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Theorem fprodabs2 40330
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 14838 . . . 4 (𝑥 = ∅ → ∏𝑘𝑥 𝐵 = ∏𝑘 ∈ ∅ 𝐵)
21fveq2d 6356 . . 3 (𝑥 = ∅ → (abs‘∏𝑘𝑥 𝐵) = (abs‘∏𝑘 ∈ ∅ 𝐵))
3 prodeq1 14838 . . 3 (𝑥 = ∅ → ∏𝑘𝑥 (abs‘𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵))
42, 3eqeq12d 2775 . 2 (𝑥 = ∅ → ((abs‘∏𝑘𝑥 𝐵) = ∏𝑘𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘 ∈ ∅ 𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵)))
5 prodeq1 14838 . . . 4 (𝑥 = 𝑦 → ∏𝑘𝑥 𝐵 = ∏𝑘𝑦 𝐵)
65fveq2d 6356 . . 3 (𝑥 = 𝑦 → (abs‘∏𝑘𝑥 𝐵) = (abs‘∏𝑘𝑦 𝐵))
7 prodeq1 14838 . . 3 (𝑥 = 𝑦 → ∏𝑘𝑥 (abs‘𝐵) = ∏𝑘𝑦 (abs‘𝐵))
86, 7eqeq12d 2775 . 2 (𝑥 = 𝑦 → ((abs‘∏𝑘𝑥 𝐵) = ∏𝑘𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵)))
9 prodeq1 14838 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → ∏𝑘𝑥 𝐵 = ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
109fveq2d 6356 . . 3 (𝑥 = (𝑦 ∪ {𝑧}) → (abs‘∏𝑘𝑥 𝐵) = (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
11 prodeq1 14838 . . 3 (𝑥 = (𝑦 ∪ {𝑧}) → ∏𝑘𝑥 (abs‘𝐵) = ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵))
1210, 11eqeq12d 2775 . 2 (𝑥 = (𝑦 ∪ {𝑧}) → ((abs‘∏𝑘𝑥 𝐵) = ∏𝑘𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵)))
13 prodeq1 14838 . . . 4 (𝑥 = 𝐴 → ∏𝑘𝑥 𝐵 = ∏𝑘𝐴 𝐵)
1413fveq2d 6356 . . 3 (𝑥 = 𝐴 → (abs‘∏𝑘𝑥 𝐵) = (abs‘∏𝑘𝐴 𝐵))
15 prodeq1 14838 . . 3 (𝑥 = 𝐴 → ∏𝑘𝑥 (abs‘𝐵) = ∏𝑘𝐴 (abs‘𝐵))
1614, 15eqeq12d 2775 . 2 (𝑥 = 𝐴 → ((abs‘∏𝑘𝑥 𝐵) = ∏𝑘𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘𝐴 𝐵) = ∏𝑘𝐴 (abs‘𝐵)))
17 abs1 14236 . . . 4 (abs‘1) = 1
18 prod0 14872 . . . . 5 𝑘 ∈ ∅ 𝐵 = 1
1918fveq2i 6355 . . . 4 (abs‘∏𝑘 ∈ ∅ 𝐵) = (abs‘1)
20 prod0 14872 . . . 4 𝑘 ∈ ∅ (abs‘𝐵) = 1
2117, 19, 203eqtr4i 2792 . . 3 (abs‘∏𝑘 ∈ ∅ 𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵)
2221a1i 11 . 2 (𝜑 → (abs‘∏𝑘 ∈ ∅ 𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵))
23 eqidd 2761 . . . 4 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ (abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵)) → (∏𝑘𝑦 (abs‘𝐵) · (abs‘𝑧 / 𝑘𝐵)) = (∏𝑘𝑦 (abs‘𝐵) · (abs‘𝑧 / 𝑘𝐵)))
24 nfv 1992 . . . . . . . 8 𝑘(𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦)))
25 nfcsb1v 3690 . . . . . . . 8 𝑘𝑧 / 𝑘𝐵
26 fprodabs2.a . . . . . . . . . . 11 (𝜑𝐴 ∈ Fin)
2726adantr 472 . . . . . . . . . 10 ((𝜑𝑦𝐴) → 𝐴 ∈ Fin)
28 simpr 479 . . . . . . . . . 10 ((𝜑𝑦𝐴) → 𝑦𝐴)
29 ssfi 8345 . . . . . . . . . 10 ((𝐴 ∈ Fin ∧ 𝑦𝐴) → 𝑦 ∈ Fin)
3027, 28, 29syl2anc 696 . . . . . . . . 9 ((𝜑𝑦𝐴) → 𝑦 ∈ Fin)
3130adantrr 755 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑦 ∈ Fin)
32 simprr 813 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑧 ∈ (𝐴𝑦))
3332eldifbd 3728 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → ¬ 𝑧𝑦)
34 simpll 807 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ 𝑘𝑦) → 𝜑)
3528sselda 3744 . . . . . . . . . 10 (((𝜑𝑦𝐴) ∧ 𝑘𝑦) → 𝑘𝐴)
3635adantlrr 759 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ 𝑘𝑦) → 𝑘𝐴)
37 fprodabs2.b . . . . . . . . 9 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
3834, 36, 37syl2anc 696 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ 𝑘𝑦) → 𝐵 ∈ ℂ)
39 csbeq1a 3683 . . . . . . . 8 (𝑘 = 𝑧𝐵 = 𝑧 / 𝑘𝐵)
40 simpl 474 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝜑)
4132eldifad 3727 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑧𝐴)
42 nfv 1992 . . . . . . . . . . 11 𝑘(𝜑𝑧𝐴)
4325nfel1 2917 . . . . . . . . . . 11 𝑘𝑧 / 𝑘𝐵 ∈ ℂ
4442, 43nfim 1974 . . . . . . . . . 10 𝑘((𝜑𝑧𝐴) → 𝑧 / 𝑘𝐵 ∈ ℂ)
45 eleq1w 2822 . . . . . . . . . . . 12 (𝑘 = 𝑧 → (𝑘𝐴𝑧𝐴))
4645anbi2d 742 . . . . . . . . . . 11 (𝑘 = 𝑧 → ((𝜑𝑘𝐴) ↔ (𝜑𝑧𝐴)))
4739eleq1d 2824 . . . . . . . . . . 11 (𝑘 = 𝑧 → (𝐵 ∈ ℂ ↔ 𝑧 / 𝑘𝐵 ∈ ℂ))
4846, 47imbi12d 333 . . . . . . . . . 10 (𝑘 = 𝑧 → (((𝜑𝑘𝐴) → 𝐵 ∈ ℂ) ↔ ((𝜑𝑧𝐴) → 𝑧 / 𝑘𝐵 ∈ ℂ)))
4944, 48, 37chvar 2407 . . . . . . . . 9 ((𝜑𝑧𝐴) → 𝑧 / 𝑘𝐵 ∈ ℂ)
5040, 41, 49syl2anc 696 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑧 / 𝑘𝐵 ∈ ℂ)
5124, 25, 31, 32, 33, 38, 39, 50fprodsplitsn 14919 . . . . . . 7 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (∏𝑘𝑦 𝐵 · 𝑧 / 𝑘𝐵))
5251adantr 472 . . . . . 6 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ (abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵)) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (∏𝑘𝑦 𝐵 · 𝑧 / 𝑘𝐵))
5352fveq2d 6356 . . . . 5 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ (abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵)) → (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (abs‘(∏𝑘𝑦 𝐵 · 𝑧 / 𝑘𝐵)))
5424, 31, 38fprodclf 14922 . . . . . . 7 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → ∏𝑘𝑦 𝐵 ∈ ℂ)
5554, 50absmuld 14392 . . . . . 6 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → (abs‘(∏𝑘𝑦 𝐵 · 𝑧 / 𝑘𝐵)) = ((abs‘∏𝑘𝑦 𝐵) · (abs‘𝑧 / 𝑘𝐵)))
5655adantr 472 . . . . 5 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ (abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵)) → (abs‘(∏𝑘𝑦 𝐵 · 𝑧 / 𝑘𝐵)) = ((abs‘∏𝑘𝑦 𝐵) · (abs‘𝑧 / 𝑘𝐵)))
57 oveq1 6820 . . . . . 6 ((abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵) → ((abs‘∏𝑘𝑦 𝐵) · (abs‘𝑧 / 𝑘𝐵)) = (∏𝑘𝑦 (abs‘𝐵) · (abs‘𝑧 / 𝑘𝐵)))
5857adantl 473 . . . . 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 2798 . . . 4 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ (abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵)) → (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (∏𝑘𝑦 (abs‘𝐵) · (abs‘𝑧 / 𝑘𝐵)))
60 nfcv 2902 . . . . . . 7 𝑘abs
6160, 25nffv 6359 . . . . . 6 𝑘(abs‘𝑧 / 𝑘𝐵)
6238abscld 14374 . . . . . . 7 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ 𝑘𝑦) → (abs‘𝐵) ∈ ℝ)
6362recnd 10260 . . . . . 6 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ 𝑘𝑦) → (abs‘𝐵) ∈ ℂ)
6439fveq2d 6356 . . . . . 6 (𝑘 = 𝑧 → (abs‘𝐵) = (abs‘𝑧 / 𝑘𝐵))
6550abscld 14374 . . . . . . 7 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → (abs‘𝑧 / 𝑘𝐵) ∈ ℝ)
6665recnd 10260 . . . . . 6 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → (abs‘𝑧 / 𝑘𝐵) ∈ ℂ)
6724, 61, 31, 32, 33, 63, 64, 66fprodsplitsn 14919 . . . . 5 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵) = (∏𝑘𝑦 (abs‘𝐵) · (abs‘𝑧 / 𝑘𝐵)))
6867adantr 472 . . . 4 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ (abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵)) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵) = (∏𝑘𝑦 (abs‘𝐵) · (abs‘𝑧 / 𝑘𝐵)))
6923, 59, 683eqtr4d 2804 . . 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 8367 1 (𝜑 → (abs‘∏𝑘𝐴 𝐵) = ∏𝑘𝐴 (abs‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  csb 3674  cdif 3712  cun 3713  wss 3715  c0 4058  {csn 4321  cfv 6049  (class class class)co 6813  Fincfn 8121  cc 10126  1c1 10129   · cmul 10133  abscabs 14173  cprod 14834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205  ax-pre-sup 10206
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-sup 8513  df-oi 8580  df-card 8955  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-div 10877  df-nn 11213  df-2 11271  df-3 11272  df-n0 11485  df-z 11570  df-uz 11880  df-rp 12026  df-fz 12520  df-fzo 12660  df-seq 12996  df-exp 13055  df-hash 13312  df-cj 14038  df-re 14039  df-im 14040  df-sqrt 14174  df-abs 14175  df-clim 14418  df-prod 14835
This theorem is referenced by:  etransclem41  40995
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