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Theorem fprodabs2 40465
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 14922 . . . 4 (𝑥 = ∅ → ∏𝑘𝑥 𝐵 = ∏𝑘 ∈ ∅ 𝐵)
21fveq2d 6379 . . 3 (𝑥 = ∅ → (abs‘∏𝑘𝑥 𝐵) = (abs‘∏𝑘 ∈ ∅ 𝐵))
3 prodeq1 14922 . . 3 (𝑥 = ∅ → ∏𝑘𝑥 (abs‘𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵))
42, 3eqeq12d 2780 . 2 (𝑥 = ∅ → ((abs‘∏𝑘𝑥 𝐵) = ∏𝑘𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘 ∈ ∅ 𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵)))
5 prodeq1 14922 . . . 4 (𝑥 = 𝑦 → ∏𝑘𝑥 𝐵 = ∏𝑘𝑦 𝐵)
65fveq2d 6379 . . 3 (𝑥 = 𝑦 → (abs‘∏𝑘𝑥 𝐵) = (abs‘∏𝑘𝑦 𝐵))
7 prodeq1 14922 . . 3 (𝑥 = 𝑦 → ∏𝑘𝑥 (abs‘𝐵) = ∏𝑘𝑦 (abs‘𝐵))
86, 7eqeq12d 2780 . 2 (𝑥 = 𝑦 → ((abs‘∏𝑘𝑥 𝐵) = ∏𝑘𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵)))
9 prodeq1 14922 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → ∏𝑘𝑥 𝐵 = ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
109fveq2d 6379 . . 3 (𝑥 = (𝑦 ∪ {𝑧}) → (abs‘∏𝑘𝑥 𝐵) = (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
11 prodeq1 14922 . . 3 (𝑥 = (𝑦 ∪ {𝑧}) → ∏𝑘𝑥 (abs‘𝐵) = ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵))
1210, 11eqeq12d 2780 . 2 (𝑥 = (𝑦 ∪ {𝑧}) → ((abs‘∏𝑘𝑥 𝐵) = ∏𝑘𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵)))
13 prodeq1 14922 . . . 4 (𝑥 = 𝐴 → ∏𝑘𝑥 𝐵 = ∏𝑘𝐴 𝐵)
1413fveq2d 6379 . . 3 (𝑥 = 𝐴 → (abs‘∏𝑘𝑥 𝐵) = (abs‘∏𝑘𝐴 𝐵))
15 prodeq1 14922 . . 3 (𝑥 = 𝐴 → ∏𝑘𝑥 (abs‘𝐵) = ∏𝑘𝐴 (abs‘𝐵))
1614, 15eqeq12d 2780 . 2 (𝑥 = 𝐴 → ((abs‘∏𝑘𝑥 𝐵) = ∏𝑘𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘𝐴 𝐵) = ∏𝑘𝐴 (abs‘𝐵)))
17 abs1 14322 . . . 4 (abs‘1) = 1
18 prod0 14956 . . . . 5 𝑘 ∈ ∅ 𝐵 = 1
1918fveq2i 6378 . . . 4 (abs‘∏𝑘 ∈ ∅ 𝐵) = (abs‘1)
20 prod0 14956 . . . 4 𝑘 ∈ ∅ (abs‘𝐵) = 1
2117, 19, 203eqtr4i 2797 . . 3 (abs‘∏𝑘 ∈ ∅ 𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵)
2221a1i 11 . 2 (𝜑 → (abs‘∏𝑘 ∈ ∅ 𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵))
23 eqidd 2766 . . . 4 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ (abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵)) → (∏𝑘𝑦 (abs‘𝐵) · (abs‘𝑧 / 𝑘𝐵)) = (∏𝑘𝑦 (abs‘𝐵) · (abs‘𝑧 / 𝑘𝐵)))
24 nfv 2009 . . . . . . . 8 𝑘(𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦)))
25 nfcsb1v 3707 . . . . . . . 8 𝑘𝑧 / 𝑘𝐵
26 fprodabs2.a . . . . . . . . . . 11 (𝜑𝐴 ∈ Fin)
2726adantr 472 . . . . . . . . . 10 ((𝜑𝑦𝐴) → 𝐴 ∈ Fin)
28 simpr 477 . . . . . . . . . 10 ((𝜑𝑦𝐴) → 𝑦𝐴)
29 ssfi 8387 . . . . . . . . . 10 ((𝐴 ∈ Fin ∧ 𝑦𝐴) → 𝑦 ∈ Fin)
3027, 28, 29syl2anc 579 . . . . . . . . 9 ((𝜑𝑦𝐴) → 𝑦 ∈ Fin)
3130adantrr 708 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑦 ∈ Fin)
32 simprr 789 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑧 ∈ (𝐴𝑦))
3332eldifbd 3745 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → ¬ 𝑧𝑦)
34 simpll 783 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ 𝑘𝑦) → 𝜑)
3528sselda 3761 . . . . . . . . . 10 (((𝜑𝑦𝐴) ∧ 𝑘𝑦) → 𝑘𝐴)
3635adantlrr 712 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ 𝑘𝑦) → 𝑘𝐴)
37 fprodabs2.b . . . . . . . . 9 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
3834, 36, 37syl2anc 579 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ 𝑘𝑦) → 𝐵 ∈ ℂ)
39 csbeq1a 3700 . . . . . . . 8 (𝑘 = 𝑧𝐵 = 𝑧 / 𝑘𝐵)
40 simpl 474 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝜑)
4132eldifad 3744 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑧𝐴)
42 nfv 2009 . . . . . . . . . . 11 𝑘(𝜑𝑧𝐴)
4325nfel1 2922 . . . . . . . . . . 11 𝑘𝑧 / 𝑘𝐵 ∈ ℂ
4442, 43nfim 1995 . . . . . . . . . 10 𝑘((𝜑𝑧𝐴) → 𝑧 / 𝑘𝐵 ∈ ℂ)
45 eleq1w 2827 . . . . . . . . . . . 12 (𝑘 = 𝑧 → (𝑘𝐴𝑧𝐴))
4645anbi2d 622 . . . . . . . . . . 11 (𝑘 = 𝑧 → ((𝜑𝑘𝐴) ↔ (𝜑𝑧𝐴)))
4739eleq1d 2829 . . . . . . . . . . 11 (𝑘 = 𝑧 → (𝐵 ∈ ℂ ↔ 𝑧 / 𝑘𝐵 ∈ ℂ))
4846, 47imbi12d 335 . . . . . . . . . 10 (𝑘 = 𝑧 → (((𝜑𝑘𝐴) → 𝐵 ∈ ℂ) ↔ ((𝜑𝑧𝐴) → 𝑧 / 𝑘𝐵 ∈ ℂ)))
4944, 48, 37chvar 2368 . . . . . . . . 9 ((𝜑𝑧𝐴) → 𝑧 / 𝑘𝐵 ∈ ℂ)
5040, 41, 49syl2anc 579 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑧 / 𝑘𝐵 ∈ ℂ)
5124, 25, 31, 32, 33, 38, 39, 50fprodsplitsn 15002 . . . . . . 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 6379 . . . . 5 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ (abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵)) → (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (abs‘(∏𝑘𝑦 𝐵 · 𝑧 / 𝑘𝐵)))
5424, 31, 38fprodclf 15005 . . . . . . 7 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → ∏𝑘𝑦 𝐵 ∈ ℂ)
5554, 50absmuld 14478 . . . . . 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 6849 . . . . . 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 2803 . . . 4 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ (abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵)) → (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (∏𝑘𝑦 (abs‘𝐵) · (abs‘𝑧 / 𝑘𝐵)))
60 nfcv 2907 . . . . . . 7 𝑘abs
6160, 25nffv 6385 . . . . . 6 𝑘(abs‘𝑧 / 𝑘𝐵)
6238abscld 14460 . . . . . . 7 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ 𝑘𝑦) → (abs‘𝐵) ∈ ℝ)
6362recnd 10322 . . . . . 6 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ 𝑘𝑦) → (abs‘𝐵) ∈ ℂ)
6439fveq2d 6379 . . . . . 6 (𝑘 = 𝑧 → (abs‘𝐵) = (abs‘𝑧 / 𝑘𝐵))
6550abscld 14460 . . . . . . 7 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → (abs‘𝑧 / 𝑘𝐵) ∈ ℝ)
6665recnd 10322 . . . . . 6 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → (abs‘𝑧 / 𝑘𝐵) ∈ ℂ)
6724, 61, 31, 32, 33, 63, 64, 66fprodsplitsn 15002 . . . . 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 2809 . . 3 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ (abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵)) → (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵))
7069ex 401 . 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 8409 1 (𝜑 → (abs‘∏𝑘𝐴 𝐵) = ∏𝑘𝐴 (abs‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  csb 3691  cdif 3729  cun 3730  wss 3732  c0 4079  {csn 4334  cfv 6068  (class class class)co 6842  Fincfn 8160  cc 10187  1c1 10190   · cmul 10194  abscabs 14259  cprod 14918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-sup 8555  df-oi 8622  df-card 9016  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-n0 11539  df-z 11625  df-uz 11887  df-rp 12029  df-fz 12534  df-fzo 12674  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14124  df-re 14125  df-im 14126  df-sqrt 14260  df-abs 14261  df-clim 14504  df-prod 14919
This theorem is referenced by:  etransclem41  41129
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