
136	131, 132, 134, 87, 135	syl22anc 1327	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
137	130, 136	breqtrd 4679	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
138		ltmuldiv 10896	. . . . . . . 8 $/\$ $/\$ $/\$
139	74, 40, 77, 94, 138	syl112anc 1330	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
140	137, 139	mpbird 247	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
141	36, 78, 40, 120, 140	lttrd 10198	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
142	27, 36, 40, 71, 141	lelttrd 10195	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
143	142	ralrimiva 2966	. . 3 $/\$ $/\$ $/\$ $/\$
144		fveq2 6191	. . . . 5
145	144	raleqdv 3144	. . . 4
146	145	rspcev 3309	. . 3 $/\$
147	18, 143, 146	syl2anc 693	. 2 $/\$ $/\$ $/\$ $/\$
148	13, 147	rexlimddv 3035	1 $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

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

wceq 1483

wcel 1990

wne 2794

wral 2912

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

cdiv 10684

cn 11020

c2 11070

cn0 11292

cz 11377

cuz 11687

crp 11832

cexp 12860

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

This theorem is referenced by: geomulcvg 14607

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