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Theorem fprodabs2 40749
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 15051 . . . 4 (𝑥 = ∅ → ∏𝑘𝑥 𝐵 = ∏𝑘 ∈ ∅ 𝐵)
21fveq2d 6452 . . 3 (𝑥 = ∅ → (abs‘∏𝑘𝑥 𝐵) = (abs‘∏𝑘 ∈ ∅ 𝐵))
3 prodeq1 15051 . . 3 (𝑥 = ∅ → ∏𝑘𝑥 (abs‘𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵))
42, 3eqeq12d 2793 . 2 (𝑥 = ∅ → ((abs‘∏𝑘𝑥 𝐵) = ∏𝑘𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘 ∈ ∅ 𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵)))
5 prodeq1 15051 . . . 4 (𝑥 = 𝑦 → ∏𝑘𝑥 𝐵 = ∏𝑘𝑦 𝐵)
65fveq2d 6452 . . 3 (𝑥 = 𝑦 → (abs‘∏𝑘𝑥 𝐵) = (abs‘∏𝑘𝑦 𝐵))
7 prodeq1 15051 . . 3 (𝑥 = 𝑦 → ∏𝑘𝑥 (abs‘𝐵) = ∏𝑘𝑦 (abs‘𝐵))
86, 7eqeq12d 2793 . 2 (𝑥 = 𝑦 → ((abs‘∏𝑘𝑥 𝐵) = ∏𝑘𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵)))
9 prodeq1 15051 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → ∏𝑘𝑥 𝐵 = ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
109fveq2d 6452 . . 3 (𝑥 = (𝑦 ∪ {𝑧}) → (abs‘∏𝑘𝑥 𝐵) = (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
11 prodeq1 15051 . . 3 (𝑥 = (𝑦 ∪ {𝑧}) → ∏𝑘𝑥 (abs‘𝐵) = ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵))
1210, 11eqeq12d 2793 . 2 (𝑥 = (𝑦 ∪ {𝑧}) → ((abs‘∏𝑘𝑥 𝐵) = ∏𝑘𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵)))
13 prodeq1 15051 . . . 4 (𝑥 = 𝐴 → ∏𝑘𝑥 𝐵 = ∏𝑘𝐴 𝐵)
1413fveq2d 6452 . . 3 (𝑥 = 𝐴 → (abs‘∏𝑘𝑥 𝐵) = (abs‘∏𝑘𝐴 𝐵))
15 prodeq1 15051 . . 3 (𝑥 = 𝐴 → ∏𝑘𝑥 (abs‘𝐵) = ∏𝑘𝐴 (abs‘𝐵))
1614, 15eqeq12d 2793 . 2 (𝑥 = 𝐴 → ((abs‘∏𝑘𝑥 𝐵) = ∏𝑘𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘𝐴 𝐵) = ∏𝑘𝐴 (abs‘𝐵)))
17 abs1 14451 . . . 4 (abs‘1) = 1
18 prod0 15085 . . . . 5 𝑘 ∈ ∅ 𝐵 = 1
1918fveq2i 6451 . . . 4 (abs‘∏𝑘 ∈ ∅ 𝐵) = (abs‘1)
20 prod0 15085 . . . 4 𝑘 ∈ ∅ (abs‘𝐵) = 1
2117, 19, 203eqtr4i 2812 . . 3 (abs‘∏𝑘 ∈ ∅ 𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵)
2221a1i 11 . 2 (𝜑 → (abs‘∏𝑘 ∈ ∅ 𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵))
23 eqidd 2779 . . . 4 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ (abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵)) → (∏𝑘𝑦 (abs‘𝐵) · (abs‘𝑧 / 𝑘𝐵)) = (∏𝑘𝑦 (abs‘𝐵) · (abs‘𝑧 / 𝑘𝐵)))
24 nfv 1957 . . . . . . . 8 𝑘(𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦)))
25 nfcsb1v 3767 . . . . . . . 8 𝑘𝑧 / 𝑘𝐵
26 fprodabs2.a . . . . . . . . . . 11 (𝜑𝐴 ∈ Fin)
2726adantr 474 . . . . . . . . . 10 ((𝜑𝑦𝐴) → 𝐴 ∈ Fin)
28 simpr 479 . . . . . . . . . 10 ((𝜑𝑦𝐴) → 𝑦𝐴)
29 ssfi 8470 . . . . . . . . . 10 ((𝐴 ∈ Fin ∧ 𝑦𝐴) → 𝑦 ∈ Fin)
3027, 28, 29syl2anc 579 . . . . . . . . 9 ((𝜑𝑦𝐴) → 𝑦 ∈ Fin)
3130adantrr 707 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑦 ∈ Fin)
32 simprr 763 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑧 ∈ (𝐴𝑦))
3332eldifbd 3805 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → ¬ 𝑧𝑦)
34 simpll 757 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ 𝑘𝑦) → 𝜑)
3528sselda 3821 . . . . . . . . . 10 (((𝜑𝑦𝐴) ∧ 𝑘𝑦) → 𝑘𝐴)
3635adantlrr 711 . . . . . . . . 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 3760 . . . . . . . 8 (𝑘 = 𝑧𝐵 = 𝑧 / 𝑘𝐵)
40 simpl 476 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝜑)
4132eldifad 3804 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑧𝐴)
42 nfv 1957 . . . . . . . . . . 11 𝑘(𝜑𝑧𝐴)
4325nfel1 2948 . . . . . . . . . . 11 𝑘𝑧 / 𝑘𝐵 ∈ ℂ
4442, 43nfim 1943 . . . . . . . . . 10 𝑘((𝜑𝑧𝐴) → 𝑧 / 𝑘𝐵 ∈ ℂ)
45 eleq1w 2842 . . . . . . . . . . . 12 (𝑘 = 𝑧 → (𝑘𝐴𝑧𝐴))
4645anbi2d 622 . . . . . . . . . . 11 (𝑘 = 𝑧 → ((𝜑𝑘𝐴) ↔ (𝜑𝑧𝐴)))
4739eleq1d 2844 . . . . . . . . . . 11 (𝑘 = 𝑧 → (𝐵 ∈ ℂ ↔ 𝑧 / 𝑘𝐵 ∈ ℂ))
4846, 47imbi12d 336 . . . . . . . . . 10 (𝑘 = 𝑧 → (((𝜑𝑘𝐴) → 𝐵 ∈ ℂ) ↔ ((𝜑𝑧𝐴) → 𝑧 / 𝑘𝐵 ∈ ℂ)))
4944, 48, 37chvar 2360 . . . . . . . . 9 ((𝜑𝑧𝐴) → 𝑧 / 𝑘𝐵 ∈ ℂ)
5040, 41, 49syl2anc 579 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑧 / 𝑘𝐵 ∈ ℂ)
5124, 25, 31, 32, 33, 38, 39, 50fprodsplitsn 15131 . . . . . . 7 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (∏𝑘𝑦 𝐵 · 𝑧 / 𝑘𝐵))
5251adantr 474 . . . . . 6 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ (abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵)) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (∏𝑘𝑦 𝐵 · 𝑧 / 𝑘𝐵))
5352fveq2d 6452 . . . . 5 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ (abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵)) → (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (abs‘(∏𝑘𝑦 𝐵 · 𝑧 / 𝑘𝐵)))
5424, 31, 38fprodclf 15134 . . . . . . 7 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → ∏𝑘𝑦 𝐵 ∈ ℂ)
5554, 50absmuld 14608 . . . . . 6 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → (abs‘(∏𝑘𝑦 𝐵 · 𝑧 / 𝑘𝐵)) = ((abs‘∏𝑘𝑦 𝐵) · (abs‘𝑧 / 𝑘𝐵)))
5655adantr 474 . . . . 5 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ (abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵)) → (abs‘(∏𝑘𝑦 𝐵 · 𝑧 / 𝑘𝐵)) = ((abs‘∏𝑘𝑦 𝐵) · (abs‘𝑧 / 𝑘𝐵)))
57 oveq1 6931 . . . . . 6 ((abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵) → ((abs‘∏𝑘𝑦 𝐵) · (abs‘𝑧 / 𝑘𝐵)) = (∏𝑘𝑦 (abs‘𝐵) · (abs‘𝑧 / 𝑘𝐵)))
5857adantl 475 . . . . 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 2818 . . . 4 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ (abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵)) → (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (∏𝑘𝑦 (abs‘𝐵) · (abs‘𝑧 / 𝑘𝐵)))
60 nfcv 2934 . . . . . . 7 𝑘abs
6160, 25nffv 6458 . . . . . 6 𝑘(abs‘𝑧 / 𝑘𝐵)
6238abscld 14590 . . . . . . 7 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ 𝑘𝑦) → (abs‘𝐵) ∈ ℝ)
6362recnd 10407 . . . . . 6 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ 𝑘𝑦) → (abs‘𝐵) ∈ ℂ)
6439fveq2d 6452 . . . . . 6 (𝑘 = 𝑧 → (abs‘𝐵) = (abs‘𝑧 / 𝑘𝐵))
6550abscld 14590 . . . . . . 7 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → (abs‘𝑧 / 𝑘𝐵) ∈ ℝ)
6665recnd 10407 . . . . . 6 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → (abs‘𝑧 / 𝑘𝐵) ∈ ℂ)
6724, 61, 31, 32, 33, 63, 64, 66fprodsplitsn 15131 . . . . 5 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵) = (∏𝑘𝑦 (abs‘𝐵) · (abs‘𝑧 / 𝑘𝐵)))
6867adantr 474 . . . 4 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ (abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵)) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵) = (∏𝑘𝑦 (abs‘𝐵) · (abs‘𝑧 / 𝑘𝐵)))
6923, 59, 683eqtr4d 2824 . . 3 (((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ (abs‘∏𝑘𝑦 𝐵) = ∏𝑘𝑦 (abs‘𝐵)) → (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵))
7069ex 403 . 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 8492 1 (𝜑 → (abs‘∏𝑘𝐴 𝐵) = ∏𝑘𝐴 (abs‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2107  csb 3751  cdif 3789  cun 3790  wss 3792  c0 4141  {csn 4398  cfv 6137  (class class class)co 6924  Fincfn 8243  cc 10272  1c1 10275   · cmul 10279  abscabs 14387  cprod 15047
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-inf2 8837  ax-cnex 10330  ax-resscn 10331  ax-1cn 10332  ax-icn 10333  ax-addcl 10334  ax-addrcl 10335  ax-mulcl 10336  ax-mulrcl 10337  ax-mulcom 10338  ax-addass 10339  ax-mulass 10340  ax-distr 10341  ax-i2m1 10342  ax-1ne0 10343  ax-1rid 10344  ax-rnegex 10345  ax-rrecex 10346  ax-cnre 10347  ax-pre-lttri 10348  ax-pre-lttrn 10349  ax-pre-ltadd 10350  ax-pre-mulgt0 10351  ax-pre-sup 10352
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-int 4713  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-se 5317  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-isom 6146  df-riota 6885  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-om 7346  df-1st 7447  df-2nd 7448  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-1o 7845  df-oadd 7849  df-er 8028  df-en 8244  df-dom 8245  df-sdom 8246  df-fin 8247  df-sup 8638  df-oi 8706  df-card 9100  df-pnf 10415  df-mnf 10416  df-xr 10417  df-ltxr 10418  df-le 10419  df-sub 10610  df-neg 10611  df-div 11036  df-nn 11380  df-2 11443  df-3 11444  df-n0 11648  df-z 11734  df-uz 11998  df-rp 12143  df-fz 12649  df-fzo 12790  df-seq 13125  df-exp 13184  df-hash 13442  df-cj 14252  df-re 14253  df-im 14254  df-sqrt 14388  df-abs 14389  df-clim 14636  df-prod 15048
This theorem is referenced by:  etransclem41  41433
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