climexp - Mathbox for Glauco Siliprandi

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Theorem climexp 45163

Description: The limit of natural powers, is the natural power of the limit. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

**Hypotheses**
Ref	Expression
climexp.1	⊢ Ⅎ𝑘𝜑
climexp.2	⊢ Ⅎ𝑘𝐹
climexp.3	⊢ Ⅎ𝑘𝐻
climexp.4	⊢ 𝑍 = (ℤ_≥‘𝑀)
climexp.5	⊢ (𝜑 → 𝑀 ∈ ℤ)
climexp.6	⊢ (𝜑 → 𝐹:𝑍⟶ℂ)
climexp.7	⊢ (𝜑 → 𝐹 ⇝ 𝐴)
climexp.8	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
climexp.9	⊢ (𝜑 → 𝐻 ∈ 𝑉)
climexp.10	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘)↑𝑁))

**Assertion**
Ref	Expression
climexp	⊢ (𝜑 → 𝐻 ⇝ (𝐴↑𝑁))

Distinct variable groups: 𝑘,𝑁 𝑘,𝑍 Allowed substitution hints: 𝜑(𝑘) 𝐴(𝑘) 𝐹(𝑘) 𝐻(𝑘) 𝑀(𝑘) 𝑉(𝑘)

Proof of Theorem climexp Dummy variables 𝑗 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		climexp.4	. . . 4 ⊢ 𝑍 = (ℤ_≥‘𝑀)
2		climexp.5	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
3		climexp.8	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
4		eqid 2725	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
5	4	expcn 24873	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
6	3, 5	syl 17	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
7	4	cncfcn1 24914	. . . . 5 ⊢ (ℂ–cn→ℂ) = ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld))
8	6, 7	eleqtrrdi 2836	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (ℂ–cn→ℂ))
9		climexp.6	. . . 4 ⊢ (𝜑 → 𝐹:𝑍⟶ℂ)
10		climexp.7	. . . 4 ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
11		climcl 15496	. . . . 5 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ)
12	10, 11	syl 17	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ)
13	1, 2, 8, 9, 10, 12	climcncf 24903	. . 3 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) ⇝ ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘𝐴))
14		eqidd 2726	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)))
15		simpr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴)
16	15	oveq1d 7438	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝑥↑𝑁) = (𝐴↑𝑁))
17	12, 3	expcld 14160	. . . 4 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℂ)
18	14, 16, 12, 17	fvmptd 7015	. . 3 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘𝐴) = (𝐴↑𝑁))
19	13, 18	breqtrd 5178	. 2 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) ⇝ (𝐴↑𝑁))
20		climexp.9	. . 3 ⊢ (𝜑 → 𝐻 ∈ 𝑉)
21		cnex 11235	. . . . 5 ⊢ ℂ ∈ V
22	21	mptex 7239	. . . 4 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ V
23	1	fvexi 6914	. . . . 5 ⊢ 𝑍 ∈ V
24		fex 7242	. . . . 5 ⊢ ((𝐹:𝑍⟶ℂ ∧ 𝑍 ∈ V) → 𝐹 ∈ V)
25	9, 23, 24	sylancl 584	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
26		coexg 7941	. . . 4 ⊢ (((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ V ∧ 𝐹 ∈ V) → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) ∈ V)
27	22, 25, 26	sylancr 585	. . 3 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) ∈ V)
28		eqidd 2726	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)))
29		simpr 483	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 = (𝐹‘𝑗)) → 𝑥 = (𝐹‘𝑗))
30	29	oveq1d 7438	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 = (𝐹‘𝑗)) → (𝑥↑𝑁) = ((𝐹‘𝑗)↑𝑁))
31	9	ffvelcdmda 7097	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℂ)
32	3	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑁 ∈ ℕ₀)
33	31, 32	expcld 14160	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝐹‘𝑗)↑𝑁) ∈ ℂ)
34	28, 30, 31, 33	fvmptd 7015	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑗)) = ((𝐹‘𝑗)↑𝑁))
35		fvco3 7000	. . . . 5 ⊢ ((𝐹:𝑍⟶ℂ ∧ 𝑗 ∈ 𝑍) → (((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹)‘𝑗) = ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑗)))
36	9, 35	sylan 578	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹)‘𝑗) = ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑗)))
37		climexp.1	. . . . . . 7 ⊢ Ⅎ𝑘𝜑
38		nfv 1909	. . . . . . 7 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍
39	37, 38	nfan 1894	. . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍)
40		climexp.3	. . . . . . . 8 ⊢ Ⅎ𝑘𝐻
41		nfcv 2891	. . . . . . . 8 ⊢ Ⅎ𝑘𝑗
42	40, 41	nffv 6910	. . . . . . 7 ⊢ Ⅎ𝑘(𝐻‘𝑗)
43		climexp.2	. . . . . . . . 9 ⊢ Ⅎ𝑘𝐹
44	43, 41	nffv 6910	. . . . . . . 8 ⊢ Ⅎ𝑘(𝐹‘𝑗)
45		nfcv 2891	. . . . . . . 8 ⊢ Ⅎ𝑘↑
46		nfcv 2891	. . . . . . . 8 ⊢ Ⅎ𝑘𝑁
47	44, 45, 46	nfov 7453	. . . . . . 7 ⊢ Ⅎ𝑘((𝐹‘𝑗)↑𝑁)
48	42, 47	nfeq 2905	. . . . . 6 ⊢ Ⅎ𝑘(𝐻‘𝑗) = ((𝐹‘𝑗)↑𝑁)
49	39, 48	nfim 1891	. . . . 5 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝐹‘𝑗)↑𝑁))
50		eleq1w 2808	. . . . . . 7 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍))
51	50	anbi2d 628	. . . . . 6 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍)))
52		fveq2 6900	. . . . . . 7 ⊢ (𝑘 = 𝑗 → (𝐻‘𝑘) = (𝐻‘𝑗))
53		fveq2 6900	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗))
54	53	oveq1d 7438	. . . . . . 7 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘)↑𝑁) = ((𝐹‘𝑗)↑𝑁))
55	52, 54	eqeq12d 2741	. . . . . 6 ⊢ (𝑘 = 𝑗 → ((𝐻‘𝑘) = ((𝐹‘𝑘)↑𝑁) ↔ (𝐻‘𝑗) = ((𝐹‘𝑗)↑𝑁)))
56	51, 55	imbi12d 343	. . . . 5 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘)↑𝑁)) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝐹‘𝑗)↑𝑁))))
57		climexp.10	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘)↑𝑁))
58	49, 56, 57	chvarfv 2228	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝐹‘𝑗)↑𝑁))
59	34, 36, 58	3eqtr4rd 2776	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = (((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹)‘𝑗))
60	1, 20, 27, 2, 59	climeq 15564	. 2 ⊢ (𝜑 → (𝐻 ⇝ (𝐴↑𝑁) ↔ ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) ⇝ (𝐴↑𝑁)))
61	19, 60	mpbird 256	1 ⊢ (𝜑 → 𝐻 ⇝ (𝐴↑𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 Ⅎwnf 1777 ∈ wcel 2098 Ⅎwnfc 2875 Vcvv 3461 class class class wbr 5152 ↦ cmpt 5235 ∘ ccom 5685 ⟶wf 6549 ‘cfv 6553 (class class class)co 7423 ℂcc 11152 ℕ₀cn0 12519 ℤcz 12605 ℤ_≥cuz 12869 ↑cexp 14076 ⇝ cli 15481 TopOpenctopn 17431 ℂ_fldccnfld 21335 Cn ccn 23211 –cn→ccncf 24879

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 ax-pre-sup 11232

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-se 5637 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-of 7689 df-om 7876 df-1st 8002 df-2nd 8003 df-supp 8174 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-2o 8496 df-er 8733 df-map 8856 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9402 df-fi 9450 df-sup 9481 df-inf 9482 df-oi 9549 df-card 9978 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-div 11918 df-nn 12260 df-2 12322 df-3 12323 df-4 12324 df-5 12325 df-6 12326 df-7 12327 df-8 12328 df-9 12329 df-n0 12520 df-z 12606 df-dec 12725 df-uz 12870 df-q 12980 df-rp 13024 df-xneg 13141 df-xadd 13142 df-xmul 13143 df-icc 13380 df-fz 13534 df-fzo 13677 df-seq 14017 df-exp 14077 df-hash 14343 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-clim 15485 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-starv 17276 df-sca 17277 df-vsca 17278 df-ip 17279 df-tset 17280 df-ple 17281 df-ds 17283 df-unif 17284 df-hom 17285 df-cco 17286 df-rest 17432 df-topn 17433 df-0g 17451 df-gsum 17452 df-topgen 17453 df-pt 17454 df-prds 17457 df-xrs 17512 df-qtop 17517 df-imas 17518 df-xps 17520 df-mre 17594 df-mrc 17595 df-acs 17597 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-submnd 18769 df-mulg 19057 df-cntz 19306 df-cmn 19775 df-psmet 21327 df-xmet 21328 df-met 21329 df-bl 21330 df-mopn 21331 df-cnfld 21336 df-top 22879 df-topon 22896 df-topsp 22918 df-bases 22932 df-cn 23214 df-cnp 23215 df-tx 23549 df-hmeo 23742 df-xms 24309 df-ms 24310 df-tms 24311 df-cncf 24881

This theorem is referenced by: stirlinglem8 45639

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