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Theorem fprod2d 15168
Description: Write a double product as a product over a two-dimensional region. Compare fsum2d 14959. (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 3910 . 2 𝐴𝐴
2 fprod2d.2 . . 3 (𝜑𝐴 ∈ Fin)
3 sseq1 3913 . . . . . 6 (𝑤 = ∅ → (𝑤𝐴 ↔ ∅ ⊆ 𝐴))
4 prodeq1 15096 . . . . . . 7 (𝑤 = ∅ → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑗 ∈ ∅ ∏𝑘𝐵 𝐶)
5 iuneq1 4840 . . . . . . . . 9 (𝑤 = ∅ → 𝑗𝑤 ({𝑗} × 𝐵) = 𝑗 ∈ ∅ ({𝑗} × 𝐵))
6 0iun 4885 . . . . . . . . 9 𝑗 ∈ ∅ ({𝑗} × 𝐵) = ∅
75, 6syl6eq 2847 . . . . . . . 8 (𝑤 = ∅ → 𝑗𝑤 ({𝑗} × 𝐵) = ∅)
87prodeq1d 15108 . . . . . . 7 (𝑤 = ∅ → ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷 = ∏𝑧 ∈ ∅ 𝐷)
94, 8eqeq12d 2810 . . . . . 6 (𝑤 = ∅ → (∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷 ↔ ∏𝑗 ∈ ∅ ∏𝑘𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷))
103, 9imbi12d 346 . . . . 5 (𝑤 = ∅ → ((𝑤𝐴 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷) ↔ (∅ ⊆ 𝐴 → ∏𝑗 ∈ ∅ ∏𝑘𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷)))
1110imbi2d 342 . . . 4 (𝑤 = ∅ → ((𝜑 → (𝑤𝐴 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷)) ↔ (𝜑 → (∅ ⊆ 𝐴 → ∏𝑗 ∈ ∅ ∏𝑘𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷))))
12 sseq1 3913 . . . . . 6 (𝑤 = 𝑥 → (𝑤𝐴𝑥𝐴))
13 prodeq1 15096 . . . . . . 7 (𝑤 = 𝑥 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑗𝑥𝑘𝐵 𝐶)
14 iuneq1 4840 . . . . . . . 8 (𝑤 = 𝑥 𝑗𝑤 ({𝑗} × 𝐵) = 𝑗𝑥 ({𝑗} × 𝐵))
1514prodeq1d 15108 . . . . . . 7 (𝑤 = 𝑥 → ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷)
1613, 15eqeq12d 2810 . . . . . 6 (𝑤 = 𝑥 → (∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷 ↔ ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷))
1712, 16imbi12d 346 . . . . 5 (𝑤 = 𝑥 → ((𝑤𝐴 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷) ↔ (𝑥𝐴 → ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷)))
1817imbi2d 342 . . . 4 (𝑤 = 𝑥 → ((𝜑 → (𝑤𝐴 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷)) ↔ (𝜑 → (𝑥𝐴 → ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷))))
19 sseq1 3913 . . . . . 6 (𝑤 = (𝑥 ∪ {𝑦}) → (𝑤𝐴 ↔ (𝑥 ∪ {𝑦}) ⊆ 𝐴))
20 prodeq1 15096 . . . . . . 7 (𝑤 = (𝑥 ∪ {𝑦}) → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘𝐵 𝐶)
21 iuneq1 4840 . . . . . . . 8 (𝑤 = (𝑥 ∪ {𝑦}) → 𝑗𝑤 ({𝑗} × 𝐵) = 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵))
2221prodeq1d 15108 . . . . . . 7 (𝑤 = (𝑥 ∪ {𝑦}) → ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷 = ∏𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)
2320, 22eqeq12d 2810 . . . . . 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 346 . . . . 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 342 . . . 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 3913 . . . . . 6 (𝑤 = 𝐴 → (𝑤𝐴𝐴𝐴))
27 prodeq1 15096 . . . . . . 7 (𝑤 = 𝐴 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑗𝐴𝑘𝐵 𝐶)
28 iuneq1 4840 . . . . . . . 8 (𝑤 = 𝐴 𝑗𝑤 ({𝑗} × 𝐵) = 𝑗𝐴 ({𝑗} × 𝐵))
2928prodeq1d 15108 . . . . . . 7 (𝑤 = 𝐴 → ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷 = ∏𝑧 𝑗𝐴 ({𝑗} × 𝐵)𝐷)
3027, 29eqeq12d 2810 . . . . . 6 (𝑤 = 𝐴 → (∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷 ↔ ∏𝑗𝐴𝑘𝐵 𝐶 = ∏𝑧 𝑗𝐴 ({𝑗} × 𝐵)𝐷))
3126, 30imbi12d 346 . . . . 5 (𝑤 = 𝐴 → ((𝑤𝐴 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷) ↔ (𝐴𝐴 → ∏𝑗𝐴𝑘𝐵 𝐶 = ∏𝑧 𝑗𝐴 ({𝑗} × 𝐵)𝐷)))
3231imbi2d 342 . . . 4 (𝑤 = 𝐴 → ((𝜑 → (𝑤𝐴 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷)) ↔ (𝜑 → (𝐴𝐴 → ∏𝑗𝐴𝑘𝐵 𝐶 = ∏𝑧 𝑗𝐴 ({𝑗} × 𝐵)𝐷))))
33 prod0 15130 . . . . . 6 𝑗 ∈ ∅ ∏𝑘𝐵 𝐶 = 1
34 prod0 15130 . . . . . 6 𝑧 ∈ ∅ 𝐷 = 1
3533, 34eqtr4i 2822 . . . . 5 𝑗 ∈ ∅ ∏𝑘𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷
36352a1i 12 . . . 4 (𝜑 → (∅ ⊆ 𝐴 → ∏𝑗 ∈ ∅ ∏𝑘𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷))
37 ssun1 4069 . . . . . . . . . 10 𝑥 ⊆ (𝑥 ∪ {𝑦})
38 sstr 3897 . . . . . . . . . 10 ((𝑥 ⊆ (𝑥 ∪ {𝑦}) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑥𝐴)
3937, 38mpan 686 . . . . . . . . 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 722 . . . . . . . . . . 11 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝐴 ∈ Fin)
43 fprod2d.3 . . . . . . . . . . . 12 ((𝜑𝑗𝐴) → 𝐵 ∈ Fin)
4443ad4ant14 748 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑗𝐴) → 𝐵 ∈ Fin)
45 fprod2d.4 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ ℂ)
4645ad4ant14 748 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ ℂ)
47 simplr 765 . . . . . . . . . . 11 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ¬ 𝑦𝑥)
48 simpr 485 . . . . . . . . . . 11 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∪ {𝑦}) ⊆ 𝐴)
49 biid 262 . . . . . . . . . . 11 (∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷 ↔ ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷)
5041, 42, 44, 46, 47, 48, 49fprod2dlem 15167 . . . . . . . . . 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 420 . . . . . . . . 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 414 . . . . . 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 482 . . . 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 8605 . . 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 396   = wceq 1522  wcel 2081  cun 3857  wss 3859  c0 4211  {csn 4472  cop 4478   ciun 4825   × cxp 5441  Fincfn 8357  cc 10381  1c1 10384  cprod 15092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-inf2 8950  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460  ax-pre-sup 10461
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-isom 6234  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-oadd 7957  df-er 8139  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-sup 8752  df-oi 8820  df-card 9214  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-div 11146  df-nn 11487  df-2 11548  df-3 11549  df-n0 11746  df-z 11830  df-uz 12094  df-rp 12240  df-fz 12743  df-fzo 12884  df-seq 13220  df-exp 13280  df-hash 13541  df-cj 14292  df-re 14293  df-im 14294  df-sqrt 14428  df-abs 14429  df-clim 14679  df-prod 15093
This theorem is referenced by:  fprodxp  15169  fprodcom2  15171
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