Theorem iprodclim3 15926

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 15480	. . 3 ⊢ (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ ( ⇝ ‘𝐹))
3	1, 2	sylib 217	. 2 ⊢ (𝜑 → 𝐹 ⇝ ( ⇝ ‘𝐹))
4		prodfc 15871	. . . 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 2732	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))
9		iprodclim3.5	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)
10	9	fmpttd 7099	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ 𝐴):𝑍⟶ℂ)
11	10	ffvelcdmda 7071	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) ∈ ℂ)
12	5, 6, 7, 8, 11	iprod 15864	. . . 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 13950	. . . . . . 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 6897	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (𝑀...𝑗) → (((𝑘 ∈ 𝑍 ↦ 𝐴) ↾ (𝑀...𝑗))‘𝑚) = ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))
18		fzssuz 13524	. . . . . . . . . . . . . 14 ⊢ (𝑀...𝑗) ⊆ (ℤ≥‘𝑀)
19	18, 5	sseqtrri 4015	. . . . . . . . . . . . 13 ⊢ (𝑀...𝑗) ⊆ 𝑍
20		resmpt 6027	. . . . . . . . . . . . 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 6879	. . . . . . . . . . 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 15845	. . . . . . . . 9 ⊢ ∏𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ∏𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)‘𝑚)
25		prodfc 15871	. . . . . . . . 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 2732	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑚 ∈ (𝑀...𝑗)) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))
28		simpr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍)
29	28, 5	eleqtrdi 2842	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ≥‘𝑀))
30		elfzuz 13479	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (𝑀...𝑗) → 𝑚 ∈ (ℤ≥‘𝑀))
31	30, 5	eleqtrrdi 2843	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (𝑀...𝑗) → 𝑚 ∈ 𝑍)
32	31, 11	sylan2 593	. . . . . . . . . 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 15875	. . . . . . . 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 15493	. . . . 5 ⊢ (𝜑 → (seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇝ 𝑥 ↔ 𝐹 ⇝ 𝑥))
38	37	iotabidv 6516	. . . 4 ⊢ (𝜑 → (℩𝑥seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇝ 𝑥) = (℩𝑥𝐹 ⇝ 𝑥))
39		df-fv 6540	. . . 4 ⊢ ( ⇝ ‘seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴))) = (℩𝑥seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇝ 𝑥)
40		df-fv 6540	. . . 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 5169	1 ⊢ (𝜑 → 𝐹 ⇝ ∏𝑘 ∈ 𝑍 𝐴)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106   ≠ wne 2939  ∃wrex 3069  Vcvv 3473   ⊆ wss 3944   class class class wbr 5141   ↦ cmpt 5224  dom cdm 5669   ↾ cres 5671  ℩cio 6482  ‘cfv 6532  (class class class)co 7393  ℂcc 11090  0cc0 11092   · cmul 11097  ℤcz 12540  ℤ_≥cuz 12804  ...cfz 13466  seqcseq 13948   ⇝ cli 15410  ∏cprod 15831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-inf2 9618  ax-cnex 11148  ax-resscn 11149  ax-1cn 11150  ax-icn 11151  ax-addcl 11152  ax-addrcl 11153  ax-mulcl 11154  ax-mulrcl 11155  ax-mulcom 11156  ax-addass 11157  ax-mulass 11158  ax-distr 11159  ax-i2m1 11160  ax-1ne0 11161  ax-1rid 11162  ax-rnegex 11163  ax-rrecex 11164  ax-cnre 11165  ax-pre-lttri 11166  ax-pre-lttrn 11167  ax-pre-ltadd 11168  ax-pre-mulgt0 11169  ax-pre-sup 11170

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-isom 6541  df-riota 7349  df-ov 7396  df-oprab 7397  df-mpo 7398  df-om 7839  df-1st 7957  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-1o 8448  df-er 8686  df-en 8923  df-dom 8924  df-sdom 8925  df-fin 8926  df-sup 9419  df-oi 9487  df-card 9916  df-pnf 11232  df-mnf 11233  df-xr 11234  df-ltxr 11235  df-le 11236  df-sub 11428  df-neg 11429  df-div 11854  df-nn 12195  df-2 12257  df-3 12258  df-n0 12455  df-z 12541  df-uz 12805  df-rp 12957  df-fz 13467  df-fzo 13610  df-seq 13949  df-exp 14010  df-hash 14273  df-cj 15028  df-re 15029  df-im 15030  df-sqrt 15164  df-abs 15165  df-clim 15414  df-prod 15832

This theorem is referenced by: (None)