climexp - Mathbox for Glauco Siliprandi

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Theorem climexp 42821

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 2737	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
5	4	expcn 23769	. . . . . 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 23808	. . . . 5 ⊢ (ℂ–cn→ℂ) = ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld))
8	6, 7	eleqtrrdi 2849	. . . 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 15060	. . . . 5 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ)
12	10, 11	syl 17	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ)
13	1, 2, 8, 9, 10, 12	climcncf 23797	. . 3 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) ⇝ ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘𝐴))
14		eqidd 2738	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)))
15		simpr 488	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴)
16	15	oveq1d 7228	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝑥↑𝑁) = (𝐴↑𝑁))
17	12, 3	expcld 13716	. . . 4 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℂ)
18	14, 16, 12, 17	fvmptd 6825	. . 3 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘𝐴) = (𝐴↑𝑁))
19	13, 18	breqtrd 5079	. 2 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) ⇝ (𝐴↑𝑁))
20		climexp.9	. . 3 ⊢ (𝜑 → 𝐻 ∈ 𝑉)
21		cnex 10810	. . . . 5 ⊢ ℂ ∈ V
22	21	mptex 7039	. . . 4 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ V
23	1	fvexi 6731	. . . . 5 ⊢ 𝑍 ∈ V
24		fex 7042	. . . . 5 ⊢ ((𝐹:𝑍⟶ℂ ∧ 𝑍 ∈ V) → 𝐹 ∈ V)
25	9, 23, 24	sylancl 589	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
26		coexg 7707	. . . 4 ⊢ (((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ V ∧ 𝐹 ∈ V) → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) ∈ V)
27	22, 25, 26	sylancr 590	. . 3 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) ∈ V)
28		eqidd 2738	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)))
29		simpr 488	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 = (𝐹‘𝑗)) → 𝑥 = (𝐹‘𝑗))
30	29	oveq1d 7228	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 = (𝐹‘𝑗)) → (𝑥↑𝑁) = ((𝐹‘𝑗)↑𝑁))
31	9	ffvelrnda 6904	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℂ)
32	3	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑁 ∈ ℕ₀)
33	31, 32	expcld 13716	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝐹‘𝑗)↑𝑁) ∈ ℂ)
34	28, 30, 31, 33	fvmptd 6825	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑗)) = ((𝐹‘𝑗)↑𝑁))
35		fvco3 6810	. . . . 5 ⊢ ((𝐹:𝑍⟶ℂ ∧ 𝑗 ∈ 𝑍) → (((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹)‘𝑗) = ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑗)))
36	9, 35	sylan 583	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹)‘𝑗) = ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑗)))
37		climexp.1	. . . . . . 7 ⊢ Ⅎ𝑘𝜑
38		nfv 1922	. . . . . . 7 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍
39	37, 38	nfan 1907	. . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍)
40		climexp.3	. . . . . . . 8 ⊢ Ⅎ𝑘𝐻
41		nfcv 2904	. . . . . . . 8 ⊢ Ⅎ𝑘𝑗
42	40, 41	nffv 6727	. . . . . . 7 ⊢ Ⅎ𝑘(𝐻‘𝑗)
43		climexp.2	. . . . . . . . 9 ⊢ Ⅎ𝑘𝐹
44	43, 41	nffv 6727	. . . . . . . 8 ⊢ Ⅎ𝑘(𝐹‘𝑗)
45		nfcv 2904	. . . . . . . 8 ⊢ Ⅎ𝑘↑
46		nfcv 2904	. . . . . . . 8 ⊢ Ⅎ𝑘𝑁
47	44, 45, 46	nfov 7243	. . . . . . 7 ⊢ Ⅎ𝑘((𝐹‘𝑗)↑𝑁)
48	42, 47	nfeq 2917	. . . . . 6 ⊢ Ⅎ𝑘(𝐻‘𝑗) = ((𝐹‘𝑗)↑𝑁)
49	39, 48	nfim 1904	. . . . 5 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝐹‘𝑗)↑𝑁))
50		eleq1w 2820	. . . . . . 7 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍))
51	50	anbi2d 632	. . . . . 6 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍)))
52		fveq2 6717	. . . . . . 7 ⊢ (𝑘 = 𝑗 → (𝐻‘𝑘) = (𝐻‘𝑗))
53		fveq2 6717	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗))
54	53	oveq1d 7228	. . . . . . 7 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘)↑𝑁) = ((𝐹‘𝑗)↑𝑁))
55	52, 54	eqeq12d 2753	. . . . . 6 ⊢ (𝑘 = 𝑗 → ((𝐻‘𝑘) = ((𝐹‘𝑘)↑𝑁) ↔ (𝐻‘𝑗) = ((𝐹‘𝑗)↑𝑁)))
56	51, 55	imbi12d 348	. . . . 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 2238	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝐹‘𝑗)↑𝑁))
59	34, 36, 58	3eqtr4rd 2788	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = (((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹)‘𝑗))
60	1, 20, 27, 2, 59	climeq 15128	. 2 ⊢ (𝜑 → (𝐻 ⇝ (𝐴↑𝑁) ↔ ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) ⇝ (𝐴↑𝑁)))
61	19, 60	mpbird 260	1 ⊢ (𝜑 → 𝐻 ⇝ (𝐴↑𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 Ⅎwnf 1791 ∈ wcel 2110 Ⅎwnfc 2884 Vcvv 3408 class class class wbr 5053 ↦ cmpt 5135 ∘ ccom 5555 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 ℂcc 10727 ℕ₀cn0 12090 ℤcz 12176 ℤ_≥cuz 12438 ↑cexp 13635 ⇝ cli 15045 TopOpenctopn 16926 ℂ_fldccnfld 20363 Cn ccn 22121 –cn→ccncf 23773

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 ax-mulf 10809

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-er 8391 df-map 8510 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-fi 9027 df-sup 9058 df-inf 9059 df-oi 9126 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-q 12545 df-rp 12587 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-icc 12942 df-fz 13096 df-fzo 13239 df-seq 13575 df-exp 13636 df-hash 13897 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-clim 15049 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-starv 16817 df-sca 16818 df-vsca 16819 df-ip 16820 df-tset 16821 df-ple 16822 df-ds 16824 df-unif 16825 df-hom 16826 df-cco 16827 df-rest 16927 df-topn 16928 df-0g 16946 df-gsum 16947 df-topgen 16948 df-pt 16949 df-prds 16952 df-xrs 17007 df-qtop 17012 df-imas 17013 df-xps 17015 df-mre 17089 df-mrc 17090 df-acs 17092 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-mulg 18489 df-cntz 18711 df-cmn 19172 df-psmet 20355 df-xmet 20356 df-met 20357 df-bl 20358 df-mopn 20359 df-cnfld 20364 df-top 21791 df-topon 21808 df-topsp 21830 df-bases 21843 df-cn 22124 df-cnp 22125 df-tx 22459 df-hmeo 22652 df-xms 23218 df-ms 23219 df-tms 23220 df-cncf 23775

This theorem is referenced by: stirlinglem8 43297

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