MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fprod2d Structured version   Visualization version   GIF version

Theorem fprod2d 16017
Description: Write a double product as a product over a two-dimensional region. Compare fsum2d 15807. (Contributed by Scott Fenton, 30-Jan-2018.)
Hypotheses
Ref Expression
fprod2d.1 (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)
fprod2d.2 (𝜑𝐴 ∈ Fin)
fprod2d.3 ((𝜑𝑗𝐴) → 𝐵 ∈ Fin)
fprod2d.4 ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ ℂ)
Assertion
Ref Expression
fprod2d (𝜑 → ∏𝑗𝐴𝑘𝐵 𝐶 = ∏𝑧 𝑗𝐴 ({𝑗} × 𝐵)𝐷)
Distinct variable groups:   𝐴,𝑗,𝑘,𝑧   𝐷,𝑗,𝑘   𝐵,𝑘,𝑧   𝑧,𝐶   𝜑,𝑗,𝑘,𝑧
Allowed substitution hints:   𝐵(𝑗)   𝐶(𝑗,𝑘)   𝐷(𝑧)

Proof of Theorem fprod2d
Dummy variables 𝑤 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssid 4006 . 2 𝐴𝐴
2 fprod2d.2 . . 3 (𝜑𝐴 ∈ Fin)
3 sseq1 4009 . . . . . 6 (𝑤 = ∅ → (𝑤𝐴 ↔ ∅ ⊆ 𝐴))
4 prodeq1 15943 . . . . . . 7 (𝑤 = ∅ → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑗 ∈ ∅ ∏𝑘𝐵 𝐶)
5 iuneq1 5008 . . . . . . . . 9 (𝑤 = ∅ → 𝑗𝑤 ({𝑗} × 𝐵) = 𝑗 ∈ ∅ ({𝑗} × 𝐵))
6 0iun 5063 . . . . . . . . 9 𝑗 ∈ ∅ ({𝑗} × 𝐵) = ∅
75, 6eqtrdi 2793 . . . . . . . 8 (𝑤 = ∅ → 𝑗𝑤 ({𝑗} × 𝐵) = ∅)
87prodeq1d 15956 . . . . . . 7 (𝑤 = ∅ → ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷 = ∏𝑧 ∈ ∅ 𝐷)
94, 8eqeq12d 2753 . . . . . 6 (𝑤 = ∅ → (∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷 ↔ ∏𝑗 ∈ ∅ ∏𝑘𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷))
103, 9imbi12d 344 . . . . 5 (𝑤 = ∅ → ((𝑤𝐴 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷) ↔ (∅ ⊆ 𝐴 → ∏𝑗 ∈ ∅ ∏𝑘𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷)))
1110imbi2d 340 . . . 4 (𝑤 = ∅ → ((𝜑 → (𝑤𝐴 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷)) ↔ (𝜑 → (∅ ⊆ 𝐴 → ∏𝑗 ∈ ∅ ∏𝑘𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷))))
12 sseq1 4009 . . . . . 6 (𝑤 = 𝑥 → (𝑤𝐴𝑥𝐴))
13 prodeq1 15943 . . . . . . 7 (𝑤 = 𝑥 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑗𝑥𝑘𝐵 𝐶)
14 iuneq1 5008 . . . . . . . 8 (𝑤 = 𝑥 𝑗𝑤 ({𝑗} × 𝐵) = 𝑗𝑥 ({𝑗} × 𝐵))
1514prodeq1d 15956 . . . . . . 7 (𝑤 = 𝑥 → ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷)
1613, 15eqeq12d 2753 . . . . . 6 (𝑤 = 𝑥 → (∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷 ↔ ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷))
1712, 16imbi12d 344 . . . . 5 (𝑤 = 𝑥 → ((𝑤𝐴 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷) ↔ (𝑥𝐴 → ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷)))
1817imbi2d 340 . . . 4 (𝑤 = 𝑥 → ((𝜑 → (𝑤𝐴 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷)) ↔ (𝜑 → (𝑥𝐴 → ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷))))
19 sseq1 4009 . . . . . 6 (𝑤 = (𝑥 ∪ {𝑦}) → (𝑤𝐴 ↔ (𝑥 ∪ {𝑦}) ⊆ 𝐴))
20 prodeq1 15943 . . . . . . 7 (𝑤 = (𝑥 ∪ {𝑦}) → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘𝐵 𝐶)
21 iuneq1 5008 . . . . . . . 8 (𝑤 = (𝑥 ∪ {𝑦}) → 𝑗𝑤 ({𝑗} × 𝐵) = 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵))
2221prodeq1d 15956 . . . . . . 7 (𝑤 = (𝑥 ∪ {𝑦}) → ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷 = ∏𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)
2320, 22eqeq12d 2753 . . . . . 6 (𝑤 = (𝑥 ∪ {𝑦}) → (∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷 ↔ ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘𝐵 𝐶 = ∏𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷))
2419, 23imbi12d 344 . . . . 5 (𝑤 = (𝑥 ∪ {𝑦}) → ((𝑤𝐴 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷) ↔ ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘𝐵 𝐶 = ∏𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)))
2524imbi2d 340 . . . 4 (𝑤 = (𝑥 ∪ {𝑦}) → ((𝜑 → (𝑤𝐴 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷)) ↔ (𝜑 → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘𝐵 𝐶 = ∏𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷))))
26 sseq1 4009 . . . . . 6 (𝑤 = 𝐴 → (𝑤𝐴𝐴𝐴))
27 prodeq1 15943 . . . . . . 7 (𝑤 = 𝐴 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑗𝐴𝑘𝐵 𝐶)
28 iuneq1 5008 . . . . . . . 8 (𝑤 = 𝐴 𝑗𝑤 ({𝑗} × 𝐵) = 𝑗𝐴 ({𝑗} × 𝐵))
2928prodeq1d 15956 . . . . . . 7 (𝑤 = 𝐴 → ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷 = ∏𝑧 𝑗𝐴 ({𝑗} × 𝐵)𝐷)
3027, 29eqeq12d 2753 . . . . . 6 (𝑤 = 𝐴 → (∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷 ↔ ∏𝑗𝐴𝑘𝐵 𝐶 = ∏𝑧 𝑗𝐴 ({𝑗} × 𝐵)𝐷))
3126, 30imbi12d 344 . . . . 5 (𝑤 = 𝐴 → ((𝑤𝐴 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷) ↔ (𝐴𝐴 → ∏𝑗𝐴𝑘𝐵 𝐶 = ∏𝑧 𝑗𝐴 ({𝑗} × 𝐵)𝐷)))
3231imbi2d 340 . . . 4 (𝑤 = 𝐴 → ((𝜑 → (𝑤𝐴 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷)) ↔ (𝜑 → (𝐴𝐴 → ∏𝑗𝐴𝑘𝐵 𝐶 = ∏𝑧 𝑗𝐴 ({𝑗} × 𝐵)𝐷))))
33 prod0 15979 . . . . . 6 𝑗 ∈ ∅ ∏𝑘𝐵 𝐶 = 1
34 prod0 15979 . . . . . 6 𝑧 ∈ ∅ 𝐷 = 1
3533, 34eqtr4i 2768 . . . . 5 𝑗 ∈ ∅ ∏𝑘𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷
36352a1i 12 . . . 4 (𝜑 → (∅ ⊆ 𝐴 → ∏𝑗 ∈ ∅ ∏𝑘𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷))
37 ssun1 4178 . . . . . . . . . 10 𝑥 ⊆ (𝑥 ∪ {𝑦})
38 sstr 3992 . . . . . . . . . 10 ((𝑥 ⊆ (𝑥 ∪ {𝑦}) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑥𝐴)
3937, 38mpan 690 . . . . . . . . 9 ((𝑥 ∪ {𝑦}) ⊆ 𝐴𝑥𝐴)
4039imim1i 63 . . . . . . . 8 ((𝑥𝐴 → ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷) → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷))
41 fprod2d.1 . . . . . . . . . . 11 (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)
422ad2antrr 726 . . . . . . . . . . 11 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝐴 ∈ Fin)
43 fprod2d.3 . . . . . . . . . . . 12 ((𝜑𝑗𝐴) → 𝐵 ∈ Fin)
4443ad4ant14 752 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑗𝐴) → 𝐵 ∈ Fin)
45 fprod2d.4 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ ℂ)
4645ad4ant14 752 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ ℂ)
47 simplr 769 . . . . . . . . . . 11 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ¬ 𝑦𝑥)
48 simpr 484 . . . . . . . . . . 11 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∪ {𝑦}) ⊆ 𝐴)
49 biid 261 . . . . . . . . . . 11 (∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷 ↔ ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷)
5041, 42, 44, 46, 47, 48, 49fprod2dlem 16016 . . . . . . . . . 10 ((((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷) → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘𝐵 𝐶 = ∏𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)
5150exp31 419 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝑦𝑥) → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → (∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷 → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘𝐵 𝐶 = ∏𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)))
5251a2d 29 . . . . . . . 8 ((𝜑 ∧ ¬ 𝑦𝑥) → (((𝑥 ∪ {𝑦}) ⊆ 𝐴 → ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷) → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘𝐵 𝐶 = ∏𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)))
5340, 52syl5 34 . . . . . . 7 ((𝜑 ∧ ¬ 𝑦𝑥) → ((𝑥𝐴 → ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷) → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘𝐵 𝐶 = ∏𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)))
5453expcom 413 . . . . . 6 𝑦𝑥 → (𝜑 → ((𝑥𝐴 → ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷) → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘𝐵 𝐶 = ∏𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷))))
5554a2d 29 . . . . 5 𝑦𝑥 → ((𝜑 → (𝑥𝐴 → ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷)) → (𝜑 → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘𝐵 𝐶 = ∏𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷))))
5655adantl 481 . . . 4 ((𝑥 ∈ Fin ∧ ¬ 𝑦𝑥) → ((𝜑 → (𝑥𝐴 → ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷)) → (𝜑 → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘𝐵 𝐶 = ∏𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷))))
5711, 18, 25, 32, 36, 56findcard2s 9205 . . 3 (𝐴 ∈ Fin → (𝜑 → (𝐴𝐴 → ∏𝑗𝐴𝑘𝐵 𝐶 = ∏𝑧 𝑗𝐴 ({𝑗} × 𝐵)𝐷)))
582, 57mpcom 38 . 2 (𝜑 → (𝐴𝐴 → ∏𝑗𝐴𝑘𝐵 𝐶 = ∏𝑧 𝑗𝐴 ({𝑗} × 𝐵)𝐷))
591, 58mpi 20 1 (𝜑 → ∏𝑗𝐴𝑘𝐵 𝐶 = ∏𝑧 𝑗𝐴 ({𝑗} × 𝐵)𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2108  cun 3949  wss 3951  c0 4333  {csn 4626  cop 4632   ciun 4991   × cxp 5683  Fincfn 8985  cc 11153  1c1 11156  cprod 15939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-fz 13548  df-fzo 13695  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-prod 15940
This theorem is referenced by:  fprodxp  16018  fprodcom2  16020
  Copyright terms: Public domain W3C validator