climexp - Mathbox for Glauco Siliprandi

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Theorem climexp 43146

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 2738	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
5	4	expcn 24035	. . . . . 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 24074	. . . . 5 ⊢ (ℂ–cn→ℂ) = ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld))
8	6, 7	eleqtrrdi 2850	. . . 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 15208	. . . . 5 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ)
12	10, 11	syl 17	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ)
13	1, 2, 8, 9, 10, 12	climcncf 24063	. . 3 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) ⇝ ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘𝐴))
14		eqidd 2739	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)))
15		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴)
16	15	oveq1d 7290	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝑥↑𝑁) = (𝐴↑𝑁))
17	12, 3	expcld 13864	. . . 4 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℂ)
18	14, 16, 12, 17	fvmptd 6882	. . 3 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘𝐴) = (𝐴↑𝑁))
19	13, 18	breqtrd 5100	. 2 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) ⇝ (𝐴↑𝑁))
20		climexp.9	. . 3 ⊢ (𝜑 → 𝐻 ∈ 𝑉)
21		cnex 10952	. . . . 5 ⊢ ℂ ∈ V
22	21	mptex 7099	. . . 4 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ V
23	1	fvexi 6788	. . . . 5 ⊢ 𝑍 ∈ V
24		fex 7102	. . . . 5 ⊢ ((𝐹:𝑍⟶ℂ ∧ 𝑍 ∈ V) → 𝐹 ∈ V)
25	9, 23, 24	sylancl 586	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
26		coexg 7776	. . . 4 ⊢ (((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ V ∧ 𝐹 ∈ V) → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) ∈ V)
27	22, 25, 26	sylancr 587	. . 3 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) ∈ V)
28		eqidd 2739	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)))
29		simpr 485	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 = (𝐹‘𝑗)) → 𝑥 = (𝐹‘𝑗))
30	29	oveq1d 7290	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 = (𝐹‘𝑗)) → (𝑥↑𝑁) = ((𝐹‘𝑗)↑𝑁))
31	9	ffvelrnda 6961	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℂ)
32	3	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑁 ∈ ℕ₀)
33	31, 32	expcld 13864	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝐹‘𝑗)↑𝑁) ∈ ℂ)
34	28, 30, 31, 33	fvmptd 6882	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑗)) = ((𝐹‘𝑗)↑𝑁))
35		fvco3 6867	. . . . 5 ⊢ ((𝐹:𝑍⟶ℂ ∧ 𝑗 ∈ 𝑍) → (((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹)‘𝑗) = ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑗)))
36	9, 35	sylan 580	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹)‘𝑗) = ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑗)))
37		climexp.1	. . . . . . 7 ⊢ Ⅎ𝑘𝜑
38		nfv 1917	. . . . . . 7 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍
39	37, 38	nfan 1902	. . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍)
40		climexp.3	. . . . . . . 8 ⊢ Ⅎ𝑘𝐻
41		nfcv 2907	. . . . . . . 8 ⊢ Ⅎ𝑘𝑗
42	40, 41	nffv 6784	. . . . . . 7 ⊢ Ⅎ𝑘(𝐻‘𝑗)
43		climexp.2	. . . . . . . . 9 ⊢ Ⅎ𝑘𝐹
44	43, 41	nffv 6784	. . . . . . . 8 ⊢ Ⅎ𝑘(𝐹‘𝑗)
45		nfcv 2907	. . . . . . . 8 ⊢ Ⅎ𝑘↑
46		nfcv 2907	. . . . . . . 8 ⊢ Ⅎ𝑘𝑁
47	44, 45, 46	nfov 7305	. . . . . . 7 ⊢ Ⅎ𝑘((𝐹‘𝑗)↑𝑁)
48	42, 47	nfeq 2920	. . . . . 6 ⊢ Ⅎ𝑘(𝐻‘𝑗) = ((𝐹‘𝑗)↑𝑁)
49	39, 48	nfim 1899	. . . . 5 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝐹‘𝑗)↑𝑁))
50		eleq1w 2821	. . . . . . 7 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍))
51	50	anbi2d 629	. . . . . 6 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍)))
52		fveq2 6774	. . . . . . 7 ⊢ (𝑘 = 𝑗 → (𝐻‘𝑘) = (𝐻‘𝑗))
53		fveq2 6774	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗))
54	53	oveq1d 7290	. . . . . . 7 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘)↑𝑁) = ((𝐹‘𝑗)↑𝑁))
55	52, 54	eqeq12d 2754	. . . . . 6 ⊢ (𝑘 = 𝑗 → ((𝐻‘𝑘) = ((𝐹‘𝑘)↑𝑁) ↔ (𝐻‘𝑗) = ((𝐹‘𝑗)↑𝑁)))
56	51, 55	imbi12d 345	. . . . 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 2233	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝐹‘𝑗)↑𝑁))
59	34, 36, 58	3eqtr4rd 2789	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = (((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹)‘𝑗))
60	1, 20, 27, 2, 59	climeq 15276	. 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 396 = wceq 1539 Ⅎwnf 1786 ∈ wcel 2106 Ⅎwnfc 2887 Vcvv 3432 class class class wbr 5074 ↦ cmpt 5157 ∘ ccom 5593 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 ℕ₀cn0 12233 ℤcz 12319 ℤ_≥cuz 12582 ↑cexp 13782 ⇝ cli 15193 TopOpenctopn 17132 ℂ_fldccnfld 20597 Cn ccn 22375 –cn→ccncf 24039

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-icc 13086 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cn 22378 df-cnp 22379 df-tx 22713 df-hmeo 22906 df-xms 23473 df-ms 23474 df-tms 23475 df-cncf 24041

This theorem is referenced by: stirlinglem8 43622

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