ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fprodrec GIF version

Theorem fprodrec 11632
Description: The finite product of reciprocals is the reciprocal of the product. (Contributed by Jim Kingdon, 28-Aug-2024.)
Hypotheses
Ref Expression
fprodrec.a (𝜑𝐴 ∈ Fin)
fprodrec.ccl ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
fprodrec.cap ((𝜑𝑘𝐴) → 𝐵 # 0)
Assertion
Ref Expression
fprodrec (𝜑 → ∏𝑘𝐴 (1 / 𝐵) = (1 / ∏𝑘𝐴 𝐵))
Distinct variable groups:   𝐴,𝑘   𝜑,𝑘
Allowed substitution hint:   𝐵(𝑘)

Proof of Theorem fprodrec
Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prodeq1 11556 . . 3 (𝑤 = ∅ → ∏𝑘𝑤 (1 / 𝐵) = ∏𝑘 ∈ ∅ (1 / 𝐵))
2 prodeq1 11556 . . . 4 (𝑤 = ∅ → ∏𝑘𝑤 𝐵 = ∏𝑘 ∈ ∅ 𝐵)
32oveq2d 5890 . . 3 (𝑤 = ∅ → (1 / ∏𝑘𝑤 𝐵) = (1 / ∏𝑘 ∈ ∅ 𝐵))
41, 3eqeq12d 2192 . 2 (𝑤 = ∅ → (∏𝑘𝑤 (1 / 𝐵) = (1 / ∏𝑘𝑤 𝐵) ↔ ∏𝑘 ∈ ∅ (1 / 𝐵) = (1 / ∏𝑘 ∈ ∅ 𝐵)))
5 prodeq1 11556 . . 3 (𝑤 = 𝑦 → ∏𝑘𝑤 (1 / 𝐵) = ∏𝑘𝑦 (1 / 𝐵))
6 prodeq1 11556 . . . 4 (𝑤 = 𝑦 → ∏𝑘𝑤 𝐵 = ∏𝑘𝑦 𝐵)
76oveq2d 5890 . . 3 (𝑤 = 𝑦 → (1 / ∏𝑘𝑤 𝐵) = (1 / ∏𝑘𝑦 𝐵))
85, 7eqeq12d 2192 . 2 (𝑤 = 𝑦 → (∏𝑘𝑤 (1 / 𝐵) = (1 / ∏𝑘𝑤 𝐵) ↔ ∏𝑘𝑦 (1 / 𝐵) = (1 / ∏𝑘𝑦 𝐵)))
9 prodeq1 11556 . . 3 (𝑤 = (𝑦 ∪ {𝑧}) → ∏𝑘𝑤 (1 / 𝐵) = ∏𝑘 ∈ (𝑦 ∪ {𝑧})(1 / 𝐵))
10 prodeq1 11556 . . . 4 (𝑤 = (𝑦 ∪ {𝑧}) → ∏𝑘𝑤 𝐵 = ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
1110oveq2d 5890 . . 3 (𝑤 = (𝑦 ∪ {𝑧}) → (1 / ∏𝑘𝑤 𝐵) = (1 / ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
129, 11eqeq12d 2192 . 2 (𝑤 = (𝑦 ∪ {𝑧}) → (∏𝑘𝑤 (1 / 𝐵) = (1 / ∏𝑘𝑤 𝐵) ↔ ∏𝑘 ∈ (𝑦 ∪ {𝑧})(1 / 𝐵) = (1 / ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)))
13 prodeq1 11556 . . 3 (𝑤 = 𝐴 → ∏𝑘𝑤 (1 / 𝐵) = ∏𝑘𝐴 (1 / 𝐵))
14 prodeq1 11556 . . . 4 (𝑤 = 𝐴 → ∏𝑘𝑤 𝐵 = ∏𝑘𝐴 𝐵)
1514oveq2d 5890 . . 3 (𝑤 = 𝐴 → (1 / ∏𝑘𝑤 𝐵) = (1 / ∏𝑘𝐴 𝐵))
1613, 15eqeq12d 2192 . 2 (𝑤 = 𝐴 → (∏𝑘𝑤 (1 / 𝐵) = (1 / ∏𝑘𝑤 𝐵) ↔ ∏𝑘𝐴 (1 / 𝐵) = (1 / ∏𝑘𝐴 𝐵)))
17 1div1e1 8659 . . . 4 (1 / 1) = 1
18 prod0 11588 . . . . 5 𝑘 ∈ ∅ 𝐵 = 1
1918oveq2i 5885 . . . 4 (1 / ∏𝑘 ∈ ∅ 𝐵) = (1 / 1)
20 prod0 11588 . . . 4 𝑘 ∈ ∅ (1 / 𝐵) = 1
2117, 19, 203eqtr4ri 2209 . . 3 𝑘 ∈ ∅ (1 / 𝐵) = (1 / ∏𝑘 ∈ ∅ 𝐵)
2221a1i 9 . 2 (𝜑 → ∏𝑘 ∈ ∅ (1 / 𝐵) = (1 / ∏𝑘 ∈ ∅ 𝐵))
23 simpr 110 . . . . . 6 ((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ ∏𝑘𝑦 (1 / 𝐵) = (1 / ∏𝑘𝑦 𝐵)) → ∏𝑘𝑦 (1 / 𝐵) = (1 / ∏𝑘𝑦 𝐵))
2423oveq1d 5889 . . . . 5 ((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ ∏𝑘𝑦 (1 / 𝐵) = (1 / ∏𝑘𝑦 𝐵)) → (∏𝑘𝑦 (1 / 𝐵) · (1 / 𝑧 / 𝑘𝐵)) = ((1 / ∏𝑘𝑦 𝐵) · (1 / 𝑧 / 𝑘𝐵)))
25 1cnd 7972 . . . . . . 7 ((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ ∏𝑘𝑦 (1 / 𝐵) = (1 / ∏𝑘𝑦 𝐵)) → 1 ∈ ℂ)
26 simplr 528 . . . . . . . . 9 (((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑦 ∈ Fin)
27 simplll 533 . . . . . . . . . 10 ((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ 𝑘𝑦) → 𝜑)
28 simplrl 535 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ 𝑘𝑦) → 𝑦𝐴)
29 simpr 110 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ 𝑘𝑦) → 𝑘𝑦)
3028, 29sseldd 3156 . . . . . . . . . 10 ((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ 𝑘𝑦) → 𝑘𝐴)
31 fprodrec.ccl . . . . . . . . . 10 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
3227, 30, 31syl2anc 411 . . . . . . . . 9 ((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ 𝑘𝑦) → 𝐵 ∈ ℂ)
3326, 32fprodcl 11610 . . . . . . . 8 (((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → ∏𝑘𝑦 𝐵 ∈ ℂ)
3433adantr 276 . . . . . . 7 ((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ ∏𝑘𝑦 (1 / 𝐵) = (1 / ∏𝑘𝑦 𝐵)) → ∏𝑘𝑦 𝐵 ∈ ℂ)
35 simprr 531 . . . . . . . . . 10 (((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑧 ∈ (𝐴𝑦))
3635eldifad 3140 . . . . . . . . 9 (((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑧𝐴)
3731ralrimiva 2550 . . . . . . . . . 10 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
3837ad2antrr 488 . . . . . . . . 9 (((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → ∀𝑘𝐴 𝐵 ∈ ℂ)
39 nfcsb1v 3090 . . . . . . . . . . 11 𝑘𝑧 / 𝑘𝐵
4039nfel1 2330 . . . . . . . . . 10 𝑘𝑧 / 𝑘𝐵 ∈ ℂ
41 csbeq1a 3066 . . . . . . . . . . 11 (𝑘 = 𝑧𝐵 = 𝑧 / 𝑘𝐵)
4241eleq1d 2246 . . . . . . . . . 10 (𝑘 = 𝑧 → (𝐵 ∈ ℂ ↔ 𝑧 / 𝑘𝐵 ∈ ℂ))
4340, 42rspc 2835 . . . . . . . . 9 (𝑧𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → 𝑧 / 𝑘𝐵 ∈ ℂ))
4436, 38, 43sylc 62 . . . . . . . 8 (((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑧 / 𝑘𝐵 ∈ ℂ)
4544adantr 276 . . . . . . 7 ((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ ∏𝑘𝑦 (1 / 𝐵) = (1 / ∏𝑘𝑦 𝐵)) → 𝑧 / 𝑘𝐵 ∈ ℂ)
46 fprodrec.cap . . . . . . . . . 10 ((𝜑𝑘𝐴) → 𝐵 # 0)
4727, 30, 46syl2anc 411 . . . . . . . . 9 ((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ 𝑘𝑦) → 𝐵 # 0)
4826, 32, 47fprodap0 11624 . . . . . . . 8 (((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → ∏𝑘𝑦 𝐵 # 0)
4948adantr 276 . . . . . . 7 ((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ ∏𝑘𝑦 (1 / 𝐵) = (1 / ∏𝑘𝑦 𝐵)) → ∏𝑘𝑦 𝐵 # 0)
5046ralrimiva 2550 . . . . . . . . . 10 (𝜑 → ∀𝑘𝐴 𝐵 # 0)
5150ad2antrr 488 . . . . . . . . 9 (((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → ∀𝑘𝐴 𝐵 # 0)
52 nfcv 2319 . . . . . . . . . . 11 𝑘 #
53 nfcv 2319 . . . . . . . . . . 11 𝑘0
5439, 52, 53nfbr 4049 . . . . . . . . . 10 𝑘𝑧 / 𝑘𝐵 # 0
5541breq1d 4013 . . . . . . . . . 10 (𝑘 = 𝑧 → (𝐵 # 0 ↔ 𝑧 / 𝑘𝐵 # 0))
5654, 55rspc 2835 . . . . . . . . 9 (𝑧𝐴 → (∀𝑘𝐴 𝐵 # 0 → 𝑧 / 𝑘𝐵 # 0))
5736, 51, 56sylc 62 . . . . . . . 8 (((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑧 / 𝑘𝐵 # 0)
5857adantr 276 . . . . . . 7 ((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ ∏𝑘𝑦 (1 / 𝐵) = (1 / ∏𝑘𝑦 𝐵)) → 𝑧 / 𝑘𝐵 # 0)
5925, 34, 25, 45, 49, 58divmuldivapd 8787 . . . . . 6 ((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ ∏𝑘𝑦 (1 / 𝐵) = (1 / ∏𝑘𝑦 𝐵)) → ((1 / ∏𝑘𝑦 𝐵) · (1 / 𝑧 / 𝑘𝐵)) = ((1 · 1) / (∏𝑘𝑦 𝐵 · 𝑧 / 𝑘𝐵)))
60 1t1e1 9069 . . . . . . 7 (1 · 1) = 1
6160oveq1i 5884 . . . . . 6 ((1 · 1) / (∏𝑘𝑦 𝐵 · 𝑧 / 𝑘𝐵)) = (1 / (∏𝑘𝑦 𝐵 · 𝑧 / 𝑘𝐵))
6259, 61eqtrdi 2226 . . . . 5 ((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ ∏𝑘𝑦 (1 / 𝐵) = (1 / ∏𝑘𝑦 𝐵)) → ((1 / ∏𝑘𝑦 𝐵) · (1 / 𝑧 / 𝑘𝐵)) = (1 / (∏𝑘𝑦 𝐵 · 𝑧 / 𝑘𝐵)))
6324, 62eqtrd 2210 . . . 4 ((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ ∏𝑘𝑦 (1 / 𝐵) = (1 / ∏𝑘𝑦 𝐵)) → (∏𝑘𝑦 (1 / 𝐵) · (1 / 𝑧 / 𝑘𝐵)) = (1 / (∏𝑘𝑦 𝐵 · 𝑧 / 𝑘𝐵)))
64 nfcv 2319 . . . . . . 7 𝑘1
65 nfcv 2319 . . . . . . 7 𝑘 /
6664, 65, 39nfov 5904 . . . . . 6 𝑘(1 / 𝑧 / 𝑘𝐵)
6735eldifbd 3141 . . . . . 6 (((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → ¬ 𝑧𝑦)
6832, 47recclapd 8736 . . . . . 6 ((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ 𝑘𝑦) → (1 / 𝐵) ∈ ℂ)
6944, 57recclapd 8736 . . . . . 6 (((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → (1 / 𝑧 / 𝑘𝐵) ∈ ℂ)
7041oveq2d 5890 . . . . . 6 (𝑘 = 𝑧 → (1 / 𝐵) = (1 / 𝑧 / 𝑘𝐵))
7166, 26, 35, 67, 68, 69, 70fprodunsn 11607 . . . . 5 (((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})(1 / 𝐵) = (∏𝑘𝑦 (1 / 𝐵) · (1 / 𝑧 / 𝑘𝐵)))
7271adantr 276 . . . 4 ((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ ∏𝑘𝑦 (1 / 𝐵) = (1 / ∏𝑘𝑦 𝐵)) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})(1 / 𝐵) = (∏𝑘𝑦 (1 / 𝐵) · (1 / 𝑧 / 𝑘𝐵)))
7339, 26, 35, 67, 32, 44, 41fprodunsn 11607 . . . . . 6 (((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (∏𝑘𝑦 𝐵 · 𝑧 / 𝑘𝐵))
7473oveq2d 5890 . . . . 5 (((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → (1 / ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (1 / (∏𝑘𝑦 𝐵 · 𝑧 / 𝑘𝐵)))
7574adantr 276 . . . 4 ((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ ∏𝑘𝑦 (1 / 𝐵) = (1 / ∏𝑘𝑦 𝐵)) → (1 / ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (1 / (∏𝑘𝑦 𝐵 · 𝑧 / 𝑘𝐵)))
7663, 72, 753eqtr4d 2220 . . 3 ((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ ∏𝑘𝑦 (1 / 𝐵) = (1 / ∏𝑘𝑦 𝐵)) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})(1 / 𝐵) = (1 / ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
7776ex 115 . 2 (((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → (∏𝑘𝑦 (1 / 𝐵) = (1 / ∏𝑘𝑦 𝐵) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})(1 / 𝐵) = (1 / ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)))
78 fprodrec.a . 2 (𝜑𝐴 ∈ Fin)
794, 8, 12, 16, 22, 77, 78findcard2sd 6891 1 (𝜑 → ∏𝑘𝐴 (1 / 𝐵) = (1 / ∏𝑘𝐴 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  wral 2455  csb 3057  cdif 3126  cun 3127  wss 3129  c0 3422  {csn 3592   class class class wbr 4003  (class class class)co 5874  Fincfn 6739  cc 7808  0cc0 7810  1c1 7811   · cmul 7815   # cap 8536   / cdiv 8627  cprod 11553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929  ax-caucvg 7930
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-isom 5225  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-frec 6391  df-1o 6416  df-oadd 6420  df-er 6534  df-en 6740  df-dom 6741  df-fin 6742  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-reap 8530  df-ap 8537  df-div 8628  df-inn 8918  df-2 8976  df-3 8977  df-4 8978  df-n0 9175  df-z 9252  df-uz 9527  df-q 9618  df-rp 9652  df-fz 10007  df-fzo 10140  df-seqfrec 10443  df-exp 10517  df-ihash 10751  df-cj 10846  df-re 10847  df-im 10848  df-rsqrt 11002  df-abs 11003  df-clim 11282  df-proddc 11554
This theorem is referenced by:  fproddivap  11633
  Copyright terms: Public domain W3C validator