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Theorem fprod2d 14755
Description: Write a double product as a product over a two-dimensional region. Compare fsum2d 14546. (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   𝐵,𝑘,𝑧   𝑧,𝐶   𝐷,𝑗,𝑘   𝜑,𝑗,𝑧,𝑘
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 3657 . 2 𝐴𝐴
2 fprod2d.2 . . 3 (𝜑𝐴 ∈ Fin)
3 sseq1 3659 . . . . . 6 (𝑤 = ∅ → (𝑤𝐴 ↔ ∅ ⊆ 𝐴))
4 prodeq1 14683 . . . . . . 7 (𝑤 = ∅ → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑗 ∈ ∅ ∏𝑘𝐵 𝐶)
5 iuneq1 4566 . . . . . . . . 9 (𝑤 = ∅ → 𝑗𝑤 ({𝑗} × 𝐵) = 𝑗 ∈ ∅ ({𝑗} × 𝐵))
6 0iun 4609 . . . . . . . . 9 𝑗 ∈ ∅ ({𝑗} × 𝐵) = ∅
75, 6syl6eq 2701 . . . . . . . 8 (𝑤 = ∅ → 𝑗𝑤 ({𝑗} × 𝐵) = ∅)
87prodeq1d 14695 . . . . . . 7 (𝑤 = ∅ → ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷 = ∏𝑧 ∈ ∅ 𝐷)
94, 8eqeq12d 2666 . . . . . 6 (𝑤 = ∅ → (∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷 ↔ ∏𝑗 ∈ ∅ ∏𝑘𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷))
103, 9imbi12d 333 . . . . 5 (𝑤 = ∅ → ((𝑤𝐴 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷) ↔ (∅ ⊆ 𝐴 → ∏𝑗 ∈ ∅ ∏𝑘𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷)))
1110imbi2d 329 . . . 4 (𝑤 = ∅ → ((𝜑 → (𝑤𝐴 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷)) ↔ (𝜑 → (∅ ⊆ 𝐴 → ∏𝑗 ∈ ∅ ∏𝑘𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷))))
12 sseq1 3659 . . . . . 6 (𝑤 = 𝑥 → (𝑤𝐴𝑥𝐴))
13 prodeq1 14683 . . . . . . 7 (𝑤 = 𝑥 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑗𝑥𝑘𝐵 𝐶)
14 iuneq1 4566 . . . . . . . 8 (𝑤 = 𝑥 𝑗𝑤 ({𝑗} × 𝐵) = 𝑗𝑥 ({𝑗} × 𝐵))
1514prodeq1d 14695 . . . . . . 7 (𝑤 = 𝑥 → ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷)
1613, 15eqeq12d 2666 . . . . . 6 (𝑤 = 𝑥 → (∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷 ↔ ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷))
1712, 16imbi12d 333 . . . . 5 (𝑤 = 𝑥 → ((𝑤𝐴 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷) ↔ (𝑥𝐴 → ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷)))
1817imbi2d 329 . . . 4 (𝑤 = 𝑥 → ((𝜑 → (𝑤𝐴 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷)) ↔ (𝜑 → (𝑥𝐴 → ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷))))
19 sseq1 3659 . . . . . 6 (𝑤 = (𝑥 ∪ {𝑦}) → (𝑤𝐴 ↔ (𝑥 ∪ {𝑦}) ⊆ 𝐴))
20 prodeq1 14683 . . . . . . 7 (𝑤 = (𝑥 ∪ {𝑦}) → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘𝐵 𝐶)
21 iuneq1 4566 . . . . . . . 8 (𝑤 = (𝑥 ∪ {𝑦}) → 𝑗𝑤 ({𝑗} × 𝐵) = 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵))
2221prodeq1d 14695 . . . . . . 7 (𝑤 = (𝑥 ∪ {𝑦}) → ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷 = ∏𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)
2320, 22eqeq12d 2666 . . . . . 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 333 . . . . 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 329 . . . 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 3659 . . . . . 6 (𝑤 = 𝐴 → (𝑤𝐴𝐴𝐴))
27 prodeq1 14683 . . . . . . 7 (𝑤 = 𝐴 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑗𝐴𝑘𝐵 𝐶)
28 iuneq1 4566 . . . . . . . 8 (𝑤 = 𝐴 𝑗𝑤 ({𝑗} × 𝐵) = 𝑗𝐴 ({𝑗} × 𝐵))
2928prodeq1d 14695 . . . . . . 7 (𝑤 = 𝐴 → ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷 = ∏𝑧 𝑗𝐴 ({𝑗} × 𝐵)𝐷)
3027, 29eqeq12d 2666 . . . . . 6 (𝑤 = 𝐴 → (∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷 ↔ ∏𝑗𝐴𝑘𝐵 𝐶 = ∏𝑧 𝑗𝐴 ({𝑗} × 𝐵)𝐷))
3126, 30imbi12d 333 . . . . 5 (𝑤 = 𝐴 → ((𝑤𝐴 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷) ↔ (𝐴𝐴 → ∏𝑗𝐴𝑘𝐵 𝐶 = ∏𝑧 𝑗𝐴 ({𝑗} × 𝐵)𝐷)))
3231imbi2d 329 . . . 4 (𝑤 = 𝐴 → ((𝜑 → (𝑤𝐴 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷)) ↔ (𝜑 → (𝐴𝐴 → ∏𝑗𝐴𝑘𝐵 𝐶 = ∏𝑧 𝑗𝐴 ({𝑗} × 𝐵)𝐷))))
33 prod0 14717 . . . . . 6 𝑗 ∈ ∅ ∏𝑘𝐵 𝐶 = 1
34 prod0 14717 . . . . . 6 𝑧 ∈ ∅ 𝐷 = 1
3533, 34eqtr4i 2676 . . . . 5 𝑗 ∈ ∅ ∏𝑘𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷
36352a1i 12 . . . 4 (𝜑 → (∅ ⊆ 𝐴 → ∏𝑗 ∈ ∅ ∏𝑘𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷))
37 ssun1 3809 . . . . . . . . . 10 𝑥 ⊆ (𝑥 ∪ {𝑦})
38 sstr 3644 . . . . . . . . . 10 ((𝑥 ⊆ (𝑥 ∪ {𝑦}) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑥𝐴)
3937, 38mpan 706 . . . . . . . . 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 762 . . . . . . . . . . 11 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝐴 ∈ Fin)
43 fprod2d.3 . . . . . . . . . . . . 13 ((𝜑𝑗𝐴) → 𝐵 ∈ Fin)
4443adantlr 751 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ 𝑗𝐴) → 𝐵 ∈ Fin)
4544adantlr 751 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑗𝐴) → 𝐵 ∈ Fin)
46 fprod2d.4 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ ℂ)
4746adantlr 751 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ ℂ)
4847adantlr 751 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ ℂ)
49 simplr 807 . . . . . . . . . . 11 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ¬ 𝑦𝑥)
50 simpr 476 . . . . . . . . . . 11 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∪ {𝑦}) ⊆ 𝐴)
51 biid 251 . . . . . . . . . . 11 (∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷 ↔ ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷)
5241, 42, 45, 48, 49, 50, 51fprod2dlem 14754 . . . . . . . . . 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)
5352exp31 629 . . . . . . . . 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)))
5453a2d 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)))
5540, 54syl5 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)))
5655expcom 450 . . . . . 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))))
5756a2d 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))))
5857adantl 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))))
5911, 18, 25, 32, 36, 58findcard2s 8242 . . 3 (𝐴 ∈ Fin → (𝜑 → (𝐴𝐴 → ∏𝑗𝐴𝑘𝐵 𝐶 = ∏𝑧 𝑗𝐴 ({𝑗} × 𝐵)𝐷)))
602, 59mpcom 38 . 2 (𝜑 → (𝐴𝐴 → ∏𝑗𝐴𝑘𝐵 𝐶 = ∏𝑧 𝑗𝐴 ({𝑗} × 𝐵)𝐷))
611, 60mpi 20 1 (𝜑 → ∏𝑗𝐴𝑘𝐵 𝐶 = ∏𝑧 𝑗𝐴 ({𝑗} × 𝐵)𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  cun 3605  wss 3607  c0 3948  {csn 4210  cop 4216   ciun 4552   × cxp 5141  Fincfn 7997  cc 9972  1c1 9975  cprod 14679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-prod 14680
This theorem is referenced by:  fprodxp  14756  fprodcom2  14758  fprodcom2OLD  14759
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