Theorem iprodclim3 16021

Description: The sequence of partial finite product of a converging infinite product converge to the infinite product of the series. Note that 𝑗 must not occur in 𝐴. (Contributed by Scott Fenton, 18-Dec-2017.)

Hypotheses

Ref	Expression
iprodclim3.1	⊢ 𝑍 = (ℤ≥‘𝑀)
iprodclim3.2	⊢ (𝜑 → 𝑀 ∈ ℤ)
iprodclim3.3	⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇝ 𝑦))
iprodclim3.4	⊢ (𝜑 → 𝐹 ∈ dom ⇝ )
iprodclim3.5	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)
iprodclim3.6	⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = ∏𝑘 ∈ (𝑀...𝑗)𝐴)

Assertion

Ref	Expression
iprodclim3	⊢ (𝜑 → 𝐹 ⇝ ∏𝑘 ∈ 𝑍 𝐴)

Distinct variable groups:   𝐴,𝑗   𝐴,𝑛,𝑦   𝑗,𝑀,𝑘   𝑦,𝑀   𝑗,𝐹   𝜑,𝑗,𝑘   𝜑,𝑛,𝑦   𝑗,𝑍,𝑘   𝑛,𝑍,𝑦,𝑘

Allowed substitution hints:   𝐴(𝑘)   𝐹(𝑦,𝑘,𝑛)   𝑀(𝑛)

Proof of Theorem iprodclim3

Dummy variables 𝑥 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iprodclim3.4	. . 3 ⊢ (𝜑 → 𝐹 ∈ dom ⇝ )
2		climdm 15575	. . 3 ⊢ (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ ( ⇝ ‘𝐹))
3	1, 2	sylib 218	. 2 ⊢ (𝜑 → 𝐹 ⇝ ( ⇝ ‘𝐹))
4		prodfc 15966	. . . 4 ⊢ ∏𝑚 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ∏𝑘 ∈ 𝑍 𝐴
5		iprodclim3.1	. . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀)
6		iprodclim3.2	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
7		iprodclim3.3	. . . . 5 ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇝ 𝑦))
8		eqidd 2737	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))
9		iprodclim3.5	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)
10	9	fmpttd 7110	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ 𝐴):𝑍⟶ℂ)
11	10	ffvelcdmda 7079	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) ∈ ℂ)
12	5, 6, 7, 8, 11	iprod 15959	. . . 4 ⊢ (𝜑 → ∏𝑚 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ( ⇝ ‘seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴))))
13	4, 12	eqtr3id 2785	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ 𝑍 𝐴 = ( ⇝ ‘seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴))))
14		seqex 14026	. . . . . . 7 ⊢ seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴)) ∈ V
15	14	a1i 11	. . . . . 6 ⊢ (𝜑 → seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴)) ∈ V)
16		iprodclim3.6	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = ∏𝑘 ∈ (𝑀...𝑗)𝐴)
17		fvres 6900	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (𝑀...𝑗) → (((𝑘 ∈ 𝑍 ↦ 𝐴) ↾ (𝑀...𝑗))‘𝑚) = ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))
18		fzssuz 13587	. . . . . . . . . . . . . 14 ⊢ (𝑀...𝑗) ⊆ (ℤ≥‘𝑀)
19	18, 5	sseqtrri 4013	. . . . . . . . . . . . 13 ⊢ (𝑀...𝑗) ⊆ 𝑍
20		resmpt 6029	. . . . . . . . . . . . 13 ⊢ ((𝑀...𝑗) ⊆ 𝑍 → ((𝑘 ∈ 𝑍 ↦ 𝐴) ↾ (𝑀...𝑗)) = (𝑘 ∈ (𝑀...𝑗) ↦ 𝐴))
21	19, 20	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ 𝑍 ↦ 𝐴) ↾ (𝑀...𝑗)) = (𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)
22	21	fveq1i 6882	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ 𝑍 ↦ 𝐴) ↾ (𝑀...𝑗))‘𝑚) = ((𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)‘𝑚)
23	17, 22	eqtr3di 2786	. . . . . . . . . 10 ⊢ (𝑚 ∈ (𝑀...𝑗) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ((𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)‘𝑚))
24	23	prodeq2i 15939	. . . . . . . . 9 ⊢ ∏𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ∏𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)‘𝑚)
25		prodfc 15966	. . . . . . . . 9 ⊢ ∏𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)‘𝑚) = ∏𝑘 ∈ (𝑀...𝑗)𝐴
26	24, 25	eqtri 2759	. . . . . . . 8 ⊢ ∏𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ∏𝑘 ∈ (𝑀...𝑗)𝐴
27		eqidd 2737	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑚 ∈ (𝑀...𝑗)) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))
28		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍)
29	28, 5	eleqtrdi 2845	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ≥‘𝑀))
30		elfzuz 13542	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (𝑀...𝑗) → 𝑚 ∈ (ℤ≥‘𝑀))
31	30, 5	eleqtrrdi 2846	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (𝑀...𝑗) → 𝑚 ∈ 𝑍)
32	31, 11	sylan2 593	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝑀...𝑗)) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) ∈ ℂ)
33	32	adantlr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑚 ∈ (𝑀...𝑗)) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) ∈ ℂ)
34	27, 29, 33	fprodser 15970	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ∏𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = (seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴))‘𝑗))
35	26, 34	eqtr3id 2785	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ∏𝑘 ∈ (𝑀...𝑗)𝐴 = (seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴))‘𝑗))
36	16, 35	eqtr2d 2772	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴))‘𝑗) = (𝐹‘𝑗))
37	5, 15, 1, 6, 36	climeq 15588	. . . . 5 ⊢ (𝜑 → (seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇝ 𝑥 ↔ 𝐹 ⇝ 𝑥))
38	37	iotabidv 6520	. . . 4 ⊢ (𝜑 → (℩𝑥seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇝ 𝑥) = (℩𝑥𝐹 ⇝ 𝑥))
39		df-fv 6544	. . . 4 ⊢ ( ⇝ ‘seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴))) = (℩𝑥seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇝ 𝑥)
40		df-fv 6544	. . . 4 ⊢ ( ⇝ ‘𝐹) = (℩𝑥𝐹 ⇝ 𝑥)
41	38, 39, 40	3eqtr4g 2796	. . 3 ⊢ (𝜑 → ( ⇝ ‘seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴))) = ( ⇝ ‘𝐹))
42	13, 41	eqtrd 2771	. 2 ⊢ (𝜑 → ∏𝑘 ∈ 𝑍 𝐴 = ( ⇝ ‘𝐹))
43	3, 42	breqtrrd 5152	1 ⊢ (𝜑 → 𝐹 ⇝ ∏𝑘 ∈ 𝑍 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2933  ∃wrex 3061  Vcvv 3464   ⊆ wss 3931   class class class wbr 5124   ↦ cmpt 5206  dom cdm 5659   ↾ cres 5661  ℩cio 6487  ‘cfv 6536  (class class class)co 7410  ℂcc 11132  0cc0 11134   · cmul 11139  ℤcz 12593  ℤ_≥cuz 12857  ...cfz 13529  seqcseq 14024   ⇝ cli 15505  ∏cprod 15924

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-inf2 9660  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211  ax-pre-sup 11212

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9459  df-oi 9529  df-card 9958  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-div 11900  df-nn 12246  df-2 12308  df-3 12309  df-n0 12507  df-z 12594  df-uz 12858  df-rp 13014  df-fz 13530  df-fzo 13677  df-seq 14025  df-exp 14085  df-hash 14354  df-cj 15123  df-re 15124  df-im 15125  df-sqrt 15259  df-abs 15260  df-clim 15509  df-prod 15925

This theorem is referenced by: (None)