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Theorem fprodabs2 42176
Description: The absolute value of a finite product . (Contributed by Glauco Siliprandi, 5-Apr-2020.)
Hypotheses
Ref Expression
fprodabs2.a (𝜑𝐴 ∈ Fin)
fprodabs2.b ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
Assertion
Ref Expression
fprodabs2 (𝜑 → (abs‘∏𝑘𝐴 𝐵) = ∏𝑘𝐴 (abs‘𝐵))
Distinct variable groups:   𝐴,𝑘   𝜑,𝑘
Allowed substitution hint:   𝐵(𝑘)

Proof of Theorem fprodabs2
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prodeq1 15254 . . . 4 (𝑥 = ∅ → ∏𝑘𝑥 𝐵 = ∏𝑘 ∈ ∅ 𝐵)
21fveq2d 6656 . . 3 (𝑥 = ∅ → (abs‘∏𝑘𝑥 𝐵) = (abs‘∏𝑘 ∈ ∅ 𝐵))
3 prodeq1 15254 . . 3 (𝑥 = ∅ → ∏𝑘𝑥 (abs‘𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵))
42, 3eqeq12d 2838 . 2 (𝑥 = ∅ → ((abs‘∏𝑘𝑥 𝐵) = ∏𝑘𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘 ∈ ∅ 𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵)))
5 prodeq1 15254 . . . 4 (𝑥 = 𝑦 → ∏𝑘𝑥 𝐵 = ∏𝑘𝑦 𝐵)
65fveq2d 6656 . . 3 (𝑥 = 𝑦 → (abs‘∏𝑘𝑥 𝐵) = (abs‘∏𝑘𝑦 𝐵))
7 prodeq1 15254 . . 3 (𝑥 = 𝑦 → ∏𝑘𝑥 (abs‘𝐵) = ∏𝑘𝑦 (abs‘𝐵))
86, 7eqeq12d 2838 . 2 (𝑥 = 𝑦 → ((abs‘∏𝑘𝑥 𝐵) = ∏𝑘𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵)))
9 prodeq1 15254 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → ∏𝑘𝑥 𝐵 = ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
109fveq2d 6656 . . 3 (𝑥 = (𝑦 ∪ {𝑧}) → (abs‘∏𝑘𝑥 𝐵) = (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
11 prodeq1 15254 . . 3 (𝑥 = (𝑦 ∪ {𝑧}) → ∏𝑘𝑥 (abs‘𝐵) = ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵))
1210, 11eqeq12d 2838 . 2 (𝑥 = (𝑦 ∪ {𝑧}) → ((abs‘∏𝑘𝑥 𝐵) = ∏𝑘𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵)))
13 prodeq1 15254 . . . 4 (𝑥 = 𝐴 → ∏𝑘𝑥 𝐵 = ∏𝑘𝐴 𝐵)
1413fveq2d 6656 . . 3 (𝑥 = 𝐴 → (abs‘∏𝑘𝑥 𝐵) = (abs‘∏𝑘𝐴 𝐵))
15 prodeq1 15254 . . 3 (𝑥 = 𝐴 → ∏𝑘𝑥 (abs‘𝐵) = ∏𝑘𝐴 (abs‘𝐵))
1614, 15eqeq12d 2838 . 2 (𝑥 = 𝐴 → ((abs‘∏𝑘𝑥 𝐵) = ∏𝑘𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘𝐴 𝐵) = ∏𝑘𝐴 (abs‘𝐵)))
17 abs1 14648 . . . 4 (abs‘1) = 1
18 prod0 15288 . . . . 5 𝑘 ∈ ∅ 𝐵 = 1
1918fveq2i 6655 . . . 4 (abs‘∏𝑘 ∈ ∅ 𝐵) = (abs‘1)
20 prod0 15288 . . . 4 𝑘 ∈ ∅ (abs‘𝐵) = 1
2117, 19, 203eqtr4i 2855 . . 3 (abs‘∏𝑘 ∈ ∅ 𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵)
2221a1i 11 . 2 (𝜑 → (abs‘∏𝑘 ∈ ∅ 𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵))
23 eqidd 2823 . . . 4 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ (abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵)) → (∏𝑘𝑦 (abs‘𝐵) · (abs‘𝑧 / 𝑘𝐵)) = (∏𝑘𝑦 (abs‘𝐵) · (abs‘𝑧 / 𝑘𝐵)))
24 nfv 1915 . . . . . . . 8 𝑘(𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦)))
25 nfcsb1v 3879 . . . . . . . 8 𝑘𝑧 / 𝑘𝐵
26 fprodabs2.a . . . . . . . . . . 11 (𝜑𝐴 ∈ Fin)
2726adantr 484 . . . . . . . . . 10 ((𝜑𝑦𝐴) → 𝐴 ∈ Fin)
28 simpr 488 . . . . . . . . . 10 ((𝜑𝑦𝐴) → 𝑦𝐴)
29 ssfi 8726 . . . . . . . . . 10 ((𝐴 ∈ Fin ∧ 𝑦𝐴) → 𝑦 ∈ Fin)
3027, 28, 29syl2anc 587 . . . . . . . . 9 ((𝜑𝑦𝐴) → 𝑦 ∈ Fin)
3130adantrr 716 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑦 ∈ Fin)
32 simprr 772 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑧 ∈ (𝐴𝑦))
3332eldifbd 3921 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → ¬ 𝑧𝑦)
34 simpll 766 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ 𝑘𝑦) → 𝜑)
3528sselda 3942 . . . . . . . . . 10 (((𝜑𝑦𝐴) ∧ 𝑘𝑦) → 𝑘𝐴)
3635adantlrr 720 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ 𝑘𝑦) → 𝑘𝐴)
37 fprodabs2.b . . . . . . . . 9 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
3834, 36, 37syl2anc 587 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ 𝑘𝑦) → 𝐵 ∈ ℂ)
39 csbeq1a 3869 . . . . . . . 8 (𝑘 = 𝑧𝐵 = 𝑧 / 𝑘𝐵)
40 simpl 486 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝜑)
4132eldifad 3920 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑧𝐴)
42 nfv 1915 . . . . . . . . . . 11 𝑘(𝜑𝑧𝐴)
4325nfel1 2995 . . . . . . . . . . 11 𝑘𝑧 / 𝑘𝐵 ∈ ℂ
4442, 43nfim 1897 . . . . . . . . . 10 𝑘((𝜑𝑧𝐴) → 𝑧 / 𝑘𝐵 ∈ ℂ)
45 eleq1w 2896 . . . . . . . . . . . 12 (𝑘 = 𝑧 → (𝑘𝐴𝑧𝐴))
4645anbi2d 631 . . . . . . . . . . 11 (𝑘 = 𝑧 → ((𝜑𝑘𝐴) ↔ (𝜑𝑧𝐴)))
4739eleq1d 2898 . . . . . . . . . . 11 (𝑘 = 𝑧 → (𝐵 ∈ ℂ ↔ 𝑧 / 𝑘𝐵 ∈ ℂ))
4846, 47imbi12d 348 . . . . . . . . . 10 (𝑘 = 𝑧 → (((𝜑𝑘𝐴) → 𝐵 ∈ ℂ) ↔ ((𝜑𝑧𝐴) → 𝑧 / 𝑘𝐵 ∈ ℂ)))
4944, 48, 37chvarfv 2243 . . . . . . . . 9 ((𝜑𝑧𝐴) → 𝑧 / 𝑘𝐵 ∈ ℂ)
5040, 41, 49syl2anc 587 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑧 / 𝑘𝐵 ∈ ℂ)
5124, 25, 31, 32, 33, 38, 39, 50fprodsplitsn 15334 . . . . . . 7 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (∏𝑘𝑦 𝐵 · 𝑧 / 𝑘𝐵))
5251adantr 484 . . . . . 6 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ (abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵)) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (∏𝑘𝑦 𝐵 · 𝑧 / 𝑘𝐵))
5352fveq2d 6656 . . . . 5 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ (abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵)) → (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (abs‘(∏𝑘𝑦 𝐵 · 𝑧 / 𝑘𝐵)))
5424, 31, 38fprodclf 15337 . . . . . . 7 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → ∏𝑘𝑦 𝐵 ∈ ℂ)
5554, 50absmuld 14805 . . . . . 6 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → (abs‘(∏𝑘𝑦 𝐵 · 𝑧 / 𝑘𝐵)) = ((abs‘∏𝑘𝑦 𝐵) · (abs‘𝑧 / 𝑘𝐵)))
5655adantr 484 . . . . 5 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ (abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵)) → (abs‘(∏𝑘𝑦 𝐵 · 𝑧 / 𝑘𝐵)) = ((abs‘∏𝑘𝑦 𝐵) · (abs‘𝑧 / 𝑘𝐵)))
57 oveq1 7147 . . . . . 6 ((abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵) → ((abs‘∏𝑘𝑦 𝐵) · (abs‘𝑧 / 𝑘𝐵)) = (∏𝑘𝑦 (abs‘𝐵) · (abs‘𝑧 / 𝑘𝐵)))
5857adantl 485 . . . . 5 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ (abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵)) → ((abs‘∏𝑘𝑦 𝐵) · (abs‘𝑧 / 𝑘𝐵)) = (∏𝑘𝑦 (abs‘𝐵) · (abs‘𝑧 / 𝑘𝐵)))
5953, 56, 583eqtrd 2861 . . . 4 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ (abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵)) → (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (∏𝑘𝑦 (abs‘𝐵) · (abs‘𝑧 / 𝑘𝐵)))
60 nfcv 2979 . . . . . . 7 𝑘abs
6160, 25nffv 6662 . . . . . 6 𝑘(abs‘𝑧 / 𝑘𝐵)
6238abscld 14787 . . . . . . 7 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ 𝑘𝑦) → (abs‘𝐵) ∈ ℝ)
6362recnd 10658 . . . . . 6 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ 𝑘𝑦) → (abs‘𝐵) ∈ ℂ)
6439fveq2d 6656 . . . . . 6 (𝑘 = 𝑧 → (abs‘𝐵) = (abs‘𝑧 / 𝑘𝐵))
6550abscld 14787 . . . . . . 7 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → (abs‘𝑧 / 𝑘𝐵) ∈ ℝ)
6665recnd 10658 . . . . . 6 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → (abs‘𝑧 / 𝑘𝐵) ∈ ℂ)
6724, 61, 31, 32, 33, 63, 64, 66fprodsplitsn 15334 . . . . 5 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵) = (∏𝑘𝑦 (abs‘𝐵) · (abs‘𝑧 / 𝑘𝐵)))
6867adantr 484 . . . 4 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ (abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵)) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵) = (∏𝑘𝑦 (abs‘𝐵) · (abs‘𝑧 / 𝑘𝐵)))
6923, 59, 683eqtr4d 2867 . . 3 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ (abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵)) → (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵))
7069ex 416 . 2 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → ((abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵) → (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵)))
714, 8, 12, 16, 22, 70, 26findcard2d 8748 1 (𝜑 → (abs‘∏𝑘𝐴 𝐵) = ∏𝑘𝐴 (abs‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  csb 3855  cdif 3905  cun 3906  wss 3908  c0 4265  {csn 4539  cfv 6334  (class class class)co 7140  Fincfn 8496  cc 10524  1c1 10527   · cmul 10531  abscabs 14584  cprod 15250
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-inf2 9092  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-se 5492  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-isom 6343  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-sup 8894  df-oi 8962  df-card 9356  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12886  df-fzo 13029  df-seq 13365  df-exp 13426  df-hash 13687  df-cj 14449  df-re 14450  df-im 14451  df-sqrt 14585  df-abs 14586  df-clim 14836  df-prod 15251
This theorem is referenced by:  etransclem41  42856
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