dvexp - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > dvexp		Unicode version

Theorem dvexp 13469

Description: Derivative of a power function. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

**Assertion**
Ref	Expression
dvexp

Distinct variable group:

Proof of Theorem dvexp Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 5861	. . . . 5
2	1	mpteq2dv 4080	. . . 4
3	2	oveq2d 5869	. . 3
4		id 19	. . . . 5
5		oveq1 5860	. . . . . 6
6	5	oveq2d 5869	. . . . 5
7	4, 6	oveq12d 5871	. . . 4
8	7	mpteq2dv 4080	. . 3
9	3, 8	eqeq12d 2185	. 2
10		oveq2 5861	. . . . 5
11	10	mpteq2dv 4080	. . . 4
12	11	oveq2d 5869	. . 3
13		id 19	. . . . 5
14		oveq1 5860	. . . . . 6
15	14	oveq2d 5869	. . . . 5
16	13, 15	oveq12d 5871	. . . 4
17	16	mpteq2dv 4080	. . 3
18	12, 17	eqeq12d 2185	. 2
19		oveq2 5861	. . . . 5
20	19	mpteq2dv 4080	. . . 4
21	20	oveq2d 5869	. . 3
22		id 19	. . . . 5
23		oveq1 5860	. . . . . 6
24	23	oveq2d 5869	. . . . 5
25	22, 24	oveq12d 5871	. . . 4
26	25	mpteq2dv 4080	. . 3
27	21, 26	eqeq12d 2185	. 2
28		oveq2 5861	. . . . 5
29	28	mpteq2dv 4080	. . . 4
30	29	oveq2d 5869	. . 3
31		id 19	. . . . 5
32		oveq1 5860	. . . . . 6
33	32	oveq2d 5869	. . . . 5
34	31, 33	oveq12d 5871	. . . 4
35	34	mpteq2dv 4080	. . 3
36	30, 35	eqeq12d 2185	. 2
37		exp1 10482	. . . . . 6
38	37	mpteq2ia 4075	. . . . 5
39		mptresid 4945	. . . . 5
40	38, 39	eqtri 2191	. . . 4
41	40	oveq2i 5864	. . 3
42		1m1e0 8947	. . . . . . . . . 10
43	42	oveq2i 5864	. . . . . . . . 9
44		exp0 10480	. . . . . . . . 9
45	43, 44	eqtrid 2215	. . . . . . . 8
46	45	oveq2d 5869	. . . . . . 7
47		1t1e1 9030	. . . . . . 7
48	46, 47	eqtrdi 2219	. . . . . 6
49	48	mpteq2ia 4075	. . . . 5
50		fconstmpt 4658	. . . . 5
51	49, 50	eqtr4i 2194	. . . 4
52		dvid 13456	. . . 4
53	51, 52	eqtr4i 2194	. . 3
54	41, 53	eqtr4i 2194	. 2
55		nncn 8886	. . . . . . . . . . . 12
56	55	adantr 274	. . . . . . . . . . 11 $/\$
57		ax-1cn 7867	. . . . . . . . . . 11
58		pncan 8125	. . . . . . . . . . 11 $/\$
59	56, 57, 58	sylancl 411	. . . . . . . . . 10 $/\$
60	59	oveq2d 5869	. . . . . . . . 9 $/\$
61	60	oveq2d 5869	. . . . . . . 8 $/\$
62	57	a1i 9	. . . . . . . . 9 $/\$
63		id 19	. . . . . . . . . 10
64		nnnn0 9142	. . . . . . . . . 10
65		expcl 10494	. . . . . . . . . 10 $/\$
66	63, 64, 65	syl2anr 288	. . . . . . . . 9 $/\$
67	56, 62, 66	adddird 7945	. . . . . . . 8 $/\$
68	66	mulid2d 7938	. . . . . . . . 9 $/\$
69	68	oveq2d 5869	. . . . . . . 8 $/\$
70	61, 67, 69	3eqtrd 2207	. . . . . . 7 $/\$
71	70	mpteq2dva 4079	. . . . . 6
72		cnex 7898	. . . . . . . 8
73	72	a1i 9	. . . . . . 7
74	56, 66	mulcld 7940	. . . . . . 7 $/\$
75		nnm1nn0 9176	. . . . . . . . . . 11
76		expcl 10494	. . . . . . . . . . 11 $/\$
77	63, 75, 76	syl2anr 288	. . . . . . . . . 10 $/\$
78	56, 77	mulcld 7940	. . . . . . . . 9 $/\$
79		simpr 109	. . . . . . . . 9 $/\$
80		eqidd 2171	. . . . . . . . 9
81	39	eqcomi 2174	. . . . . . . . . 10
82	81	a1i 9	. . . . . . . . 9
83	73, 78, 79, 80, 82	offval2 6076	. . . . . . . 8
84	56, 77, 79	mulassd 7943	. . . . . . . . . 10 $/\$
85		expm1t 10504	. . . . . . . . . . . 12 $/\$
86	85	ancoms 266	. . . . . . . . . . 11 $/\$
87	86	oveq2d 5869	. . . . . . . . . 10 $/\$
88	84, 87	eqtr4d 2206	. . . . . . . . 9 $/\$
89	88	mpteq2dva 4079	. . . . . . . 8
90	83, 89	eqtrd 2203	. . . . . . 7
91	52, 50	eqtri 2191	. . . . . . . . . 10
92	91	a1i 9	. . . . . . . . 9
93		eqidd 2171	. . . . . . . . 9
94	73, 62, 66, 92, 93	offval2 6076	. . . . . . . 8
95	68	mpteq2dva 4079	. . . . . . . 8
96	94, 95	eqtrd 2203	. . . . . . 7
97	73, 74, 66, 90, 96	offval2 6076	. . . . . 6
98	71, 97	eqtr4d 2206	. . . . 5
99		oveq1 5860	. . . . . . 7
100	99	oveq1d 5868	. . . . . 6
101	100	eqcomd 2176	. . . . 5
102	98, 101	sylan9eq 2223	. . . 4 $/\$
103		cnelprrecn 7910	. . . . . 6
104	103	a1i 9	. . . . 5 $/\$
105		ssidd 3168	. . . . 5 $/\$
106	66	fmpttd 5651	. . . . . 6
107	106	adantr 274	. . . . 5 $/\$
108		f1oi 5480	. . . . . 6
109		f1of 5442	. . . . . 6
110	108, 109	mp1i 10	. . . . 5 $/\$
111		simpr 109	. . . . . . 7 $/\$
112	111	dmeqd 4813	. . . . . 6 $/\$
113	78	fmpttd 5651	. . . . . . . 8
114	113	adantr 274	. . . . . . 7 $/\$
115	114	fdmd 5354	. . . . . 6 $/\$
116	112, 115	eqtrd 2203	. . . . 5 $/\$
117		1ex 7915	. . . . . . . . 9
118	117	fconst 5393	. . . . . . . 8
119	52	feq1i 5340	. . . . . . . 8
120	118, 119	mpbir 145	. . . . . . 7
121	120	fdmi 5355	. . . . . 6
122	121	a1i 9	. . . . 5 $/\$
123	104, 105, 107, 110, 116, 122	dvimulf 13464	. . . 4 $/\$
124	73, 66, 79, 93, 82	offval2 6076	. . . . . . 7
125		expp1 10483	. . . . . . . . 9 $/\$
126	63, 64, 125	syl2anr 288	. . . . . . . 8 $/\$
127	126	mpteq2dva 4079	. . . . . . 7
128	124, 127	eqtr4d 2206	. . . . . 6
129	128	oveq2d 5869	. . . . 5
130	129	adantr 274	. . . 4 $/\$
131	102, 123, 130	3eqtr2rd 2210	. . 3 $/\$
132	131	ex 114	. 2
133	9, 18, 27, 36, 54, 132	nnind 8894	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wceq 1348

wcel 2141

cvv 2730

csn 3583

cpr 3584

cmpt 4050

cid 4273

cxp 4609

cdm 4611

cres 4613

wf 5194

wf1o 5197 (class class class)co 5853

cof 6059

cc 7772

cr 7773

cc0 7774

c1 7775

caddc 7777

cmul 7779

cmin 8090

cn 8878

cn0 9135

cexp 10475

cdv 13418

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894 ax-addf 7896 ax-mulf 7897

This theorem depends on definitions: df-bi 116 df-stab 826 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-isom 5207 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-of 6061 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-map 6628 df-pm 6629 df-sup 6961 df-inf 6962 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-xneg 9729 df-xadd 9730 df-seqfrec 10402 df-exp 10476 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-rest 12581 df-topgen 12600 df-psmet 12781 df-xmet 12782 df-met 12783 df-bl 12784 df-mopn 12785 df-top 12790 df-topon 12803 df-bases 12835 df-ntr 12890 df-cn 12982 df-cnp 12983 df-tx 13047 df-cncf 13352 df-limced 13419 df-dvap 13420

This theorem is referenced by: dvexp2 13470

W3C validator