climexp - Mathbox for Glauco Siliprandi

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Theorem climexp 40155

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 2651	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
5	4	expcn 22722	. . . . . 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 22760	. . . . 5 ⊢ (ℂ–cn→ℂ) = ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld))
8	6, 7	syl6eleqr 2741	. . . 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 14274	. . . . 5 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ)
12	10, 11	syl 17	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ)
13	1, 2, 8, 9, 10, 12	climcncf 22750	. . 3 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) ⇝ ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘𝐴))
14		eqidd 2652	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)))
15		simpr 476	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴)
16	15	oveq1d 6705	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝑥↑𝑁) = (𝐴↑𝑁))
17	12, 3	expcld 13048	. . . 4 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℂ)
18	14, 16, 12, 17	fvmptd 6327	. . 3 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘𝐴) = (𝐴↑𝑁))
19	13, 18	breqtrd 4711	. 2 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) ⇝ (𝐴↑𝑁))
20		climexp.9	. . 3 ⊢ (𝜑 → 𝐻 ∈ 𝑉)
21		cnex 10055	. . . . 5 ⊢ ℂ ∈ V
22	21	mptex 6527	. . . 4 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ V
23		fvex 6239	. . . . . 6 ⊢ (ℤ_≥‘𝑀) ∈ V
24	1, 23	eqeltri 2726	. . . . 5 ⊢ 𝑍 ∈ V
25		fex 6530	. . . . 5 ⊢ ((𝐹:𝑍⟶ℂ ∧ 𝑍 ∈ V) → 𝐹 ∈ V)
26	9, 24, 25	sylancl 695	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
27		coexg 7159	. . . 4 ⊢ (((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ V ∧ 𝐹 ∈ V) → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) ∈ V)
28	22, 26, 27	sylancr 696	. . 3 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) ∈ V)
29		eqidd 2652	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)))
30		simpr 476	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 = (𝐹‘𝑗)) → 𝑥 = (𝐹‘𝑗))
31	30	oveq1d 6705	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 = (𝐹‘𝑗)) → (𝑥↑𝑁) = ((𝐹‘𝑗)↑𝑁))
32	9	ffvelrnda 6399	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℂ)
33	3	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑁 ∈ ℕ₀)
34	32, 33	expcld 13048	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝐹‘𝑗)↑𝑁) ∈ ℂ)
35	29, 31, 32, 34	fvmptd 6327	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑗)) = ((𝐹‘𝑗)↑𝑁))
36		fvco3 6314	. . . . 5 ⊢ ((𝐹:𝑍⟶ℂ ∧ 𝑗 ∈ 𝑍) → (((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹)‘𝑗) = ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑗)))
37	9, 36	sylan 487	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹)‘𝑗) = ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑗)))
38		climexp.1	. . . . . . 7 ⊢ Ⅎ𝑘𝜑
39		nfv 1883	. . . . . . 7 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍
40	38, 39	nfan 1868	. . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍)
41		climexp.3	. . . . . . . 8 ⊢ Ⅎ𝑘𝐻
42		nfcv 2793	. . . . . . . 8 ⊢ Ⅎ𝑘𝑗
43	41, 42	nffv 6236	. . . . . . 7 ⊢ Ⅎ𝑘(𝐻‘𝑗)
44		climexp.2	. . . . . . . . 9 ⊢ Ⅎ𝑘𝐹
45	44, 42	nffv 6236	. . . . . . . 8 ⊢ Ⅎ𝑘(𝐹‘𝑗)
46		nfcv 2793	. . . . . . . 8 ⊢ Ⅎ𝑘↑
47		nfcv 2793	. . . . . . . 8 ⊢ Ⅎ𝑘𝑁
48	45, 46, 47	nfov 6716	. . . . . . 7 ⊢ Ⅎ𝑘((𝐹‘𝑗)↑𝑁)
49	43, 48	nfeq 2805	. . . . . 6 ⊢ Ⅎ𝑘(𝐻‘𝑗) = ((𝐹‘𝑗)↑𝑁)
50	40, 49	nfim 1865	. . . . 5 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝐹‘𝑗)↑𝑁))
51		eleq1 2718	. . . . . . 7 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍))
52	51	anbi2d 740	. . . . . 6 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍)))
53		fveq2 6229	. . . . . . 7 ⊢ (𝑘 = 𝑗 → (𝐻‘𝑘) = (𝐻‘𝑗))
54		fveq2 6229	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗))
55	54	oveq1d 6705	. . . . . . 7 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘)↑𝑁) = ((𝐹‘𝑗)↑𝑁))
56	53, 55	eqeq12d 2666	. . . . . 6 ⊢ (𝑘 = 𝑗 → ((𝐻‘𝑘) = ((𝐹‘𝑘)↑𝑁) ↔ (𝐻‘𝑗) = ((𝐹‘𝑗)↑𝑁)))
57	52, 56	imbi12d 333	. . . . 5 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘)↑𝑁)) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝐹‘𝑗)↑𝑁))))
58		climexp.10	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘)↑𝑁))
59	50, 57, 58	chvar 2298	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝐹‘𝑗)↑𝑁))
60	35, 37, 59	3eqtr4rd 2696	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = (((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹)‘𝑗))
61	1, 20, 28, 2, 60	climeq 14342	. 2 ⊢ (𝜑 → (𝐻 ⇝ (𝐴↑𝑁) ↔ ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) ⇝ (𝐴↑𝑁)))
62	19, 61	mpbird 247	1 ⊢ (𝜑 → 𝐻 ⇝ (𝐴↑𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 Ⅎwnf 1748 ∈ wcel 2030 Ⅎwnfc 2780 Vcvv 3231 class class class wbr 4685 ↦ cmpt 4762 ∘ ccom 5147 ⟶wf 5922 ‘cfv 5926 (class class class)co 6690 ℂcc 9972 ℕ₀cn0 11330 ℤcz 11415 ℤ_≥cuz 11725 ↑cexp 12900 ⇝ cli 14259 TopOpenctopn 16129 ℂ_fldccnfld 19794 Cn ccn 21076 –cn→ccncf 22726

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 ax-mulf 10054

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-supp 7341 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-er 7787 df-map 7901 df-ixp 7951 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-fsupp 8317 df-fi 8358 df-sup 8389 df-inf 8390 df-oi 8456 df-card 8803 df-cda 9028 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-q 11827 df-rp 11871 df-xneg 11984 df-xadd 11985 df-xmul 11986 df-icc 12220 df-fz 12365 df-fzo 12505 df-seq 12842 df-exp 12901 df-hash 13158 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-clim 14263 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-starv 16003 df-sca 16004 df-vsca 16005 df-ip 16006 df-tset 16007 df-ple 16008 df-ds 16011 df-unif 16012 df-hom 16013 df-cco 16014 df-rest 16130 df-topn 16131 df-0g 16149 df-gsum 16150 df-topgen 16151 df-pt 16152 df-prds 16155 df-xrs 16209 df-qtop 16214 df-imas 16215 df-xps 16217 df-mre 16293 df-mrc 16294 df-acs 16296 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-submnd 17383 df-mulg 17588 df-cntz 17796 df-cmn 18241 df-psmet 19786 df-xmet 19787 df-met 19788 df-bl 19789 df-mopn 19790 df-cnfld 19795 df-top 20747 df-topon 20764 df-topsp 20785 df-bases 20798 df-cn 21079 df-cnp 21080 df-tx 21413 df-hmeo 21606 df-xms 22172 df-ms 22173 df-tms 22174 df-cncf 22728

This theorem is referenced by: stirlinglem8 40616

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