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Theorem iprodclim3 14570
 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 = (ℤ𝑀)
iprodclim3.2 (𝜑𝑀 ∈ ℤ)
iprodclim3.3 (𝜑 → ∃𝑛𝑍𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘𝑍𝐴)) ⇝ 𝑦))
iprodclim3.4 (𝜑𝐹 ∈ dom ⇝ )
iprodclim3.5 ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)
iprodclim3.6 ((𝜑𝑗𝑍) → (𝐹𝑗) = ∏𝑘 ∈ (𝑀...𝑗)𝐴)
Assertion
Ref Expression
iprodclim3 (𝜑𝐹 ⇝ ∏𝑘𝑍 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.
StepHypRef Expression
1 iprodclim3.4 . . 3 (𝜑𝐹 ∈ dom ⇝ )
2 climdm 14133 . . 3 (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ ( ⇝ ‘𝐹))
31, 2sylib 207 . 2 (𝜑𝐹 ⇝ ( ⇝ ‘𝐹))
4 prodfc 14514 . . . 4 𝑚𝑍 ((𝑘𝑍𝐴)‘𝑚) = ∏𝑘𝑍 𝐴
5 iprodclim3.1 . . . . 5 𝑍 = (ℤ𝑀)
6 iprodclim3.2 . . . . 5 (𝜑𝑀 ∈ ℤ)
7 iprodclim3.3 . . . . 5 (𝜑 → ∃𝑛𝑍𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘𝑍𝐴)) ⇝ 𝑦))
8 eqidd 2611 . . . . 5 ((𝜑𝑚𝑍) → ((𝑘𝑍𝐴)‘𝑚) = ((𝑘𝑍𝐴)‘𝑚))
9 iprodclim3.5 . . . . . . 7 ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)
10 eqid 2610 . . . . . . 7 (𝑘𝑍𝐴) = (𝑘𝑍𝐴)
119, 10fmptd 6292 . . . . . 6 (𝜑 → (𝑘𝑍𝐴):𝑍⟶ℂ)
1211ffvelrnda 6267 . . . . 5 ((𝜑𝑚𝑍) → ((𝑘𝑍𝐴)‘𝑚) ∈ ℂ)
135, 6, 7, 8, 12iprod 14507 . . . 4 (𝜑 → ∏𝑚𝑍 ((𝑘𝑍𝐴)‘𝑚) = ( ⇝ ‘seq𝑀( · , (𝑘𝑍𝐴))))
144, 13syl5eqr 2658 . . 3 (𝜑 → ∏𝑘𝑍 𝐴 = ( ⇝ ‘seq𝑀( · , (𝑘𝑍𝐴))))
15 seqex 12665 . . . . . . 7 seq𝑀( · , (𝑘𝑍𝐴)) ∈ V
1615a1i 11 . . . . . 6 (𝜑 → seq𝑀( · , (𝑘𝑍𝐴)) ∈ V)
17 iprodclim3.6 . . . . . . 7 ((𝜑𝑗𝑍) → (𝐹𝑗) = ∏𝑘 ∈ (𝑀...𝑗)𝐴)
18 fzssuz 12253 . . . . . . . . . . . . . 14 (𝑀...𝑗) ⊆ (ℤ𝑀)
1918, 5sseqtr4i 3601 . . . . . . . . . . . . 13 (𝑀...𝑗) ⊆ 𝑍
20 resmpt 5369 . . . . . . . . . . . . 13 ((𝑀...𝑗) ⊆ 𝑍 → ((𝑘𝑍𝐴) ↾ (𝑀...𝑗)) = (𝑘 ∈ (𝑀...𝑗) ↦ 𝐴))
2119, 20ax-mp 5 . . . . . . . . . . . 12 ((𝑘𝑍𝐴) ↾ (𝑀...𝑗)) = (𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)
2221fveq1i 6104 . . . . . . . . . . 11 (((𝑘𝑍𝐴) ↾ (𝑀...𝑗))‘𝑚) = ((𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)‘𝑚)
23 fvres 6117 . . . . . . . . . . 11 (𝑚 ∈ (𝑀...𝑗) → (((𝑘𝑍𝐴) ↾ (𝑀...𝑗))‘𝑚) = ((𝑘𝑍𝐴)‘𝑚))
2422, 23syl5reqr 2659 . . . . . . . . . 10 (𝑚 ∈ (𝑀...𝑗) → ((𝑘𝑍𝐴)‘𝑚) = ((𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)‘𝑚))
2524prodeq2i 14488 . . . . . . . . 9 𝑚 ∈ (𝑀...𝑗)((𝑘𝑍𝐴)‘𝑚) = ∏𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)‘𝑚)
26 prodfc 14514 . . . . . . . . 9 𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)‘𝑚) = ∏𝑘 ∈ (𝑀...𝑗)𝐴
2725, 26eqtri 2632 . . . . . . . 8 𝑚 ∈ (𝑀...𝑗)((𝑘𝑍𝐴)‘𝑚) = ∏𝑘 ∈ (𝑀...𝑗)𝐴
28 eqidd 2611 . . . . . . . . 9 (((𝜑𝑗𝑍) ∧ 𝑚 ∈ (𝑀...𝑗)) → ((𝑘𝑍𝐴)‘𝑚) = ((𝑘𝑍𝐴)‘𝑚))
29 simpr 476 . . . . . . . . . 10 ((𝜑𝑗𝑍) → 𝑗𝑍)
3029, 5syl6eleq 2698 . . . . . . . . 9 ((𝜑𝑗𝑍) → 𝑗 ∈ (ℤ𝑀))
31 elfzuz 12209 . . . . . . . . . . . 12 (𝑚 ∈ (𝑀...𝑗) → 𝑚 ∈ (ℤ𝑀))
3231, 5syl6eleqr 2699 . . . . . . . . . . 11 (𝑚 ∈ (𝑀...𝑗) → 𝑚𝑍)
3332, 12sylan2 490 . . . . . . . . . 10 ((𝜑𝑚 ∈ (𝑀...𝑗)) → ((𝑘𝑍𝐴)‘𝑚) ∈ ℂ)
3433adantlr 747 . . . . . . . . 9 (((𝜑𝑗𝑍) ∧ 𝑚 ∈ (𝑀...𝑗)) → ((𝑘𝑍𝐴)‘𝑚) ∈ ℂ)
3528, 30, 34fprodser 14518 . . . . . . . 8 ((𝜑𝑗𝑍) → ∏𝑚 ∈ (𝑀...𝑗)((𝑘𝑍𝐴)‘𝑚) = (seq𝑀( · , (𝑘𝑍𝐴))‘𝑗))
3627, 35syl5eqr 2658 . . . . . . 7 ((𝜑𝑗𝑍) → ∏𝑘 ∈ (𝑀...𝑗)𝐴 = (seq𝑀( · , (𝑘𝑍𝐴))‘𝑗))
3717, 36eqtr2d 2645 . . . . . 6 ((𝜑𝑗𝑍) → (seq𝑀( · , (𝑘𝑍𝐴))‘𝑗) = (𝐹𝑗))
385, 16, 1, 6, 37climeq 14146 . . . . 5 (𝜑 → (seq𝑀( · , (𝑘𝑍𝐴)) ⇝ 𝑥𝐹𝑥))
3938iotabidv 5789 . . . 4 (𝜑 → (℩𝑥seq𝑀( · , (𝑘𝑍𝐴)) ⇝ 𝑥) = (℩𝑥𝐹𝑥))
40 df-fv 5812 . . . 4 ( ⇝ ‘seq𝑀( · , (𝑘𝑍𝐴))) = (℩𝑥seq𝑀( · , (𝑘𝑍𝐴)) ⇝ 𝑥)
41 df-fv 5812 . . . 4 ( ⇝ ‘𝐹) = (℩𝑥𝐹𝑥)
4239, 40, 413eqtr4g 2669 . . 3 (𝜑 → ( ⇝ ‘seq𝑀( · , (𝑘𝑍𝐴))) = ( ⇝ ‘𝐹))
4314, 42eqtrd 2644 . 2 (𝜑 → ∏𝑘𝑍 𝐴 = ( ⇝ ‘𝐹))
443, 43breqtrrd 4611 1 (𝜑𝐹 ⇝ ∏𝑘𝑍 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540   class class class wbr 4583   ↦ cmpt 4643  dom cdm 5038   ↾ cres 5040  ℩cio 5766  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  0cc0 9815   · cmul 9820  ℤcz 11254  ℤ≥cuz 11563  ...cfz 12197  seqcseq 12663   ⇝ cli 14063  ∏cprod 14474 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-prod 14475 This theorem is referenced by: (None)
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