Theorem iprodclim3 15759

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,𝐹   𝑗,𝑘,𝜑   𝑘,𝑛,𝜑,𝑦   𝑗,𝑀   𝑦,𝑀   𝜑,𝑛,𝑦   𝑗,𝑍,𝑘   𝑛,𝑍,𝑦   𝑘,𝑀

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

Proof of Theorem iprodclim3

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

Step	Hyp	Ref	Expression
1		iprodclim3.4	. . 3 ⊢ (𝜑 → 𝐹 ∈ dom ⇝ )
2		climdm 15312	. . 3 ⊢ (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ ( ⇝ ‘𝐹))
3	1, 2	sylib 217	. 2 ⊢ (𝜑 → 𝐹 ⇝ ( ⇝ ‘𝐹))
4		prodfc 15704	. . . 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 7021	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ 𝐴):𝑍⟶ℂ)
11	10	ffvelcdmda 6993	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) ∈ ℂ)
12	5, 6, 7, 8, 11	iprod 15697	. . . 4 ⊢ (𝜑 → ∏𝑚 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ( ⇝ ‘seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴))))
13	4, 12	eqtr3id 2790	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ 𝑍 𝐴 = ( ⇝ ‘seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴))))
14		seqex 13773	. . . . . . 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 6823	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (𝑀...𝑗) → (((𝑘 ∈ 𝑍 ↦ 𝐴) ↾ (𝑀...𝑗))‘𝑚) = ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))
18		fzssuz 13347	. . . . . . . . . . . . . 14 ⊢ (𝑀...𝑗) ⊆ (ℤ≥‘𝑀)
19	18, 5	sseqtrri 3963	. . . . . . . . . . . . 13 ⊢ (𝑀...𝑗) ⊆ 𝑍
20		resmpt 5957	. . . . . . . . . . . . 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 6805	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ 𝑍 ↦ 𝐴) ↾ (𝑀...𝑗))‘𝑚) = ((𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)‘𝑚)
23	17, 22	eqtr3di 2791	. . . . . . . . . 10 ⊢ (𝑚 ∈ (𝑀...𝑗) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ((𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)‘𝑚))
24	23	prodeq2i 15678	. . . . . . . . 9 ⊢ ∏𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ∏𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)‘𝑚)
25		prodfc 15704	. . . . . . . . 9 ⊢ ∏𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)‘𝑚) = ∏𝑘 ∈ (𝑀...𝑗)𝐴
26	24, 25	eqtri 2764	. . . . . . . 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 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍)
29	28, 5	eleqtrdi 2847	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ≥‘𝑀))
30		elfzuz 13302	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (𝑀...𝑗) → 𝑚 ∈ (ℤ≥‘𝑀))
31	30, 5	eleqtrrdi 2848	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (𝑀...𝑗) → 𝑚 ∈ 𝑍)
32	31, 11	sylan2 594	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝑀...𝑗)) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) ∈ ℂ)
33	32	adantlr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑚 ∈ (𝑀...𝑗)) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) ∈ ℂ)
34	27, 29, 33	fprodser 15708	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ∏𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = (seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴))‘𝑗))
35	26, 34	eqtr3id 2790	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ∏𝑘 ∈ (𝑀...𝑗)𝐴 = (seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴))‘𝑗))
36	16, 35	eqtr2d 2777	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴))‘𝑗) = (𝐹‘𝑗))
37	5, 15, 1, 6, 36	climeq 15325	. . . . 5 ⊢ (𝜑 → (seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇝ 𝑥 ↔ 𝐹 ⇝ 𝑥))
38	37	iotabidv 6442	. . . 4 ⊢ (𝜑 → (℩𝑥seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇝ 𝑥) = (℩𝑥𝐹 ⇝ 𝑥))
39		df-fv 6466	. . . 4 ⊢ ( ⇝ ‘seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴))) = (℩𝑥seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇝ 𝑥)
40		df-fv 6466	. . . 4 ⊢ ( ⇝ ‘𝐹) = (℩𝑥𝐹 ⇝ 𝑥)
41	38, 39, 40	3eqtr4g 2801	. . 3 ⊢ (𝜑 → ( ⇝ ‘seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴))) = ( ⇝ ‘𝐹))
42	13, 41	eqtrd 2776	. 2 ⊢ (𝜑 → ∏𝑘 ∈ 𝑍 𝐴 = ( ⇝ ‘𝐹))
43	3, 42	breqtrrd 5109	1 ⊢ (𝜑 → 𝐹 ⇝ ∏𝑘 ∈ 𝑍 𝐴)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1539  ∃wex 1779   ∈ wcel 2104   ≠ wne 2941  ∃wrex 3071  Vcvv 3437   ⊆ wss 3892   class class class wbr 5081   ↦ cmpt 5164  dom cdm 5600   ↾ cres 5602  ℩cio 6408  ‘cfv 6458  (class class class)co 7307  ℂcc 10919  0cc0 10921   · cmul 10926  ℤcz 12369  ℤ_≥cuz 12632  ...cfz 13289  seqcseq 13771   ⇝ cli 15242  ∏cprod 15664

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620  ax-inf2 9447  ax-cnex 10977  ax-resscn 10978  ax-1cn 10979  ax-icn 10980  ax-addcl 10981  ax-addrcl 10982  ax-mulcl 10983  ax-mulrcl 10984  ax-mulcom 10985  ax-addass 10986  ax-mulass 10987  ax-distr 10988  ax-i2m1 10989  ax-1ne0 10990  ax-1rid 10991  ax-rnegex 10992  ax-rrecex 10993  ax-cnre 10994  ax-pre-lttri 10995  ax-pre-lttrn 10996  ax-pre-ltadd 10997  ax-pre-mulgt0 10998  ax-pre-sup 10999

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3304  df-reu 3305  df-rab 3306  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-int 4887  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-tr 5199  df-id 5500  df-eprel 5506  df-po 5514  df-so 5515  df-fr 5555  df-se 5556  df-we 5557  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-pred 6217  df-ord 6284  df-on 6285  df-lim 6286  df-suc 6287  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-isom 6467  df-riota 7264  df-ov 7310  df-oprab 7311  df-mpo 7312  df-om 7745  df-1st 7863  df-2nd 7864  df-frecs 8128  df-wrecs 8159  df-recs 8233  df-rdg 8272  df-1o 8328  df-er 8529  df-en 8765  df-dom 8766  df-sdom 8767  df-fin 8768  df-sup 9249  df-oi 9317  df-card 9745  df-pnf 11061  df-mnf 11062  df-xr 11063  df-ltxr 11064  df-le 11065  df-sub 11257  df-neg 11258  df-div 11683  df-nn 12024  df-2 12086  df-3 12087  df-n0 12284  df-z 12370  df-uz 12633  df-rp 12781  df-fz 13290  df-fzo 13433  df-seq 13772  df-exp 13833  df-hash 14095  df-cj 14859  df-re 14860  df-im 14861  df-sqrt 14995  df-abs 14996  df-clim 15246  df-prod 15665

This theorem is referenced by: (None)