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Theorem fprod2d 15327
 Description: Write a double product as a product over a two-dimensional region. Compare fsum2d 15118. (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 3937 . 2 𝐴𝐴
2 fprod2d.2 . . 3 (𝜑𝐴 ∈ Fin)
3 sseq1 3940 . . . . . 6 (𝑤 = ∅ → (𝑤𝐴 ↔ ∅ ⊆ 𝐴))
4 prodeq1 15255 . . . . . . 7 (𝑤 = ∅ → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑗 ∈ ∅ ∏𝑘𝐵 𝐶)
5 iuneq1 4897 . . . . . . . . 9 (𝑤 = ∅ → 𝑗𝑤 ({𝑗} × 𝐵) = 𝑗 ∈ ∅ ({𝑗} × 𝐵))
6 0iun 4949 . . . . . . . . 9 𝑗 ∈ ∅ ({𝑗} × 𝐵) = ∅
75, 6eqtrdi 2849 . . . . . . . 8 (𝑤 = ∅ → 𝑗𝑤 ({𝑗} × 𝐵) = ∅)
87prodeq1d 15267 . . . . . . 7 (𝑤 = ∅ → ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷 = ∏𝑧 ∈ ∅ 𝐷)
94, 8eqeq12d 2814 . . . . . 6 (𝑤 = ∅ → (∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷 ↔ ∏𝑗 ∈ ∅ ∏𝑘𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷))
103, 9imbi12d 348 . . . . 5 (𝑤 = ∅ → ((𝑤𝐴 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷) ↔ (∅ ⊆ 𝐴 → ∏𝑗 ∈ ∅ ∏𝑘𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷)))
1110imbi2d 344 . . . 4 (𝑤 = ∅ → ((𝜑 → (𝑤𝐴 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷)) ↔ (𝜑 → (∅ ⊆ 𝐴 → ∏𝑗 ∈ ∅ ∏𝑘𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷))))
12 sseq1 3940 . . . . . 6 (𝑤 = 𝑥 → (𝑤𝐴𝑥𝐴))
13 prodeq1 15255 . . . . . . 7 (𝑤 = 𝑥 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑗𝑥𝑘𝐵 𝐶)
14 iuneq1 4897 . . . . . . . 8 (𝑤 = 𝑥 𝑗𝑤 ({𝑗} × 𝐵) = 𝑗𝑥 ({𝑗} × 𝐵))
1514prodeq1d 15267 . . . . . . 7 (𝑤 = 𝑥 → ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷)
1613, 15eqeq12d 2814 . . . . . 6 (𝑤 = 𝑥 → (∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷 ↔ ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷))
1712, 16imbi12d 348 . . . . 5 (𝑤 = 𝑥 → ((𝑤𝐴 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷) ↔ (𝑥𝐴 → ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷)))
1817imbi2d 344 . . . 4 (𝑤 = 𝑥 → ((𝜑 → (𝑤𝐴 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷)) ↔ (𝜑 → (𝑥𝐴 → ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷))))
19 sseq1 3940 . . . . . 6 (𝑤 = (𝑥 ∪ {𝑦}) → (𝑤𝐴 ↔ (𝑥 ∪ {𝑦}) ⊆ 𝐴))
20 prodeq1 15255 . . . . . . 7 (𝑤 = (𝑥 ∪ {𝑦}) → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘𝐵 𝐶)
21 iuneq1 4897 . . . . . . . 8 (𝑤 = (𝑥 ∪ {𝑦}) → 𝑗𝑤 ({𝑗} × 𝐵) = 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵))
2221prodeq1d 15267 . . . . . . 7 (𝑤 = (𝑥 ∪ {𝑦}) → ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷 = ∏𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)
2320, 22eqeq12d 2814 . . . . . 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 348 . . . . 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 344 . . . 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 3940 . . . . . 6 (𝑤 = 𝐴 → (𝑤𝐴𝐴𝐴))
27 prodeq1 15255 . . . . . . 7 (𝑤 = 𝐴 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑗𝐴𝑘𝐵 𝐶)
28 iuneq1 4897 . . . . . . . 8 (𝑤 = 𝐴 𝑗𝑤 ({𝑗} × 𝐵) = 𝑗𝐴 ({𝑗} × 𝐵))
2928prodeq1d 15267 . . . . . . 7 (𝑤 = 𝐴 → ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷 = ∏𝑧 𝑗𝐴 ({𝑗} × 𝐵)𝐷)
3027, 29eqeq12d 2814 . . . . . 6 (𝑤 = 𝐴 → (∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷 ↔ ∏𝑗𝐴𝑘𝐵 𝐶 = ∏𝑧 𝑗𝐴 ({𝑗} × 𝐵)𝐷))
3126, 30imbi12d 348 . . . . 5 (𝑤 = 𝐴 → ((𝑤𝐴 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷) ↔ (𝐴𝐴 → ∏𝑗𝐴𝑘𝐵 𝐶 = ∏𝑧 𝑗𝐴 ({𝑗} × 𝐵)𝐷)))
3231imbi2d 344 . . . 4 (𝑤 = 𝐴 → ((𝜑 → (𝑤𝐴 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷)) ↔ (𝜑 → (𝐴𝐴 → ∏𝑗𝐴𝑘𝐵 𝐶 = ∏𝑧 𝑗𝐴 ({𝑗} × 𝐵)𝐷))))
33 prod0 15289 . . . . . 6 𝑗 ∈ ∅ ∏𝑘𝐵 𝐶 = 1
34 prod0 15289 . . . . . 6 𝑧 ∈ ∅ 𝐷 = 1
3533, 34eqtr4i 2824 . . . . 5 𝑗 ∈ ∅ ∏𝑘𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷
36352a1i 12 . . . 4 (𝜑 → (∅ ⊆ 𝐴 → ∏𝑗 ∈ ∅ ∏𝑘𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷))
37 ssun1 4099 . . . . . . . . . 10 𝑥 ⊆ (𝑥 ∪ {𝑦})
38 sstr 3923 . . . . . . . . . 10 ((𝑥 ⊆ (𝑥 ∪ {𝑦}) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑥𝐴)
3937, 38mpan 689 . . . . . . . . 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 725 . . . . . . . . . . 11 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝐴 ∈ Fin)
43 fprod2d.3 . . . . . . . . . . . 12 ((𝜑𝑗𝐴) → 𝐵 ∈ Fin)
4443ad4ant14 751 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑗𝐴) → 𝐵 ∈ Fin)
45 fprod2d.4 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ ℂ)
4645ad4ant14 751 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ ℂ)
47 simplr 768 . . . . . . . . . . 11 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ¬ 𝑦𝑥)
48 simpr 488 . . . . . . . . . . 11 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∪ {𝑦}) ⊆ 𝐴)
49 biid 264 . . . . . . . . . . 11 (∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷 ↔ ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷)
5041, 42, 44, 46, 47, 48, 49fprod2dlem 15326 . . . . . . . . . 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 423 . . . . . . . . 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 417 . . . . . 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 485 . . . 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 8743 . . 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 399   = wceq 1538   ∈ wcel 2111   ∪ cun 3879   ⊆ wss 3881  ∅c0 4243  {csn 4525  ⟨cop 4531  ∪ ciun 4881   × cxp 5517  Fincfn 8492  ℂcc 10524  1c1 10527  ∏cprod 15251 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-sup 8890  df-oi 8958  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12886  df-fzo 13029  df-seq 13365  df-exp 13426  df-hash 13687  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-clim 14837  df-prod 15252 This theorem is referenced by:  fprodxp  15328  fprodcom2  15330
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