Theorem iprodclim3 15941

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 15495	. . 3 ⊢ (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ ( ⇝ ‘𝐹))
3	1, 2	sylib 217	. 2 ⊢ (𝜑 → 𝐹 ⇝ ( ⇝ ‘𝐹))
4		prodfc 15886	. . . 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 2734	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))
9		iprodclim3.5	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)
10	9	fmpttd 7112	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ 𝐴):𝑍⟶ℂ)
11	10	ffvelcdmda 7084	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) ∈ ℂ)
12	5, 6, 7, 8, 11	iprod 15879	. . . 4 ⊢ (𝜑 → ∏𝑚 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ( ⇝ ‘seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴))))
13	4, 12	eqtr3id 2787	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ 𝑍 𝐴 = ( ⇝ ‘seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴))))
14		seqex 13965	. . . . . . 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 6908	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (𝑀...𝑗) → (((𝑘 ∈ 𝑍 ↦ 𝐴) ↾ (𝑀...𝑗))‘𝑚) = ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))
18		fzssuz 13539	. . . . . . . . . . . . . 14 ⊢ (𝑀...𝑗) ⊆ (ℤ≥‘𝑀)
19	18, 5	sseqtrri 4019	. . . . . . . . . . . . 13 ⊢ (𝑀...𝑗) ⊆ 𝑍
20		resmpt 6036	. . . . . . . . . . . . 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 6890	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ 𝑍 ↦ 𝐴) ↾ (𝑀...𝑗))‘𝑚) = ((𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)‘𝑚)
23	17, 22	eqtr3di 2788	. . . . . . . . . 10 ⊢ (𝑚 ∈ (𝑀...𝑗) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ((𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)‘𝑚))
24	23	prodeq2i 15860	. . . . . . . . 9 ⊢ ∏𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ∏𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)‘𝑚)
25		prodfc 15886	. . . . . . . . 9 ⊢ ∏𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)‘𝑚) = ∏𝑘 ∈ (𝑀...𝑗)𝐴
26	24, 25	eqtri 2761	. . . . . . . 8 ⊢ ∏𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ∏𝑘 ∈ (𝑀...𝑗)𝐴
27		eqidd 2734	. . . . . . . . 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 2844	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ≥‘𝑀))
30		elfzuz 13494	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (𝑀...𝑗) → 𝑚 ∈ (ℤ≥‘𝑀))
31	30, 5	eleqtrrdi 2845	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (𝑀...𝑗) → 𝑚 ∈ 𝑍)
32	31, 11	sylan2 594	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝑀...𝑗)) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) ∈ ℂ)
33	32	adantlr 714	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑚 ∈ (𝑀...𝑗)) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) ∈ ℂ)
34	27, 29, 33	fprodser 15890	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ∏𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = (seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴))‘𝑗))
35	26, 34	eqtr3id 2787	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ∏𝑘 ∈ (𝑀...𝑗)𝐴 = (seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴))‘𝑗))
36	16, 35	eqtr2d 2774	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴))‘𝑗) = (𝐹‘𝑗))
37	5, 15, 1, 6, 36	climeq 15508	. . . . 5 ⊢ (𝜑 → (seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇝ 𝑥 ↔ 𝐹 ⇝ 𝑥))
38	37	iotabidv 6525	. . . 4 ⊢ (𝜑 → (℩𝑥seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇝ 𝑥) = (℩𝑥𝐹 ⇝ 𝑥))
39		df-fv 6549	. . . 4 ⊢ ( ⇝ ‘seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴))) = (℩𝑥seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇝ 𝑥)
40		df-fv 6549	. . . 4 ⊢ ( ⇝ ‘𝐹) = (℩𝑥𝐹 ⇝ 𝑥)
41	38, 39, 40	3eqtr4g 2798	. . 3 ⊢ (𝜑 → ( ⇝ ‘seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴))) = ( ⇝ ‘𝐹))
42	13, 41	eqtrd 2773	. 2 ⊢ (𝜑 → ∏𝑘 ∈ 𝑍 𝐴 = ( ⇝ ‘𝐹))
43	3, 42	breqtrrd 5176	1 ⊢ (𝜑 → 𝐹 ⇝ ∏𝑘 ∈ 𝑍 𝐴)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542  ∃wex 1782   ∈ wcel 2107   ≠ wne 2941  ∃wrex 3071  Vcvv 3475   ⊆ wss 3948   class class class wbr 5148   ↦ cmpt 5231  dom cdm 5676   ↾ cres 5678  ℩cio 6491  ‘cfv 6541  (class class class)co 7406  ℂcc 11105  0cc0 11107   · cmul 11112  ℤcz 12555  ℤ_≥cuz 12819  ...cfz 13481  seqcseq 13963   ⇝ cli 15425  ∏cprod 15846

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-inf2 9633  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184  ax-pre-sup 11185

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-isom 6550  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-1st 7972  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-1o 8463  df-er 8700  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-sup 9434  df-oi 9502  df-card 9931  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-div 11869  df-nn 12210  df-2 12272  df-3 12273  df-n0 12470  df-z 12556  df-uz 12820  df-rp 12972  df-fz 13482  df-fzo 13625  df-seq 13964  df-exp 14025  df-hash 14288  df-cj 15043  df-re 15044  df-im 15045  df-sqrt 15179  df-abs 15180  df-clim 15429  df-prod 15847

This theorem is referenced by: (None)