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Theorem fprod2d 11615
Description: Write a double product as a product over a two-dimensional region. Compare fsum2d 11427. (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,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 3175 . 2 𝐴𝐴
2 fprod2d.2 . . 3 (𝜑𝐴 ∈ Fin)
3 sseq1 3178 . . . . . 6 (𝑤 = ∅ → (𝑤𝐴 ↔ ∅ ⊆ 𝐴))
4 prodeq1 11545 . . . . . . 7 (𝑤 = ∅ → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑗 ∈ ∅ ∏𝑘𝐵 𝐶)
5 iuneq1 3897 . . . . . . . . 9 (𝑤 = ∅ → 𝑗𝑤 ({𝑗} × 𝐵) = 𝑗 ∈ ∅ ({𝑗} × 𝐵))
6 0iun 3941 . . . . . . . . 9 𝑗 ∈ ∅ ({𝑗} × 𝐵) = ∅
75, 6eqtrdi 2226 . . . . . . . 8 (𝑤 = ∅ → 𝑗𝑤 ({𝑗} × 𝐵) = ∅)
87prodeq1d 11556 . . . . . . 7 (𝑤 = ∅ → ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷 = ∏𝑧 ∈ ∅ 𝐷)
94, 8eqeq12d 2192 . . . . . 6 (𝑤 = ∅ → (∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷 ↔ ∏𝑗 ∈ ∅ ∏𝑘𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷))
103, 9imbi12d 234 . . . . 5 (𝑤 = ∅ → ((𝑤𝐴 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷) ↔ (∅ ⊆ 𝐴 → ∏𝑗 ∈ ∅ ∏𝑘𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷)))
1110imbi2d 230 . . . 4 (𝑤 = ∅ → ((𝜑 → (𝑤𝐴 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷)) ↔ (𝜑 → (∅ ⊆ 𝐴 → ∏𝑗 ∈ ∅ ∏𝑘𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷))))
12 sseq1 3178 . . . . . 6 (𝑤 = 𝑥 → (𝑤𝐴𝑥𝐴))
13 prodeq1 11545 . . . . . . 7 (𝑤 = 𝑥 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑗𝑥𝑘𝐵 𝐶)
14 iuneq1 3897 . . . . . . . 8 (𝑤 = 𝑥 𝑗𝑤 ({𝑗} × 𝐵) = 𝑗𝑥 ({𝑗} × 𝐵))
1514prodeq1d 11556 . . . . . . 7 (𝑤 = 𝑥 → ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷)
1613, 15eqeq12d 2192 . . . . . 6 (𝑤 = 𝑥 → (∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷 ↔ ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷))
1712, 16imbi12d 234 . . . . 5 (𝑤 = 𝑥 → ((𝑤𝐴 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷) ↔ (𝑥𝐴 → ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷)))
1817imbi2d 230 . . . 4 (𝑤 = 𝑥 → ((𝜑 → (𝑤𝐴 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷)) ↔ (𝜑 → (𝑥𝐴 → ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷))))
19 sseq1 3178 . . . . . 6 (𝑤 = (𝑥 ∪ {𝑦}) → (𝑤𝐴 ↔ (𝑥 ∪ {𝑦}) ⊆ 𝐴))
20 prodeq1 11545 . . . . . . 7 (𝑤 = (𝑥 ∪ {𝑦}) → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘𝐵 𝐶)
21 iuneq1 3897 . . . . . . . 8 (𝑤 = (𝑥 ∪ {𝑦}) → 𝑗𝑤 ({𝑗} × 𝐵) = 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵))
2221prodeq1d 11556 . . . . . . 7 (𝑤 = (𝑥 ∪ {𝑦}) → ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷 = ∏𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)
2320, 22eqeq12d 2192 . . . . . 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 234 . . . . 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 230 . . . 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 3178 . . . . . 6 (𝑤 = 𝐴 → (𝑤𝐴𝐴𝐴))
27 prodeq1 11545 . . . . . . 7 (𝑤 = 𝐴 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑗𝐴𝑘𝐵 𝐶)
28 iuneq1 3897 . . . . . . . 8 (𝑤 = 𝐴 𝑗𝑤 ({𝑗} × 𝐵) = 𝑗𝐴 ({𝑗} × 𝐵))
2928prodeq1d 11556 . . . . . . 7 (𝑤 = 𝐴 → ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷 = ∏𝑧 𝑗𝐴 ({𝑗} × 𝐵)𝐷)
3027, 29eqeq12d 2192 . . . . . 6 (𝑤 = 𝐴 → (∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷 ↔ ∏𝑗𝐴𝑘𝐵 𝐶 = ∏𝑧 𝑗𝐴 ({𝑗} × 𝐵)𝐷))
3126, 30imbi12d 234 . . . . 5 (𝑤 = 𝐴 → ((𝑤𝐴 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷) ↔ (𝐴𝐴 → ∏𝑗𝐴𝑘𝐵 𝐶 = ∏𝑧 𝑗𝐴 ({𝑗} × 𝐵)𝐷)))
3231imbi2d 230 . . . 4 (𝑤 = 𝐴 → ((𝜑 → (𝑤𝐴 → ∏𝑗𝑤𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑤 ({𝑗} × 𝐵)𝐷)) ↔ (𝜑 → (𝐴𝐴 → ∏𝑗𝐴𝑘𝐵 𝐶 = ∏𝑧 𝑗𝐴 ({𝑗} × 𝐵)𝐷))))
33 prod0 11577 . . . . . 6 𝑗 ∈ ∅ ∏𝑘𝐵 𝐶 = 1
34 prod0 11577 . . . . . 6 𝑧 ∈ ∅ 𝐷 = 1
3533, 34eqtr4i 2201 . . . . 5 𝑗 ∈ ∅ ∏𝑘𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷
36352a1i 27 . . . 4 (𝜑 → (∅ ⊆ 𝐴 → ∏𝑗 ∈ ∅ ∏𝑘𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷))
37 ssun1 3298 . . . . . . . . 9 𝑥 ⊆ (𝑥 ∪ {𝑦})
38 sstr 3163 . . . . . . . . 9 ((𝑥 ⊆ (𝑥 ∪ {𝑦}) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑥𝐴)
3937, 38mpan 424 . . . . . . . 8 ((𝑥 ∪ {𝑦}) ⊆ 𝐴𝑥𝐴)
4039imim1i 60 . . . . . . 7 ((𝑥𝐴 → ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷) → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷))
41 fprod2d.1 . . . . . . . . . 10 (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)
422ad2antrr 488 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝐴 ∈ Fin)
43 fprod2d.3 . . . . . . . . . . 11 ((𝜑𝑗𝐴) → 𝐵 ∈ Fin)
4443ad4ant14 514 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑗𝐴) → 𝐵 ∈ Fin)
45 fprod2d.4 . . . . . . . . . . 11 ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ ℂ)
4645ad4ant14 514 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ ℂ)
47 simplrr 536 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ¬ 𝑦𝑥)
48 simpr 110 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∪ {𝑦}) ⊆ 𝐴)
49 simplrl 535 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑥 ∈ Fin)
50 biid 171 . . . . . . . . . 10 (∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷 ↔ ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷)
5141, 42, 44, 46, 47, 48, 49, 50fprod2dlemstep 11614 . . . . . . . . 9 ((((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷) → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘𝐵 𝐶 = ∏𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)
5251exp31 364 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → (∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷 → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘𝐵 𝐶 = ∏𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)))
5352a2d 26 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 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)))
5440, 53syl5 32 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) → ((𝑥𝐴 → ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷) → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘𝐵 𝐶 = ∏𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)))
5554expcom 116 . . . . 5 ((𝑥 ∈ Fin ∧ ¬ 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))))
5655a2d 26 . . . 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 6884 . . 3 (𝐴 ∈ Fin → (𝜑 → (𝐴𝐴 → ∏𝑗𝐴𝑘𝐵 𝐶 = ∏𝑧 𝑗𝐴 ({𝑗} × 𝐵)𝐷)))
582, 57mpcom 36 . 2 (𝜑 → (𝐴𝐴 → ∏𝑗𝐴𝑘𝐵 𝐶 = ∏𝑧 𝑗𝐴 ({𝑗} × 𝐵)𝐷))
591, 58mpi 15 1 (𝜑 → ∏𝑗𝐴𝑘𝐵 𝐶 = ∏𝑧 𝑗𝐴 ({𝑗} × 𝐵)𝐷)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104   = wceq 1353  wcel 2148  cun 3127  wss 3129  c0 3422  {csn 3591  cop 3594   ciun 3884   × cxp 4621  Fincfn 6734  cc 7800  1c1 7803  cprod 11542
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 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922
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 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-disj 3978  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-frec 6386  df-1o 6411  df-oadd 6415  df-er 6529  df-en 6735  df-dom 6736  df-fin 6737  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fz 9996  df-fzo 10129  df-seqfrec 10432  df-exp 10506  df-ihash 10740  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-clim 11271  df-proddc 11543
This theorem is referenced by:  fprodxp  11616  fprodcom2fi  11618
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