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Theorem fprod2dlem 14917
 Description: Lemma for fprod2d 14918- induction step. (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)
fprod2d.5 (𝜑 → ¬ 𝑦𝑥)
fprod2d.6 (𝜑 → (𝑥 ∪ {𝑦}) ⊆ 𝐴)
fprod2d.7 (𝜓 ↔ ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷)
Assertion
Ref Expression
fprod2dlem ((𝜑𝜓) → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘𝐵 𝐶 = ∏𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)
Distinct variable groups:   𝐴,𝑗,𝑘   𝐵,𝑘,𝑧   𝑧,𝐶   𝐷,𝑗,𝑘   𝜑,𝑗   𝑥,𝑗   𝑦,𝑗,𝑧   𝜑,𝑘   𝑥,𝑘   𝑦,𝑘,𝑧   𝜑,𝑧   𝑥,𝑧   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦,𝑧,𝑗,𝑘)   𝐴(𝑥,𝑦,𝑧)   𝐵(𝑥,𝑦,𝑗)   𝐶(𝑥,𝑦,𝑗,𝑘)   𝐷(𝑥,𝑦,𝑧)

Proof of Theorem fprod2dlem
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 simpr 471 . . . 4 ((𝜑𝜓) → 𝜓)
2 fprod2d.7 . . . 4 (𝜓 ↔ ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷)
31, 2sylib 208 . . 3 ((𝜑𝜓) → ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷)
4 nfcv 2913 . . . . . 6 𝑚𝑘𝐵 𝐶
5 nfcsb1v 3698 . . . . . . 7 𝑗𝑚 / 𝑗𝐵
6 nfcsb1v 3698 . . . . . . 7 𝑗𝑚 / 𝑗𝐶
75, 6nfcprod 14848 . . . . . 6 𝑗𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐶
8 csbeq1a 3691 . . . . . . 7 (𝑗 = 𝑚𝐵 = 𝑚 / 𝑗𝐵)
9 csbeq1a 3691 . . . . . . . 8 (𝑗 = 𝑚𝐶 = 𝑚 / 𝑗𝐶)
109adantr 466 . . . . . . 7 ((𝑗 = 𝑚𝑘𝐵) → 𝐶 = 𝑚 / 𝑗𝐶)
118, 10prodeq12dv 14863 . . . . . 6 (𝑗 = 𝑚 → ∏𝑘𝐵 𝐶 = ∏𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐶)
124, 7, 11cbvprodi 14854 . . . . 5 𝑗 ∈ {𝑦}∏𝑘𝐵 𝐶 = ∏𝑚 ∈ {𝑦}∏𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐶
13 fprod2d.6 . . . . . . . . 9 (𝜑 → (𝑥 ∪ {𝑦}) ⊆ 𝐴)
1413unssbd 3942 . . . . . . . 8 (𝜑 → {𝑦} ⊆ 𝐴)
15 vex 3354 . . . . . . . . 9 𝑦 ∈ V
1615snss 4451 . . . . . . . 8 (𝑦𝐴 ↔ {𝑦} ⊆ 𝐴)
1714, 16sylibr 224 . . . . . . 7 (𝜑𝑦𝐴)
18 fprod2d.3 . . . . . . . . . 10 ((𝜑𝑗𝐴) → 𝐵 ∈ Fin)
1918ralrimiva 3115 . . . . . . . . 9 (𝜑 → ∀𝑗𝐴 𝐵 ∈ Fin)
20 nfcsb1v 3698 . . . . . . . . . . 11 𝑗𝑦 / 𝑗𝐵
2120nfel1 2928 . . . . . . . . . 10 𝑗𝑦 / 𝑗𝐵 ∈ Fin
22 csbeq1a 3691 . . . . . . . . . . 11 (𝑗 = 𝑦𝐵 = 𝑦 / 𝑗𝐵)
2322eleq1d 2835 . . . . . . . . . 10 (𝑗 = 𝑦 → (𝐵 ∈ Fin ↔ 𝑦 / 𝑗𝐵 ∈ Fin))
2421, 23rspc 3454 . . . . . . . . 9 (𝑦𝐴 → (∀𝑗𝐴 𝐵 ∈ Fin → 𝑦 / 𝑗𝐵 ∈ Fin))
2517, 19, 24sylc 65 . . . . . . . 8 (𝜑𝑦 / 𝑗𝐵 ∈ Fin)
26 fprod2d.4 . . . . . . . . . . 11 ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ ℂ)
2726ralrimivva 3120 . . . . . . . . . 10 (𝜑 → ∀𝑗𝐴𝑘𝐵 𝐶 ∈ ℂ)
28 nfcsb1v 3698 . . . . . . . . . . . . 13 𝑗𝑦 / 𝑗𝐶
2928nfel1 2928 . . . . . . . . . . . 12 𝑗𝑦 / 𝑗𝐶 ∈ ℂ
3020, 29nfral 3094 . . . . . . . . . . 11 𝑗𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 ∈ ℂ
31 csbeq1a 3691 . . . . . . . . . . . . 13 (𝑗 = 𝑦𝐶 = 𝑦 / 𝑗𝐶)
3231eleq1d 2835 . . . . . . . . . . . 12 (𝑗 = 𝑦 → (𝐶 ∈ ℂ ↔ 𝑦 / 𝑗𝐶 ∈ ℂ))
3322, 32raleqbidv 3301 . . . . . . . . . . 11 (𝑗 = 𝑦 → (∀𝑘𝐵 𝐶 ∈ ℂ ↔ ∀𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 ∈ ℂ))
3430, 33rspc 3454 . . . . . . . . . 10 (𝑦𝐴 → (∀𝑗𝐴𝑘𝐵 𝐶 ∈ ℂ → ∀𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 ∈ ℂ))
3517, 27, 34sylc 65 . . . . . . . . 9 (𝜑 → ∀𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 ∈ ℂ)
3635r19.21bi 3081 . . . . . . . 8 ((𝜑𝑘𝑦 / 𝑗𝐵) → 𝑦 / 𝑗𝐶 ∈ ℂ)
3725, 36fprodcl 14889 . . . . . . 7 (𝜑 → ∏𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 ∈ ℂ)
38 csbeq1 3685 . . . . . . . . 9 (𝑚 = 𝑦𝑚 / 𝑗𝐵 = 𝑦 / 𝑗𝐵)
39 csbeq1 3685 . . . . . . . . . 10 (𝑚 = 𝑦𝑚 / 𝑗𝐶 = 𝑦 / 𝑗𝐶)
4039adantr 466 . . . . . . . . 9 ((𝑚 = 𝑦𝑘𝑚 / 𝑗𝐵) → 𝑚 / 𝑗𝐶 = 𝑦 / 𝑗𝐶)
4138, 40prodeq12dv 14863 . . . . . . . 8 (𝑚 = 𝑦 → ∏𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐶 = ∏𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶)
4241prodsn 14899 . . . . . . 7 ((𝑦𝐴 ∧ ∏𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 ∈ ℂ) → ∏𝑚 ∈ {𝑦}∏𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐶 = ∏𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶)
4317, 37, 42syl2anc 573 . . . . . 6 (𝜑 → ∏𝑚 ∈ {𝑦}∏𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐶 = ∏𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶)
44 nfcv 2913 . . . . . . . 8 𝑚𝑦 / 𝑗𝐶
45 nfcsb1v 3698 . . . . . . . 8 𝑘𝑚 / 𝑘𝑦 / 𝑗𝐶
46 csbeq1a 3691 . . . . . . . 8 (𝑘 = 𝑚𝑦 / 𝑗𝐶 = 𝑚 / 𝑘𝑦 / 𝑗𝐶)
4744, 45, 46cbvprodi 14854 . . . . . . 7 𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 = ∏𝑚 𝑦 / 𝑗𝐵𝑚 / 𝑘𝑦 / 𝑗𝐶
48 csbeq1 3685 . . . . . . . . 9 (𝑚 = (2nd𝑧) → 𝑚 / 𝑘𝑦 / 𝑗𝐶 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶)
49 snfi 8194 . . . . . . . . . 10 {𝑦} ∈ Fin
50 xpfi 8387 . . . . . . . . . 10 (({𝑦} ∈ Fin ∧ 𝑦 / 𝑗𝐵 ∈ Fin) → ({𝑦} × 𝑦 / 𝑗𝐵) ∈ Fin)
5149, 25, 50sylancr 575 . . . . . . . . 9 (𝜑 → ({𝑦} × 𝑦 / 𝑗𝐵) ∈ Fin)
52 2ndconst 7417 . . . . . . . . . 10 (𝑦𝐴 → (2nd ↾ ({𝑦} × 𝑦 / 𝑗𝐵)):({𝑦} × 𝑦 / 𝑗𝐵)–1-1-onto𝑦 / 𝑗𝐵)
5317, 52syl 17 . . . . . . . . 9 (𝜑 → (2nd ↾ ({𝑦} × 𝑦 / 𝑗𝐵)):({𝑦} × 𝑦 / 𝑗𝐵)–1-1-onto𝑦 / 𝑗𝐵)
54 fvres 6348 . . . . . . . . . 10 (𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵) → ((2nd ↾ ({𝑦} × 𝑦 / 𝑗𝐵))‘𝑧) = (2nd𝑧))
5554adantl 467 . . . . . . . . 9 ((𝜑𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)) → ((2nd ↾ ({𝑦} × 𝑦 / 𝑗𝐵))‘𝑧) = (2nd𝑧))
5645nfel1 2928 . . . . . . . . . . 11 𝑘𝑚 / 𝑘𝑦 / 𝑗𝐶 ∈ ℂ
5746eleq1d 2835 . . . . . . . . . . 11 (𝑘 = 𝑚 → (𝑦 / 𝑗𝐶 ∈ ℂ ↔ 𝑚 / 𝑘𝑦 / 𝑗𝐶 ∈ ℂ))
5856, 57rspc 3454 . . . . . . . . . 10 (𝑚𝑦 / 𝑗𝐵 → (∀𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 ∈ ℂ → 𝑚 / 𝑘𝑦 / 𝑗𝐶 ∈ ℂ))
5935, 58mpan9 496 . . . . . . . . 9 ((𝜑𝑚𝑦 / 𝑗𝐵) → 𝑚 / 𝑘𝑦 / 𝑗𝐶 ∈ ℂ)
6048, 51, 53, 55, 59fprodf1o 14883 . . . . . . . 8 (𝜑 → ∏𝑚 𝑦 / 𝑗𝐵𝑚 / 𝑘𝑦 / 𝑗𝐶 = ∏𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)(2nd𝑧) / 𝑘𝑦 / 𝑗𝐶)
61 elxp 5271 . . . . . . . . . . . 12 (𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵) ↔ ∃𝑚𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵)))
62 nfv 1995 . . . . . . . . . . . . . . 15 𝑗 𝑧 = ⟨𝑚, 𝑘
63 nfv 1995 . . . . . . . . . . . . . . . 16 𝑗 𝑚 ∈ {𝑦}
6420nfcri 2907 . . . . . . . . . . . . . . . 16 𝑗 𝑘𝑦 / 𝑗𝐵
6563, 64nfan 1980 . . . . . . . . . . . . . . 15 𝑗(𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵)
6662, 65nfan 1980 . . . . . . . . . . . . . 14 𝑗(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵))
6766nfex 2318 . . . . . . . . . . . . 13 𝑗𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵))
68 nfv 1995 . . . . . . . . . . . . 13 𝑚𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵))
69 opeq1 4539 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑗 → ⟨𝑚, 𝑘⟩ = ⟨𝑗, 𝑘⟩)
7069eqeq2d 2781 . . . . . . . . . . . . . . 15 (𝑚 = 𝑗 → (𝑧 = ⟨𝑚, 𝑘⟩ ↔ 𝑧 = ⟨𝑗, 𝑘⟩))
71 eleq1w 2833 . . . . . . . . . . . . . . . . . 18 (𝑚 = 𝑗 → (𝑚 ∈ {𝑦} ↔ 𝑗 ∈ {𝑦}))
72 velsn 4332 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ {𝑦} ↔ 𝑗 = 𝑦)
7371, 72syl6bb 276 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑗 → (𝑚 ∈ {𝑦} ↔ 𝑗 = 𝑦))
7473anbi1d 615 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑗 → ((𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵) ↔ (𝑗 = 𝑦𝑘𝑦 / 𝑗𝐵)))
7522eleq2d 2836 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑦 → (𝑘𝐵𝑘𝑦 / 𝑗𝐵))
7675pm5.32i 564 . . . . . . . . . . . . . . . 16 ((𝑗 = 𝑦𝑘𝐵) ↔ (𝑗 = 𝑦𝑘𝑦 / 𝑗𝐵))
7774, 76syl6bbr 278 . . . . . . . . . . . . . . 15 (𝑚 = 𝑗 → ((𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵) ↔ (𝑗 = 𝑦𝑘𝐵)))
7870, 77anbi12d 616 . . . . . . . . . . . . . 14 (𝑚 = 𝑗 → ((𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵)) ↔ (𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵))))
7978exbidv 2002 . . . . . . . . . . . . 13 (𝑚 = 𝑗 → (∃𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵)) ↔ ∃𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵))))
8067, 68, 79cbvex 2433 . . . . . . . . . . . 12 (∃𝑚𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵)) ↔ ∃𝑗𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵)))
8161, 80bitri 264 . . . . . . . . . . 11 (𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵) ↔ ∃𝑗𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵)))
82 nfv 1995 . . . . . . . . . . . 12 𝑗𝜑
83 nfcv 2913 . . . . . . . . . . . . . 14 𝑗(2nd𝑧)
8483, 28nfcsb 3700 . . . . . . . . . . . . 13 𝑗(2nd𝑧) / 𝑘𝑦 / 𝑗𝐶
8584nfeq2 2929 . . . . . . . . . . . 12 𝑗 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶
86 nfv 1995 . . . . . . . . . . . . 13 𝑘𝜑
87 nfcsb1v 3698 . . . . . . . . . . . . . 14 𝑘(2nd𝑧) / 𝑘𝑦 / 𝑗𝐶
8887nfeq2 2929 . . . . . . . . . . . . 13 𝑘 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶
89 fprod2d.1 . . . . . . . . . . . . . . . 16 (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)
9089ad2antlr 706 . . . . . . . . . . . . . . 15 (((𝜑𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝐷 = 𝐶)
9131ad2antrl 707 . . . . . . . . . . . . . . 15 (((𝜑𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝐶 = 𝑦 / 𝑗𝐶)
92 fveq2 6332 . . . . . . . . . . . . . . . . . 18 (𝑧 = ⟨𝑗, 𝑘⟩ → (2nd𝑧) = (2nd ‘⟨𝑗, 𝑘⟩))
93 vex 3354 . . . . . . . . . . . . . . . . . . 19 𝑗 ∈ V
94 vex 3354 . . . . . . . . . . . . . . . . . . 19 𝑘 ∈ V
9593, 94op2nd 7324 . . . . . . . . . . . . . . . . . 18 (2nd ‘⟨𝑗, 𝑘⟩) = 𝑘
9692, 95syl6req 2822 . . . . . . . . . . . . . . . . 17 (𝑧 = ⟨𝑗, 𝑘⟩ → 𝑘 = (2nd𝑧))
9796ad2antlr 706 . . . . . . . . . . . . . . . 16 (((𝜑𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝑘 = (2nd𝑧))
98 csbeq1a 3691 . . . . . . . . . . . . . . . 16 (𝑘 = (2nd𝑧) → 𝑦 / 𝑗𝐶 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶)
9997, 98syl 17 . . . . . . . . . . . . . . 15 (((𝜑𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝑦 / 𝑗𝐶 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶)
10090, 91, 993eqtrd 2809 . . . . . . . . . . . . . 14 (((𝜑𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶)
101100expl 445 . . . . . . . . . . . . 13 (𝜑 → ((𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶))
10286, 88, 101exlimd 2243 . . . . . . . . . . . 12 (𝜑 → (∃𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶))
10382, 85, 102exlimd 2243 . . . . . . . . . . 11 (𝜑 → (∃𝑗𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶))
10481, 103syl5bi 232 . . . . . . . . . 10 (𝜑 → (𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵) → 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶))
105104imp 393 . . . . . . . . 9 ((𝜑𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)) → 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶)
106105prodeq2dv 14860 . . . . . . . 8 (𝜑 → ∏𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷 = ∏𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)(2nd𝑧) / 𝑘𝑦 / 𝑗𝐶)
10760, 106eqtr4d 2808 . . . . . . 7 (𝜑 → ∏𝑚 𝑦 / 𝑗𝐵𝑚 / 𝑘𝑦 / 𝑗𝐶 = ∏𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷)
10847, 107syl5eq 2817 . . . . . 6 (𝜑 → ∏𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 = ∏𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷)
10943, 108eqtrd 2805 . . . . 5 (𝜑 → ∏𝑚 ∈ {𝑦}∏𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐶 = ∏𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷)
11012, 109syl5eq 2817 . . . 4 (𝜑 → ∏𝑗 ∈ {𝑦}∏𝑘𝐵 𝐶 = ∏𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷)
111110adantr 466 . . 3 ((𝜑𝜓) → ∏𝑗 ∈ {𝑦}∏𝑘𝐵 𝐶 = ∏𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷)
1123, 111oveq12d 6811 . 2 ((𝜑𝜓) → (∏𝑗𝑥𝑘𝐵 𝐶 · ∏𝑗 ∈ {𝑦}∏𝑘𝐵 𝐶) = (∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷 · ∏𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷))
113 fprod2d.5 . . . . 5 (𝜑 → ¬ 𝑦𝑥)
114 disjsn 4383 . . . . 5 ((𝑥 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦𝑥)
115113, 114sylibr 224 . . . 4 (𝜑 → (𝑥 ∩ {𝑦}) = ∅)
116 eqidd 2772 . . . 4 (𝜑 → (𝑥 ∪ {𝑦}) = (𝑥 ∪ {𝑦}))
117 fprod2d.2 . . . . 5 (𝜑𝐴 ∈ Fin)
118 ssfi 8336 . . . . 5 ((𝐴 ∈ Fin ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∪ {𝑦}) ∈ Fin)
119117, 13, 118syl2anc 573 . . . 4 (𝜑 → (𝑥 ∪ {𝑦}) ∈ Fin)
12013sselda 3752 . . . . 5 ((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) → 𝑗𝐴)
12126anassrs 458 . . . . . 6 (((𝜑𝑗𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℂ)
12218, 121fprodcl 14889 . . . . 5 ((𝜑𝑗𝐴) → ∏𝑘𝐵 𝐶 ∈ ℂ)
123120, 122syldan 579 . . . 4 ((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) → ∏𝑘𝐵 𝐶 ∈ ℂ)
124115, 116, 119, 123fprodsplit 14903 . . 3 (𝜑 → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘𝐵 𝐶 = (∏𝑗𝑥𝑘𝐵 𝐶 · ∏𝑗 ∈ {𝑦}∏𝑘𝐵 𝐶))
125124adantr 466 . 2 ((𝜑𝜓) → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘𝐵 𝐶 = (∏𝑗𝑥𝑘𝐵 𝐶 · ∏𝑗 ∈ {𝑦}∏𝑘𝐵 𝐶))
126 eliun 4658 . . . . . . . . . 10 (𝑧 𝑗𝑥 ({𝑗} × 𝐵) ↔ ∃𝑗𝑥 𝑧 ∈ ({𝑗} × 𝐵))
127 xp1st 7347 . . . . . . . . . . . . . 14 (𝑧 ∈ ({𝑗} × 𝐵) → (1st𝑧) ∈ {𝑗})
128 elsni 4333 . . . . . . . . . . . . . 14 ((1st𝑧) ∈ {𝑗} → (1st𝑧) = 𝑗)
129127, 128syl 17 . . . . . . . . . . . . 13 (𝑧 ∈ ({𝑗} × 𝐵) → (1st𝑧) = 𝑗)
130129eleq1d 2835 . . . . . . . . . . . 12 (𝑧 ∈ ({𝑗} × 𝐵) → ((1st𝑧) ∈ 𝑥𝑗𝑥))
131130biimparc 465 . . . . . . . . . . 11 ((𝑗𝑥𝑧 ∈ ({𝑗} × 𝐵)) → (1st𝑧) ∈ 𝑥)
132131rexlimiva 3176 . . . . . . . . . 10 (∃𝑗𝑥 𝑧 ∈ ({𝑗} × 𝐵) → (1st𝑧) ∈ 𝑥)
133126, 132sylbi 207 . . . . . . . . 9 (𝑧 𝑗𝑥 ({𝑗} × 𝐵) → (1st𝑧) ∈ 𝑥)
134 xp1st 7347 . . . . . . . . 9 (𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵) → (1st𝑧) ∈ {𝑦})
135133, 134anim12i 600 . . . . . . . 8 ((𝑧 𝑗𝑥 ({𝑗} × 𝐵) ∧ 𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)) → ((1st𝑧) ∈ 𝑥 ∧ (1st𝑧) ∈ {𝑦}))
136 elin 3947 . . . . . . . 8 (𝑧 ∈ ( 𝑗𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × 𝑦 / 𝑗𝐵)) ↔ (𝑧 𝑗𝑥 ({𝑗} × 𝐵) ∧ 𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)))
137 elin 3947 . . . . . . . 8 ((1st𝑧) ∈ (𝑥 ∩ {𝑦}) ↔ ((1st𝑧) ∈ 𝑥 ∧ (1st𝑧) ∈ {𝑦}))
138135, 136, 1373imtr4i 281 . . . . . . 7 (𝑧 ∈ ( 𝑗𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × 𝑦 / 𝑗𝐵)) → (1st𝑧) ∈ (𝑥 ∩ {𝑦}))
139115eleq2d 2836 . . . . . . . 8 (𝜑 → ((1st𝑧) ∈ (𝑥 ∩ {𝑦}) ↔ (1st𝑧) ∈ ∅))
140 noel 4067 . . . . . . . . 9 ¬ (1st𝑧) ∈ ∅
141140pm2.21i 117 . . . . . . . 8 ((1st𝑧) ∈ ∅ → 𝑧 ∈ ∅)
142139, 141syl6bi 243 . . . . . . 7 (𝜑 → ((1st𝑧) ∈ (𝑥 ∩ {𝑦}) → 𝑧 ∈ ∅))
143138, 142syl5 34 . . . . . 6 (𝜑 → (𝑧 ∈ ( 𝑗𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × 𝑦 / 𝑗𝐵)) → 𝑧 ∈ ∅))
144143ssrdv 3758 . . . . 5 (𝜑 → ( 𝑗𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × 𝑦 / 𝑗𝐵)) ⊆ ∅)
145 ss0 4118 . . . . 5 (( 𝑗𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × 𝑦 / 𝑗𝐵)) ⊆ ∅ → ( 𝑗𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × 𝑦 / 𝑗𝐵)) = ∅)
146144, 145syl 17 . . . 4 (𝜑 → ( 𝑗𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × 𝑦 / 𝑗𝐵)) = ∅)
147 iunxun 4739 . . . . . 6 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) = ( 𝑗𝑥 ({𝑗} × 𝐵) ∪ 𝑗 ∈ {𝑦} ({𝑗} × 𝐵))
148 nfcv 2913 . . . . . . . . 9 𝑚({𝑗} × 𝐵)
149 nfcv 2913 . . . . . . . . . 10 𝑗{𝑚}
150149, 5nfxp 5282 . . . . . . . . 9 𝑗({𝑚} × 𝑚 / 𝑗𝐵)
151 sneq 4326 . . . . . . . . . 10 (𝑗 = 𝑚 → {𝑗} = {𝑚})
152151, 8xpeq12d 5280 . . . . . . . . 9 (𝑗 = 𝑚 → ({𝑗} × 𝐵) = ({𝑚} × 𝑚 / 𝑗𝐵))
153148, 150, 152cbviun 4691 . . . . . . . 8 𝑗 ∈ {𝑦} ({𝑗} × 𝐵) = 𝑚 ∈ {𝑦} ({𝑚} × 𝑚 / 𝑗𝐵)
154 sneq 4326 . . . . . . . . . 10 (𝑚 = 𝑦 → {𝑚} = {𝑦})
155154, 38xpeq12d 5280 . . . . . . . . 9 (𝑚 = 𝑦 → ({𝑚} × 𝑚 / 𝑗𝐵) = ({𝑦} × 𝑦 / 𝑗𝐵))
15615, 155iunxsn 4737 . . . . . . . 8 𝑚 ∈ {𝑦} ({𝑚} × 𝑚 / 𝑗𝐵) = ({𝑦} × 𝑦 / 𝑗𝐵)
157153, 156eqtri 2793 . . . . . . 7 𝑗 ∈ {𝑦} ({𝑗} × 𝐵) = ({𝑦} × 𝑦 / 𝑗𝐵)
158157uneq2i 3915 . . . . . 6 ( 𝑗𝑥 ({𝑗} × 𝐵) ∪ 𝑗 ∈ {𝑦} ({𝑗} × 𝐵)) = ( 𝑗𝑥 ({𝑗} × 𝐵) ∪ ({𝑦} × 𝑦 / 𝑗𝐵))
159147, 158eqtri 2793 . . . . 5 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) = ( 𝑗𝑥 ({𝑗} × 𝐵) ∪ ({𝑦} × 𝑦 / 𝑗𝐵))
160159a1i 11 . . . 4 (𝜑 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) = ( 𝑗𝑥 ({𝑗} × 𝐵) ∪ ({𝑦} × 𝑦 / 𝑗𝐵)))
161 snfi 8194 . . . . . . 7 {𝑗} ∈ Fin
162120, 18syldan 579 . . . . . . 7 ((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) → 𝐵 ∈ Fin)
163 xpfi 8387 . . . . . . 7 (({𝑗} ∈ Fin ∧ 𝐵 ∈ Fin) → ({𝑗} × 𝐵) ∈ Fin)
164161, 162, 163sylancr 575 . . . . . 6 ((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) → ({𝑗} × 𝐵) ∈ Fin)
165164ralrimiva 3115 . . . . 5 (𝜑 → ∀𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin)
166 iunfi 8410 . . . . 5 (((𝑥 ∪ {𝑦}) ∈ Fin ∧ ∀𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin) → 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin)
167119, 165, 166syl2anc 573 . . . 4 (𝜑 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin)
168 eliun 4658 . . . . . 6 (𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ↔ ∃𝑗 ∈ (𝑥 ∪ {𝑦})𝑧 ∈ ({𝑗} × 𝐵))
169 elxp 5271 . . . . . . . 8 (𝑧 ∈ ({𝑗} × 𝐵) ↔ ∃𝑚𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵)))
170 simprl 754 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝑧 = ⟨𝑚, 𝑘⟩)
171 simprrl 766 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝑚 ∈ {𝑗})
172 elsni 4333 . . . . . . . . . . . . . . 15 (𝑚 ∈ {𝑗} → 𝑚 = 𝑗)
173171, 172syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝑚 = 𝑗)
174173opeq1d 4545 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → ⟨𝑚, 𝑘⟩ = ⟨𝑗, 𝑘⟩)
175170, 174eqtrd 2805 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝑧 = ⟨𝑗, 𝑘⟩)
176175, 89syl 17 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝐷 = 𝐶)
177 simpll 750 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝜑)
178120adantr 466 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝑗𝐴)
179 simprrr 767 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝑘𝐵)
180177, 178, 179, 26syl12anc 1474 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝐶 ∈ ℂ)
181176, 180eqeltrd 2850 . . . . . . . . . 10 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝐷 ∈ ℂ)
182181ex 397 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) → ((𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵)) → 𝐷 ∈ ℂ))
183182exlimdvv 2014 . . . . . . . 8 ((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) → (∃𝑚𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵)) → 𝐷 ∈ ℂ))
184169, 183syl5bi 232 . . . . . . 7 ((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) → (𝑧 ∈ ({𝑗} × 𝐵) → 𝐷 ∈ ℂ))
185184rexlimdva 3179 . . . . . 6 (𝜑 → (∃𝑗 ∈ (𝑥 ∪ {𝑦})𝑧 ∈ ({𝑗} × 𝐵) → 𝐷 ∈ ℂ))
186168, 185syl5bi 232 . . . . 5 (𝜑 → (𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) → 𝐷 ∈ ℂ))
187186imp 393 . . . 4 ((𝜑𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)) → 𝐷 ∈ ℂ)
188146, 160, 167, 187fprodsplit 14903 . . 3 (𝜑 → ∏𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷 = (∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷 · ∏𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷))
189188adantr 466 . 2 ((𝜑𝜓) → ∏𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷 = (∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷 · ∏𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷))
190112, 125, 1893eqtr4d 2815 1 ((𝜑𝜓) → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘𝐵 𝐶 = ∏𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1631  ∃wex 1852   ∈ wcel 2145  ∀wral 3061  ∃wrex 3062  ⦋csb 3682   ∪ cun 3721   ∩ cin 3722   ⊆ wss 3723  ∅c0 4063  {csn 4316  ⟨cop 4322  ∪ ciun 4654   × cxp 5247   ↾ cres 5251  –1-1-onto→wf1o 6030  ‘cfv 6031  (class class class)co 6793  1st c1st 7313  2nd c2nd 7314  Fincfn 8109  ℂcc 10136   · cmul 10143  ∏cprod 14842 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-pre-sup 10216 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-sup 8504  df-oi 8571  df-card 8965  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-div 10887  df-nn 11223  df-2 11281  df-3 11282  df-n0 11495  df-z 11580  df-uz 11889  df-rp 12036  df-fz 12534  df-fzo 12674  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-clim 14427  df-prod 14843 This theorem is referenced by:  fprod2d  14918
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