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Theorem fprod2dlem 15334
Description: Lemma for fprod2d 15335- 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 488 . . . 4 ((𝜑𝜓) → 𝜓)
2 fprod2d.7 . . . 4 (𝜓 ↔ ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷)
31, 2sylib 221 . . 3 ((𝜑𝜓) → ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷)
4 nfcv 2982 . . . . . 6 𝑚𝑘𝐵 𝐶
5 nfcsb1v 3890 . . . . . . 7 𝑗𝑚 / 𝑗𝐵
6 nfcsb1v 3890 . . . . . . 7 𝑗𝑚 / 𝑗𝐶
75, 6nfcprod 15265 . . . . . 6 𝑗𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐶
8 csbeq1a 3880 . . . . . . 7 (𝑗 = 𝑚𝐵 = 𝑚 / 𝑗𝐵)
9 csbeq1a 3880 . . . . . . . 8 (𝑗 = 𝑚𝐶 = 𝑚 / 𝑗𝐶)
109adantr 484 . . . . . . 7 ((𝑗 = 𝑚𝑘𝐵) → 𝐶 = 𝑚 / 𝑗𝐶)
118, 10prodeq12dv 15280 . . . . . 6 (𝑗 = 𝑚 → ∏𝑘𝐵 𝐶 = ∏𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐶)
124, 7, 11cbvprodi 15271 . . . . 5 𝑗 ∈ {𝑦}∏𝑘𝐵 𝐶 = ∏𝑚 ∈ {𝑦}∏𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐶
13 fprod2d.6 . . . . . . . . 9 (𝜑 → (𝑥 ∪ {𝑦}) ⊆ 𝐴)
1413unssbd 4150 . . . . . . . 8 (𝜑 → {𝑦} ⊆ 𝐴)
15 vex 3483 . . . . . . . . 9 𝑦 ∈ V
1615snss 4703 . . . . . . . 8 (𝑦𝐴 ↔ {𝑦} ⊆ 𝐴)
1714, 16sylibr 237 . . . . . . 7 (𝜑𝑦𝐴)
18 fprod2d.3 . . . . . . . . . 10 ((𝜑𝑗𝐴) → 𝐵 ∈ Fin)
1918ralrimiva 3177 . . . . . . . . 9 (𝜑 → ∀𝑗𝐴 𝐵 ∈ Fin)
20 nfcsb1v 3890 . . . . . . . . . . 11 𝑗𝑦 / 𝑗𝐵
2120nfel1 2998 . . . . . . . . . 10 𝑗𝑦 / 𝑗𝐵 ∈ Fin
22 csbeq1a 3880 . . . . . . . . . . 11 (𝑗 = 𝑦𝐵 = 𝑦 / 𝑗𝐵)
2322eleq1d 2900 . . . . . . . . . 10 (𝑗 = 𝑦 → (𝐵 ∈ Fin ↔ 𝑦 / 𝑗𝐵 ∈ Fin))
2421, 23rspc 3597 . . . . . . . . 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 3186 . . . . . . . . . 10 (𝜑 → ∀𝑗𝐴𝑘𝐵 𝐶 ∈ ℂ)
28 nfcsb1v 3890 . . . . . . . . . . . . 13 𝑗𝑦 / 𝑗𝐶
2928nfel1 2998 . . . . . . . . . . . 12 𝑗𝑦 / 𝑗𝐶 ∈ ℂ
3020, 29nfralw 3219 . . . . . . . . . . 11 𝑗𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 ∈ ℂ
31 csbeq1a 3880 . . . . . . . . . . . . 13 (𝑗 = 𝑦𝐶 = 𝑦 / 𝑗𝐶)
3231eleq1d 2900 . . . . . . . . . . . 12 (𝑗 = 𝑦 → (𝐶 ∈ ℂ ↔ 𝑦 / 𝑗𝐶 ∈ ℂ))
3322, 32raleqbidv 3392 . . . . . . . . . . 11 (𝑗 = 𝑦 → (∀𝑘𝐵 𝐶 ∈ ℂ ↔ ∀𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 ∈ ℂ))
3430, 33rspc 3597 . . . . . . . . . 10 (𝑦𝐴 → (∀𝑗𝐴𝑘𝐵 𝐶 ∈ ℂ → ∀𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 ∈ ℂ))
3517, 27, 34sylc 65 . . . . . . . . 9 (𝜑 → ∀𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 ∈ ℂ)
3635r19.21bi 3203 . . . . . . . 8 ((𝜑𝑘𝑦 / 𝑗𝐵) → 𝑦 / 𝑗𝐶 ∈ ℂ)
3725, 36fprodcl 15306 . . . . . . 7 (𝜑 → ∏𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 ∈ ℂ)
38 csbeq1 3869 . . . . . . . . 9 (𝑚 = 𝑦𝑚 / 𝑗𝐵 = 𝑦 / 𝑗𝐵)
39 csbeq1 3869 . . . . . . . . . 10 (𝑚 = 𝑦𝑚 / 𝑗𝐶 = 𝑦 / 𝑗𝐶)
4039adantr 484 . . . . . . . . 9 ((𝑚 = 𝑦𝑘𝑚 / 𝑗𝐵) → 𝑚 / 𝑗𝐶 = 𝑦 / 𝑗𝐶)
4138, 40prodeq12dv 15280 . . . . . . . 8 (𝑚 = 𝑦 → ∏𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐶 = ∏𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶)
4241prodsn 15316 . . . . . . 7 ((𝑦𝐴 ∧ ∏𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 ∈ ℂ) → ∏𝑚 ∈ {𝑦}∏𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐶 = ∏𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶)
4317, 37, 42syl2anc 587 . . . . . 6 (𝜑 → ∏𝑚 ∈ {𝑦}∏𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐶 = ∏𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶)
44 nfcv 2982 . . . . . . . 8 𝑚𝑦 / 𝑗𝐶
45 nfcsb1v 3890 . . . . . . . 8 𝑘𝑚 / 𝑘𝑦 / 𝑗𝐶
46 csbeq1a 3880 . . . . . . . 8 (𝑘 = 𝑚𝑦 / 𝑗𝐶 = 𝑚 / 𝑘𝑦 / 𝑗𝐶)
4744, 45, 46cbvprodi 15271 . . . . . . 7 𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 = ∏𝑚 𝑦 / 𝑗𝐵𝑚 / 𝑘𝑦 / 𝑗𝐶
48 csbeq1 3869 . . . . . . . . 9 (𝑚 = (2nd𝑧) → 𝑚 / 𝑘𝑦 / 𝑗𝐶 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶)
49 snfi 8590 . . . . . . . . . 10 {𝑦} ∈ Fin
50 xpfi 8786 . . . . . . . . . 10 (({𝑦} ∈ Fin ∧ 𝑦 / 𝑗𝐵 ∈ Fin) → ({𝑦} × 𝑦 / 𝑗𝐵) ∈ Fin)
5149, 25, 50sylancr 590 . . . . . . . . 9 (𝜑 → ({𝑦} × 𝑦 / 𝑗𝐵) ∈ Fin)
52 2ndconst 7792 . . . . . . . . . 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 6680 . . . . . . . . . 10 (𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵) → ((2nd ↾ ({𝑦} × 𝑦 / 𝑗𝐵))‘𝑧) = (2nd𝑧))
5554adantl 485 . . . . . . . . 9 ((𝜑𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)) → ((2nd ↾ ({𝑦} × 𝑦 / 𝑗𝐵))‘𝑧) = (2nd𝑧))
5645nfel1 2998 . . . . . . . . . . 11 𝑘𝑚 / 𝑘𝑦 / 𝑗𝐶 ∈ ℂ
5746eleq1d 2900 . . . . . . . . . . 11 (𝑘 = 𝑚 → (𝑦 / 𝑗𝐶 ∈ ℂ ↔ 𝑚 / 𝑘𝑦 / 𝑗𝐶 ∈ ℂ))
5856, 57rspc 3597 . . . . . . . . . 10 (𝑚𝑦 / 𝑗𝐵 → (∀𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 ∈ ℂ → 𝑚 / 𝑘𝑦 / 𝑗𝐶 ∈ ℂ))
5935, 58mpan9 510 . . . . . . . . 9 ((𝜑𝑚𝑦 / 𝑗𝐵) → 𝑚 / 𝑘𝑦 / 𝑗𝐶 ∈ ℂ)
6048, 51, 53, 55, 59fprodf1o 15300 . . . . . . . 8 (𝜑 → ∏𝑚 𝑦 / 𝑗𝐵𝑚 / 𝑘𝑦 / 𝑗𝐶 = ∏𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)(2nd𝑧) / 𝑘𝑦 / 𝑗𝐶)
61 elxp 5565 . . . . . . . . . . . 12 (𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵) ↔ ∃𝑚𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵)))
62 nfv 1916 . . . . . . . . . . . . . . 15 𝑗 𝑧 = ⟨𝑚, 𝑘
63 nfv 1916 . . . . . . . . . . . . . . . 16 𝑗 𝑚 ∈ {𝑦}
6420nfcri 2969 . . . . . . . . . . . . . . . 16 𝑗 𝑘𝑦 / 𝑗𝐵
6563, 64nfan 1901 . . . . . . . . . . . . . . 15 𝑗(𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵)
6662, 65nfan 1901 . . . . . . . . . . . . . 14 𝑗(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵))
6766nfex 2345 . . . . . . . . . . . . 13 𝑗𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵))
68 nfv 1916 . . . . . . . . . . . . 13 𝑚𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵))
69 opeq1 4787 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑗 → ⟨𝑚, 𝑘⟩ = ⟨𝑗, 𝑘⟩)
7069eqeq2d 2835 . . . . . . . . . . . . . . 15 (𝑚 = 𝑗 → (𝑧 = ⟨𝑚, 𝑘⟩ ↔ 𝑧 = ⟨𝑗, 𝑘⟩))
71 eleq1w 2898 . . . . . . . . . . . . . . . . . 18 (𝑚 = 𝑗 → (𝑚 ∈ {𝑦} ↔ 𝑗 ∈ {𝑦}))
72 velsn 4566 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ {𝑦} ↔ 𝑗 = 𝑦)
7371, 72syl6bb 290 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑗 → (𝑚 ∈ {𝑦} ↔ 𝑗 = 𝑦))
7473anbi1d 632 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑗 → ((𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵) ↔ (𝑗 = 𝑦𝑘𝑦 / 𝑗𝐵)))
7522eleq2d 2901 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑦 → (𝑘𝐵𝑘𝑦 / 𝑗𝐵))
7675pm5.32i 578 . . . . . . . . . . . . . . . 16 ((𝑗 = 𝑦𝑘𝐵) ↔ (𝑗 = 𝑦𝑘𝑦 / 𝑗𝐵))
7774, 76syl6bbr 292 . . . . . . . . . . . . . . 15 (𝑚 = 𝑗 → ((𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵) ↔ (𝑗 = 𝑦𝑘𝐵)))
7870, 77anbi12d 633 . . . . . . . . . . . . . 14 (𝑚 = 𝑗 → ((𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵)) ↔ (𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵))))
7978exbidv 1923 . . . . . . . . . . . . 13 (𝑚 = 𝑗 → (∃𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵)) ↔ ∃𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵))))
8067, 68, 79cbvexv1 2364 . . . . . . . . . . . 12 (∃𝑚𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵)) ↔ ∃𝑗𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵)))
8161, 80bitri 278 . . . . . . . . . . 11 (𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵) ↔ ∃𝑗𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵)))
82 nfv 1916 . . . . . . . . . . . 12 𝑗𝜑
83 nfcv 2982 . . . . . . . . . . . . . 14 𝑗(2nd𝑧)
8483, 28nfcsbw 3892 . . . . . . . . . . . . 13 𝑗(2nd𝑧) / 𝑘𝑦 / 𝑗𝐶
8584nfeq2 2999 . . . . . . . . . . . 12 𝑗 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶
86 nfv 1916 . . . . . . . . . . . . 13 𝑘𝜑
87 nfcsb1v 3890 . . . . . . . . . . . . . 14 𝑘(2nd𝑧) / 𝑘𝑦 / 𝑗𝐶
8887nfeq2 2999 . . . . . . . . . . . . 13 𝑘 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶
89 fprod2d.1 . . . . . . . . . . . . . . . 16 (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)
9089ad2antlr 726 . . . . . . . . . . . . . . 15 (((𝜑𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝐷 = 𝐶)
9131ad2antrl 727 . . . . . . . . . . . . . . 15 (((𝜑𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝐶 = 𝑦 / 𝑗𝐶)
92 fveq2 6661 . . . . . . . . . . . . . . . . . 18 (𝑧 = ⟨𝑗, 𝑘⟩ → (2nd𝑧) = (2nd ‘⟨𝑗, 𝑘⟩))
93 vex 3483 . . . . . . . . . . . . . . . . . . 19 𝑗 ∈ V
94 vex 3483 . . . . . . . . . . . . . . . . . . 19 𝑘 ∈ V
9593, 94op2nd 7693 . . . . . . . . . . . . . . . . . 18 (2nd ‘⟨𝑗, 𝑘⟩) = 𝑘
9692, 95syl6req 2876 . . . . . . . . . . . . . . . . 17 (𝑧 = ⟨𝑗, 𝑘⟩ → 𝑘 = (2nd𝑧))
9796ad2antlr 726 . . . . . . . . . . . . . . . 16 (((𝜑𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝑘 = (2nd𝑧))
98 csbeq1a 3880 . . . . . . . . . . . . . . . 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 2863 . . . . . . . . . . . . . 14 (((𝜑𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶)
101100expl 461 . . . . . . . . . . . . 13 (𝜑 → ((𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶))
10286, 88, 101exlimd 2220 . . . . . . . . . . . 12 (𝜑 → (∃𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶))
10382, 85, 102exlimd 2220 . . . . . . . . . . 11 (𝜑 → (∃𝑗𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶))
10481, 103syl5bi 245 . . . . . . . . . 10 (𝜑 → (𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵) → 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶))
105104imp 410 . . . . . . . . 9 ((𝜑𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)) → 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶)
106105prodeq2dv 15277 . . . . . . . 8 (𝜑 → ∏𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷 = ∏𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)(2nd𝑧) / 𝑘𝑦 / 𝑗𝐶)
10760, 106eqtr4d 2862 . . . . . . 7 (𝜑 → ∏𝑚 𝑦 / 𝑗𝐵𝑚 / 𝑘𝑦 / 𝑗𝐶 = ∏𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷)
10847, 107syl5eq 2871 . . . . . 6 (𝜑 → ∏𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 = ∏𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷)
10943, 108eqtrd 2859 . . . . 5 (𝜑 → ∏𝑚 ∈ {𝑦}∏𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐶 = ∏𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷)
11012, 109syl5eq 2871 . . . 4 (𝜑 → ∏𝑗 ∈ {𝑦}∏𝑘𝐵 𝐶 = ∏𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷)
111110adantr 484 . . 3 ((𝜑𝜓) → ∏𝑗 ∈ {𝑦}∏𝑘𝐵 𝐶 = ∏𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷)
1123, 111oveq12d 7167 . 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 4632 . . . . 5 ((𝑥 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦𝑥)
115113, 114sylibr 237 . . . 4 (𝜑 → (𝑥 ∩ {𝑦}) = ∅)
116 eqidd 2825 . . . 4 (𝜑 → (𝑥 ∪ {𝑦}) = (𝑥 ∪ {𝑦}))
117 fprod2d.2 . . . . 5 (𝜑𝐴 ∈ Fin)
118117, 13ssfid 8738 . . . 4 (𝜑 → (𝑥 ∪ {𝑦}) ∈ Fin)
11913sselda 3953 . . . . 5 ((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) → 𝑗𝐴)
12026anassrs 471 . . . . . 6 (((𝜑𝑗𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℂ)
12118, 120fprodcl 15306 . . . . 5 ((𝜑𝑗𝐴) → ∏𝑘𝐵 𝐶 ∈ ℂ)
122119, 121syldan 594 . . . 4 ((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) → ∏𝑘𝐵 𝐶 ∈ ℂ)
123115, 116, 118, 122fprodsplit 15320 . . 3 (𝜑 → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘𝐵 𝐶 = (∏𝑗𝑥𝑘𝐵 𝐶 · ∏𝑗 ∈ {𝑦}∏𝑘𝐵 𝐶))
124123adantr 484 . 2 ((𝜑𝜓) → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘𝐵 𝐶 = (∏𝑗𝑥𝑘𝐵 𝐶 · ∏𝑗 ∈ {𝑦}∏𝑘𝐵 𝐶))
125 eliun 4909 . . . . . . . . . 10 (𝑧 𝑗𝑥 ({𝑗} × 𝐵) ↔ ∃𝑗𝑥 𝑧 ∈ ({𝑗} × 𝐵))
126 xp1st 7716 . . . . . . . . . . . . . 14 (𝑧 ∈ ({𝑗} × 𝐵) → (1st𝑧) ∈ {𝑗})
127 elsni 4567 . . . . . . . . . . . . . 14 ((1st𝑧) ∈ {𝑗} → (1st𝑧) = 𝑗)
128126, 127syl 17 . . . . . . . . . . . . 13 (𝑧 ∈ ({𝑗} × 𝐵) → (1st𝑧) = 𝑗)
129128eleq1d 2900 . . . . . . . . . . . 12 (𝑧 ∈ ({𝑗} × 𝐵) → ((1st𝑧) ∈ 𝑥𝑗𝑥))
130129biimparc 483 . . . . . . . . . . 11 ((𝑗𝑥𝑧 ∈ ({𝑗} × 𝐵)) → (1st𝑧) ∈ 𝑥)
131130rexlimiva 3273 . . . . . . . . . 10 (∃𝑗𝑥 𝑧 ∈ ({𝑗} × 𝐵) → (1st𝑧) ∈ 𝑥)
132125, 131sylbi 220 . . . . . . . . 9 (𝑧 𝑗𝑥 ({𝑗} × 𝐵) → (1st𝑧) ∈ 𝑥)
133 xp1st 7716 . . . . . . . . 9 (𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵) → (1st𝑧) ∈ {𝑦})
134132, 133anim12i 615 . . . . . . . 8 ((𝑧 𝑗𝑥 ({𝑗} × 𝐵) ∧ 𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)) → ((1st𝑧) ∈ 𝑥 ∧ (1st𝑧) ∈ {𝑦}))
135 elin 3935 . . . . . . . 8 (𝑧 ∈ ( 𝑗𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × 𝑦 / 𝑗𝐵)) ↔ (𝑧 𝑗𝑥 ({𝑗} × 𝐵) ∧ 𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)))
136 elin 3935 . . . . . . . 8 ((1st𝑧) ∈ (𝑥 ∩ {𝑦}) ↔ ((1st𝑧) ∈ 𝑥 ∧ (1st𝑧) ∈ {𝑦}))
137134, 135, 1363imtr4i 295 . . . . . . 7 (𝑧 ∈ ( 𝑗𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × 𝑦 / 𝑗𝐵)) → (1st𝑧) ∈ (𝑥 ∩ {𝑦}))
138115eleq2d 2901 . . . . . . . 8 (𝜑 → ((1st𝑧) ∈ (𝑥 ∩ {𝑦}) ↔ (1st𝑧) ∈ ∅))
139 noel 4280 . . . . . . . . 9 ¬ (1st𝑧) ∈ ∅
140139pm2.21i 119 . . . . . . . 8 ((1st𝑧) ∈ ∅ → 𝑧 ∈ ∅)
141138, 140syl6bi 256 . . . . . . 7 (𝜑 → ((1st𝑧) ∈ (𝑥 ∩ {𝑦}) → 𝑧 ∈ ∅))
142137, 141syl5 34 . . . . . 6 (𝜑 → (𝑧 ∈ ( 𝑗𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × 𝑦 / 𝑗𝐵)) → 𝑧 ∈ ∅))
143142ssrdv 3959 . . . . 5 (𝜑 → ( 𝑗𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × 𝑦 / 𝑗𝐵)) ⊆ ∅)
144 ss0 4335 . . . . 5 (( 𝑗𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × 𝑦 / 𝑗𝐵)) ⊆ ∅ → ( 𝑗𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × 𝑦 / 𝑗𝐵)) = ∅)
145143, 144syl 17 . . . 4 (𝜑 → ( 𝑗𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × 𝑦 / 𝑗𝐵)) = ∅)
146 iunxun 5002 . . . . . 6 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) = ( 𝑗𝑥 ({𝑗} × 𝐵) ∪ 𝑗 ∈ {𝑦} ({𝑗} × 𝐵))
147 nfcv 2982 . . . . . . . . 9 𝑚({𝑗} × 𝐵)
148 nfcv 2982 . . . . . . . . . 10 𝑗{𝑚}
149148, 5nfxp 5575 . . . . . . . . 9 𝑗({𝑚} × 𝑚 / 𝑗𝐵)
150 sneq 4560 . . . . . . . . . 10 (𝑗 = 𝑚 → {𝑗} = {𝑚})
151150, 8xpeq12d 5573 . . . . . . . . 9 (𝑗 = 𝑚 → ({𝑗} × 𝐵) = ({𝑚} × 𝑚 / 𝑗𝐵))
152147, 149, 151cbviun 4947 . . . . . . . 8 𝑗 ∈ {𝑦} ({𝑗} × 𝐵) = 𝑚 ∈ {𝑦} ({𝑚} × 𝑚 / 𝑗𝐵)
153 sneq 4560 . . . . . . . . . 10 (𝑚 = 𝑦 → {𝑚} = {𝑦})
154153, 38xpeq12d 5573 . . . . . . . . 9 (𝑚 = 𝑦 → ({𝑚} × 𝑚 / 𝑗𝐵) = ({𝑦} × 𝑦 / 𝑗𝐵))
15515, 154iunxsn 4999 . . . . . . . 8 𝑚 ∈ {𝑦} ({𝑚} × 𝑚 / 𝑗𝐵) = ({𝑦} × 𝑦 / 𝑗𝐵)
156152, 155eqtri 2847 . . . . . . 7 𝑗 ∈ {𝑦} ({𝑗} × 𝐵) = ({𝑦} × 𝑦 / 𝑗𝐵)
157156uneq2i 4122 . . . . . 6 ( 𝑗𝑥 ({𝑗} × 𝐵) ∪ 𝑗 ∈ {𝑦} ({𝑗} × 𝐵)) = ( 𝑗𝑥 ({𝑗} × 𝐵) ∪ ({𝑦} × 𝑦 / 𝑗𝐵))
158146, 157eqtri 2847 . . . . 5 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) = ( 𝑗𝑥 ({𝑗} × 𝐵) ∪ ({𝑦} × 𝑦 / 𝑗𝐵))
159158a1i 11 . . . 4 (𝜑 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) = ( 𝑗𝑥 ({𝑗} × 𝐵) ∪ ({𝑦} × 𝑦 / 𝑗𝐵)))
160 snfi 8590 . . . . . . 7 {𝑗} ∈ Fin
161119, 18syldan 594 . . . . . . 7 ((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) → 𝐵 ∈ Fin)
162 xpfi 8786 . . . . . . 7 (({𝑗} ∈ Fin ∧ 𝐵 ∈ Fin) → ({𝑗} × 𝐵) ∈ Fin)
163160, 161, 162sylancr 590 . . . . . 6 ((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) → ({𝑗} × 𝐵) ∈ Fin)
164163ralrimiva 3177 . . . . 5 (𝜑 → ∀𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin)
165 iunfi 8809 . . . . 5 (((𝑥 ∪ {𝑦}) ∈ Fin ∧ ∀𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin) → 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin)
166118, 164, 165syl2anc 587 . . . 4 (𝜑 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin)
167 eliun 4909 . . . . . 6 (𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ↔ ∃𝑗 ∈ (𝑥 ∪ {𝑦})𝑧 ∈ ({𝑗} × 𝐵))
168 elxp 5565 . . . . . . . 8 (𝑧 ∈ ({𝑗} × 𝐵) ↔ ∃𝑚𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵)))
169 simprl 770 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝑧 = ⟨𝑚, 𝑘⟩)
170 simprrl 780 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝑚 ∈ {𝑗})
171 elsni 4567 . . . . . . . . . . . . . . 15 (𝑚 ∈ {𝑗} → 𝑚 = 𝑗)
172170, 171syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝑚 = 𝑗)
173172opeq1d 4795 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → ⟨𝑚, 𝑘⟩ = ⟨𝑗, 𝑘⟩)
174169, 173eqtrd 2859 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝑧 = ⟨𝑗, 𝑘⟩)
175174, 89syl 17 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝐷 = 𝐶)
176 simpll 766 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝜑)
177119adantr 484 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝑗𝐴)
178 simprrr 781 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝑘𝐵)
179176, 177, 178, 26syl12anc 835 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝐶 ∈ ℂ)
180175, 179eqeltrd 2916 . . . . . . . . . 10 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝐷 ∈ ℂ)
181180ex 416 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) → ((𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵)) → 𝐷 ∈ ℂ))
182181exlimdvv 1936 . . . . . . . 8 ((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) → (∃𝑚𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵)) → 𝐷 ∈ ℂ))
183168, 182syl5bi 245 . . . . . . 7 ((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) → (𝑧 ∈ ({𝑗} × 𝐵) → 𝐷 ∈ ℂ))
184183rexlimdva 3276 . . . . . 6 (𝜑 → (∃𝑗 ∈ (𝑥 ∪ {𝑦})𝑧 ∈ ({𝑗} × 𝐵) → 𝐷 ∈ ℂ))
185167, 184syl5bi 245 . . . . 5 (𝜑 → (𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) → 𝐷 ∈ ℂ))
186185imp 410 . . . 4 ((𝜑𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)) → 𝐷 ∈ ℂ)
187145, 159, 166, 186fprodsplit 15320 . . 3 (𝜑 → ∏𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷 = (∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷 · ∏𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷))
188187adantr 484 . 2 ((𝜑𝜓) → ∏𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷 = (∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷 · ∏𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷))
189112, 124, 1883eqtr4d 2869 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 209  wa 399   = wceq 1538  wex 1781  wcel 2115  wral 3133  wrex 3134  csb 3866  cun 3917  cin 3918  wss 3919  c0 4276  {csn 4550  cop 4556   ciun 4905   × cxp 5540  cres 5544  1-1-ontowf1o 6342  cfv 6343  (class class class)co 7149  1st c1st 7682  2nd c2nd 7683  Fincfn 8505  cc 10533   · cmul 10540  cprod 15259
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-inf2 9101  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612  ax-pre-sup 10613
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8285  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-sup 8903  df-oi 8971  df-card 9365  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-div 11296  df-nn 11635  df-2 11697  df-3 11698  df-n0 11895  df-z 11979  df-uz 12241  df-rp 12387  df-fz 12895  df-fzo 13038  df-seq 13374  df-exp 13435  df-hash 13696  df-cj 14458  df-re 14459  df-im 14460  df-sqrt 14594  df-abs 14595  df-clim 14845  df-prod 15260
This theorem is referenced by:  fprod2d  15335
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