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Theorem fprodabs2 38662
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 14478 . . . 4 (𝑥 = ∅ → ∏𝑘𝑥 𝐵 = ∏𝑘 ∈ ∅ 𝐵)
21fveq2d 6107 . . 3 (𝑥 = ∅ → (abs‘∏𝑘𝑥 𝐵) = (abs‘∏𝑘 ∈ ∅ 𝐵))
3 prodeq1 14478 . . 3 (𝑥 = ∅ → ∏𝑘𝑥 (abs‘𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵))
42, 3eqeq12d 2625 . 2 (𝑥 = ∅ → ((abs‘∏𝑘𝑥 𝐵) = ∏𝑘𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘 ∈ ∅ 𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵)))
5 prodeq1 14478 . . . 4 (𝑥 = 𝑦 → ∏𝑘𝑥 𝐵 = ∏𝑘𝑦 𝐵)
65fveq2d 6107 . . 3 (𝑥 = 𝑦 → (abs‘∏𝑘𝑥 𝐵) = (abs‘∏𝑘𝑦 𝐵))
7 prodeq1 14478 . . 3 (𝑥 = 𝑦 → ∏𝑘𝑥 (abs‘𝐵) = ∏𝑘𝑦 (abs‘𝐵))
86, 7eqeq12d 2625 . 2 (𝑥 = 𝑦 → ((abs‘∏𝑘𝑥 𝐵) = ∏𝑘𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵)))
9 prodeq1 14478 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → ∏𝑘𝑥 𝐵 = ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
109fveq2d 6107 . . 3 (𝑥 = (𝑦 ∪ {𝑧}) → (abs‘∏𝑘𝑥 𝐵) = (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
11 prodeq1 14478 . . 3 (𝑥 = (𝑦 ∪ {𝑧}) → ∏𝑘𝑥 (abs‘𝐵) = ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵))
1210, 11eqeq12d 2625 . 2 (𝑥 = (𝑦 ∪ {𝑧}) → ((abs‘∏𝑘𝑥 𝐵) = ∏𝑘𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵)))
13 prodeq1 14478 . . . 4 (𝑥 = 𝐴 → ∏𝑘𝑥 𝐵 = ∏𝑘𝐴 𝐵)
1413fveq2d 6107 . . 3 (𝑥 = 𝐴 → (abs‘∏𝑘𝑥 𝐵) = (abs‘∏𝑘𝐴 𝐵))
15 prodeq1 14478 . . 3 (𝑥 = 𝐴 → ∏𝑘𝑥 (abs‘𝐵) = ∏𝑘𝐴 (abs‘𝐵))
1614, 15eqeq12d 2625 . 2 (𝑥 = 𝐴 → ((abs‘∏𝑘𝑥 𝐵) = ∏𝑘𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘𝐴 𝐵) = ∏𝑘𝐴 (abs‘𝐵)))
17 abs1 13885 . . . 4 (abs‘1) = 1
18 prod0 14512 . . . . 5 𝑘 ∈ ∅ 𝐵 = 1
1918fveq2i 6106 . . . 4 (abs‘∏𝑘 ∈ ∅ 𝐵) = (abs‘1)
20 prod0 14512 . . . 4 𝑘 ∈ ∅ (abs‘𝐵) = 1
2117, 19, 203eqtr4i 2642 . . 3 (abs‘∏𝑘 ∈ ∅ 𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵)
2221a1i 11 . 2 (𝜑 → (abs‘∏𝑘 ∈ ∅ 𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵))
23 eqidd 2611 . . . 4 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ (abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵)) → (∏𝑘𝑦 (abs‘𝐵) · (abs‘𝑧 / 𝑘𝐵)) = (∏𝑘𝑦 (abs‘𝐵) · (abs‘𝑧 / 𝑘𝐵)))
24 nfv 1830 . . . . . . . 8 𝑘(𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦)))
25 nfcsb1v 3515 . . . . . . . 8 𝑘𝑧 / 𝑘𝐵
26 fprodabs2.a . . . . . . . . . . 11 (𝜑𝐴 ∈ Fin)
2726adantr 480 . . . . . . . . . 10 ((𝜑𝑦𝐴) → 𝐴 ∈ Fin)
28 simpr 476 . . . . . . . . . 10 ((𝜑𝑦𝐴) → 𝑦𝐴)
29 ssfi 8065 . . . . . . . . . 10 ((𝐴 ∈ Fin ∧ 𝑦𝐴) → 𝑦 ∈ Fin)
3027, 28, 29syl2anc 691 . . . . . . . . 9 ((𝜑𝑦𝐴) → 𝑦 ∈ Fin)
3130adantrr 749 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑦 ∈ Fin)
32 simprr 792 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑧 ∈ (𝐴𝑦))
3332eldifbd 3553 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → ¬ 𝑧𝑦)
34 simpll 786 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ 𝑘𝑦) → 𝜑)
3528sselda 3568 . . . . . . . . . 10 (((𝜑𝑦𝐴) ∧ 𝑘𝑦) → 𝑘𝐴)
3635adantlrr 753 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ 𝑘𝑦) → 𝑘𝐴)
37 fprodabs2.b . . . . . . . . 9 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
3834, 36, 37syl2anc 691 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ 𝑘𝑦) → 𝐵 ∈ ℂ)
39 csbeq1a 3508 . . . . . . . 8 (𝑘 = 𝑧𝐵 = 𝑧 / 𝑘𝐵)
40 simpl 472 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝜑)
4132eldifad 3552 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑧𝐴)
42 nfv 1830 . . . . . . . . . . 11 𝑘(𝜑𝑧𝐴)
4325nfel1 2765 . . . . . . . . . . 11 𝑘𝑧 / 𝑘𝐵 ∈ ℂ
4442, 43nfim 1813 . . . . . . . . . 10 𝑘((𝜑𝑧𝐴) → 𝑧 / 𝑘𝐵 ∈ ℂ)
45 eleq1 2676 . . . . . . . . . . . 12 (𝑘 = 𝑧 → (𝑘𝐴𝑧𝐴))
4645anbi2d 736 . . . . . . . . . . 11 (𝑘 = 𝑧 → ((𝜑𝑘𝐴) ↔ (𝜑𝑧𝐴)))
4739eleq1d 2672 . . . . . . . . . . 11 (𝑘 = 𝑧 → (𝐵 ∈ ℂ ↔ 𝑧 / 𝑘𝐵 ∈ ℂ))
4846, 47imbi12d 333 . . . . . . . . . 10 (𝑘 = 𝑧 → (((𝜑𝑘𝐴) → 𝐵 ∈ ℂ) ↔ ((𝜑𝑧𝐴) → 𝑧 / 𝑘𝐵 ∈ ℂ)))
4944, 48, 37chvar 2250 . . . . . . . . 9 ((𝜑𝑧𝐴) → 𝑧 / 𝑘𝐵 ∈ ℂ)
5040, 41, 49syl2anc 691 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑧 / 𝑘𝐵 ∈ ℂ)
5124, 25, 31, 32, 33, 38, 39, 50fprodsplitsn 14559 . . . . . . 7 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (∏𝑘𝑦 𝐵 · 𝑧 / 𝑘𝐵))
5251adantr 480 . . . . . 6 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ (abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵)) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (∏𝑘𝑦 𝐵 · 𝑧 / 𝑘𝐵))
5352fveq2d 6107 . . . . 5 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ (abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵)) → (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (abs‘(∏𝑘𝑦 𝐵 · 𝑧 / 𝑘𝐵)))
5424, 31, 38fprodclf 14562 . . . . . . 7 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → ∏𝑘𝑦 𝐵 ∈ ℂ)
5554, 50absmuld 14041 . . . . . 6 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → (abs‘(∏𝑘𝑦 𝐵 · 𝑧 / 𝑘𝐵)) = ((abs‘∏𝑘𝑦 𝐵) · (abs‘𝑧 / 𝑘𝐵)))
5655adantr 480 . . . . 5 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ (abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵)) → (abs‘(∏𝑘𝑦 𝐵 · 𝑧 / 𝑘𝐵)) = ((abs‘∏𝑘𝑦 𝐵) · (abs‘𝑧 / 𝑘𝐵)))
57 oveq1 6556 . . . . . 6 ((abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵) → ((abs‘∏𝑘𝑦 𝐵) · (abs‘𝑧 / 𝑘𝐵)) = (∏𝑘𝑦 (abs‘𝐵) · (abs‘𝑧 / 𝑘𝐵)))
5857adantl 481 . . . . 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 2648 . . . 4 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ (abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵)) → (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (∏𝑘𝑦 (abs‘𝐵) · (abs‘𝑧 / 𝑘𝐵)))
60 nfcv 2751 . . . . . . 7 𝑘abs
6160, 25nffv 6110 . . . . . 6 𝑘(abs‘𝑧 / 𝑘𝐵)
6238abscld 14023 . . . . . . 7 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ 𝑘𝑦) → (abs‘𝐵) ∈ ℝ)
6362recnd 9947 . . . . . 6 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ 𝑘𝑦) → (abs‘𝐵) ∈ ℂ)
6439fveq2d 6107 . . . . . 6 (𝑘 = 𝑧 → (abs‘𝐵) = (abs‘𝑧 / 𝑘𝐵))
6550abscld 14023 . . . . . . 7 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → (abs‘𝑧 / 𝑘𝐵) ∈ ℝ)
6665recnd 9947 . . . . . 6 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → (abs‘𝑧 / 𝑘𝐵) ∈ ℂ)
6724, 61, 31, 32, 33, 63, 64, 66fprodsplitsn 14559 . . . . 5 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵) = (∏𝑘𝑦 (abs‘𝐵) · (abs‘𝑧 / 𝑘𝐵)))
6867adantr 480 . . . 4 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ (abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵)) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵) = (∏𝑘𝑦 (abs‘𝐵) · (abs‘𝑧 / 𝑘𝐵)))
6923, 59, 683eqtr4d 2654 . . 3 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ (abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵)) → (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵))
7069ex 449 . 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 8087 1 (𝜑 → (abs‘∏𝑘𝐴 𝐵) = ∏𝑘𝐴 (abs‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  csb 3499  cdif 3537  cun 3538  wss 3540  c0 3874  {csn 4125  cfv 5804  (class class class)co 6549  Fincfn 7841  cc 9813  1c1 9816   · cmul 9820  abscabs 13822  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:  etransclem41  39168
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