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Theorem fprodcom2OLD 14554
 Description: Obsolete proof of fprodcom2 14553 as of 2-Aug-2021. (Contributed by Scott Fenton, 1-Feb-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
fprodcom2.1 (𝜑𝐴 ∈ Fin)
fprodcom2.2 (𝜑𝐶 ∈ Fin)
fprodcom2.3 ((𝜑𝑗𝐴) → 𝐵 ∈ Fin)
fprodcom2.4 (𝜑 → ((𝑗𝐴𝑘𝐵) ↔ (𝑘𝐶𝑗𝐷)))
fprodcom2.5 ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐸 ∈ ℂ)
Assertion
Ref Expression
fprodcom2OLD (𝜑 → ∏𝑗𝐴𝑘𝐵 𝐸 = ∏𝑘𝐶𝑗𝐷 𝐸)
Distinct variable groups:   𝐴,𝑗,𝑘   𝐵,𝑘   𝐶,𝑗,𝑘   𝐷,𝑗   𝜑,𝑗,𝑘
Allowed substitution hints:   𝐵(𝑗)   𝐷(𝑘)   𝐸(𝑗,𝑘)

Proof of Theorem fprodcom2OLD
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relxp 5150 . . . . . . . . 9 Rel ({𝑗} × 𝐵)
21rgenw 2908 . . . . . . . 8 𝑗𝐴 Rel ({𝑗} × 𝐵)
3 reliun 5162 . . . . . . . 8 (Rel 𝑗𝐴 ({𝑗} × 𝐵) ↔ ∀𝑗𝐴 Rel ({𝑗} × 𝐵))
42, 3mpbir 220 . . . . . . 7 Rel 𝑗𝐴 ({𝑗} × 𝐵)
5 relcnv 5422 . . . . . . 7 Rel 𝑘𝐶 ({𝑘} × 𝐷)
6 ancom 465 . . . . . . . . . . . 12 ((𝑥 = 𝑗𝑦 = 𝑘) ↔ (𝑦 = 𝑘𝑥 = 𝑗))
7 vex 3176 . . . . . . . . . . . . 13 𝑥 ∈ V
8 vex 3176 . . . . . . . . . . . . 13 𝑦 ∈ V
97, 8opth 4871 . . . . . . . . . . . 12 (⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ↔ (𝑥 = 𝑗𝑦 = 𝑘))
108, 7opth 4871 . . . . . . . . . . . 12 (⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ↔ (𝑦 = 𝑘𝑥 = 𝑗))
116, 9, 103bitr4i 291 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ↔ ⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩)
1211a1i 11 . . . . . . . . . 10 (𝜑 → (⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ↔ ⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩))
13 fprodcom2.4 . . . . . . . . . 10 (𝜑 → ((𝑗𝐴𝑘𝐵) ↔ (𝑘𝐶𝑗𝐷)))
1412, 13anbi12d 743 . . . . . . . . 9 (𝜑 → ((⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗𝐴𝑘𝐵)) ↔ (⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘𝐶𝑗𝐷))))
15142exbidv 1839 . . . . . . . 8 (𝜑 → (∃𝑗𝑘(⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗𝐴𝑘𝐵)) ↔ ∃𝑗𝑘(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘𝐶𝑗𝐷))))
16 eliunxp 5181 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗𝑘(⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗𝐴𝑘𝐵)))
177, 8opelcnv 5226 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷) ↔ ⟨𝑦, 𝑥⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷))
18 eliunxp 5181 . . . . . . . . 9 (⟨𝑦, 𝑥⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷) ↔ ∃𝑘𝑗(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘𝐶𝑗𝐷)))
19 excom 2029 . . . . . . . . 9 (∃𝑘𝑗(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘𝐶𝑗𝐷)) ↔ ∃𝑗𝑘(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘𝐶𝑗𝐷)))
2017, 18, 193bitri 285 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷) ↔ ∃𝑗𝑘(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘𝐶𝑗𝐷)))
2115, 16, 203bitr4g 302 . . . . . . 7 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷)))
224, 5, 21eqrelrdv 5139 . . . . . 6 (𝜑 𝑗𝐴 ({𝑗} × 𝐵) = 𝑘𝐶 ({𝑘} × 𝐷))
23 nfcv 2751 . . . . . . 7 𝑚({𝑗} × 𝐵)
24 nfcv 2751 . . . . . . . 8 𝑗{𝑚}
25 nfcsb1v 3515 . . . . . . . 8 𝑗𝑚 / 𝑗𝐵
2624, 25nfxp 5066 . . . . . . 7 𝑗({𝑚} × 𝑚 / 𝑗𝐵)
27 sneq 4135 . . . . . . . 8 (𝑗 = 𝑚 → {𝑗} = {𝑚})
28 csbeq1a 3508 . . . . . . . 8 (𝑗 = 𝑚𝐵 = 𝑚 / 𝑗𝐵)
2927, 28xpeq12d 5064 . . . . . . 7 (𝑗 = 𝑚 → ({𝑗} × 𝐵) = ({𝑚} × 𝑚 / 𝑗𝐵))
3023, 26, 29cbviun 4493 . . . . . 6 𝑗𝐴 ({𝑗} × 𝐵) = 𝑚𝐴 ({𝑚} × 𝑚 / 𝑗𝐵)
31 nfcv 2751 . . . . . . . 8 𝑛({𝑘} × 𝐷)
32 nfcv 2751 . . . . . . . . 9 𝑘{𝑛}
33 nfcsb1v 3515 . . . . . . . . 9 𝑘𝑛 / 𝑘𝐷
3432, 33nfxp 5066 . . . . . . . 8 𝑘({𝑛} × 𝑛 / 𝑘𝐷)
35 sneq 4135 . . . . . . . . 9 (𝑘 = 𝑛 → {𝑘} = {𝑛})
36 csbeq1a 3508 . . . . . . . . 9 (𝑘 = 𝑛𝐷 = 𝑛 / 𝑘𝐷)
3735, 36xpeq12d 5064 . . . . . . . 8 (𝑘 = 𝑛 → ({𝑘} × 𝐷) = ({𝑛} × 𝑛 / 𝑘𝐷))
3831, 34, 37cbviun 4493 . . . . . . 7 𝑘𝐶 ({𝑘} × 𝐷) = 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)
3938cnveqi 5219 . . . . . 6 𝑘𝐶 ({𝑘} × 𝐷) = 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)
4022, 30, 393eqtr3g 2667 . . . . 5 (𝜑 𝑚𝐴 ({𝑚} × 𝑚 / 𝑗𝐵) = 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷))
4140prodeq1d 14490 . . . 4 (𝜑 → ∏𝑧 𝑚𝐴 ({𝑚} × 𝑚 / 𝑗𝐵)(2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸 = ∏𝑧 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)(2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸)
42 vex 3176 . . . . . . . 8 𝑛 ∈ V
43 vex 3176 . . . . . . . 8 𝑚 ∈ V
4442, 43op1std 7069 . . . . . . 7 (𝑤 = ⟨𝑛, 𝑚⟩ → (1st𝑤) = 𝑛)
4544csbeq1d 3506 . . . . . 6 (𝑤 = ⟨𝑛, 𝑚⟩ → (1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸 = 𝑛 / 𝑘(2nd𝑤) / 𝑗𝐸)
4642, 43op2ndd 7070 . . . . . . . 8 (𝑤 = ⟨𝑛, 𝑚⟩ → (2nd𝑤) = 𝑚)
4746csbeq1d 3506 . . . . . . 7 (𝑤 = ⟨𝑛, 𝑚⟩ → (2nd𝑤) / 𝑗𝐸 = 𝑚 / 𝑗𝐸)
4847csbeq2dv 3944 . . . . . 6 (𝑤 = ⟨𝑛, 𝑚⟩ → 𝑛 / 𝑘(2nd𝑤) / 𝑗𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
4945, 48eqtrd 2644 . . . . 5 (𝑤 = ⟨𝑛, 𝑚⟩ → (1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
5043, 42op2ndd 7070 . . . . . . 7 (𝑧 = ⟨𝑚, 𝑛⟩ → (2nd𝑧) = 𝑛)
5150csbeq1d 3506 . . . . . 6 (𝑧 = ⟨𝑚, 𝑛⟩ → (2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸 = 𝑛 / 𝑘(1st𝑧) / 𝑗𝐸)
5243, 42op1std 7069 . . . . . . . 8 (𝑧 = ⟨𝑚, 𝑛⟩ → (1st𝑧) = 𝑚)
5352csbeq1d 3506 . . . . . . 7 (𝑧 = ⟨𝑚, 𝑛⟩ → (1st𝑧) / 𝑗𝐸 = 𝑚 / 𝑗𝐸)
5453csbeq2dv 3944 . . . . . 6 (𝑧 = ⟨𝑚, 𝑛⟩ → 𝑛 / 𝑘(1st𝑧) / 𝑗𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
5551, 54eqtrd 2644 . . . . 5 (𝑧 = ⟨𝑚, 𝑛⟩ → (2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
56 fprodcom2.2 . . . . . 6 (𝜑𝐶 ∈ Fin)
57 snfi 7923 . . . . . . . 8 {𝑛} ∈ Fin
58 fprodcom2.1 . . . . . . . . . 10 (𝜑𝐴 ∈ Fin)
5958adantr 480 . . . . . . . . 9 ((𝜑𝑛𝐶) → 𝐴 ∈ Fin)
6033nfcri 2745 . . . . . . . . . . . . . . . . . 18 𝑘 𝑚𝑛 / 𝑘𝐷
61 id 22 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑛𝑘 = 𝑛)
62 vsnid 4156 . . . . . . . . . . . . . . . . . . . . . 22 𝑘 ∈ {𝑘}
6361, 62syl6eqelr 2697 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑛𝑛 ∈ {𝑘})
6463biantrurd 528 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑛 → (𝑚𝐷 ↔ (𝑛 ∈ {𝑘} ∧ 𝑚𝐷)))
65 opelxp 5070 . . . . . . . . . . . . . . . . . . . 20 (⟨𝑛, 𝑚⟩ ∈ ({𝑘} × 𝐷) ↔ (𝑛 ∈ {𝑘} ∧ 𝑚𝐷))
6664, 65syl6rbbr 278 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑛 → (⟨𝑛, 𝑚⟩ ∈ ({𝑘} × 𝐷) ↔ 𝑚𝐷))
6736eleq2d 2673 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑛 → (𝑚𝐷𝑚𝑛 / 𝑘𝐷))
6866, 67bitrd 267 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑛 → (⟨𝑛, 𝑚⟩ ∈ ({𝑘} × 𝐷) ↔ 𝑚𝑛 / 𝑘𝐷))
6960, 68rspce 3277 . . . . . . . . . . . . . . . . 17 ((𝑛𝐶𝑚𝑛 / 𝑘𝐷) → ∃𝑘𝐶𝑛, 𝑚⟩ ∈ ({𝑘} × 𝐷))
70 eliun 4460 . . . . . . . . . . . . . . . . 17 (⟨𝑛, 𝑚⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷) ↔ ∃𝑘𝐶𝑛, 𝑚⟩ ∈ ({𝑘} × 𝐷))
7169, 70sylibr 223 . . . . . . . . . . . . . . . 16 ((𝑛𝐶𝑚𝑛 / 𝑘𝐷) → ⟨𝑛, 𝑚⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷))
7243, 42opelcnv 5226 . . . . . . . . . . . . . . . 16 (⟨𝑚, 𝑛⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷) ↔ ⟨𝑛, 𝑚⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷))
7371, 72sylibr 223 . . . . . . . . . . . . . . 15 ((𝑛𝐶𝑚𝑛 / 𝑘𝐷) → ⟨𝑚, 𝑛⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷))
7473adantl 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → ⟨𝑚, 𝑛⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷))
7522adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → 𝑗𝐴 ({𝑗} × 𝐵) = 𝑘𝐶 ({𝑘} × 𝐷))
7674, 75eleqtrrd 2691 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → ⟨𝑚, 𝑛⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐵))
77 eliun 4460 . . . . . . . . . . . . 13 (⟨𝑚, 𝑛⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗𝐴𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵))
7876, 77sylib 207 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → ∃𝑗𝐴𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵))
79 simpr 476 . . . . . . . . . . . . . . . . 17 ((𝑗𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵))
80 opelxp 5070 . . . . . . . . . . . . . . . . 17 (⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵) ↔ (𝑚 ∈ {𝑗} ∧ 𝑛𝐵))
8179, 80sylib 207 . . . . . . . . . . . . . . . 16 ((𝑗𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → (𝑚 ∈ {𝑗} ∧ 𝑛𝐵))
8281simpld 474 . . . . . . . . . . . . . . 15 ((𝑗𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → 𝑚 ∈ {𝑗})
83 elsni 4142 . . . . . . . . . . . . . . 15 (𝑚 ∈ {𝑗} → 𝑚 = 𝑗)
8482, 83syl 17 . . . . . . . . . . . . . 14 ((𝑗𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → 𝑚 = 𝑗)
85 simpl 472 . . . . . . . . . . . . . 14 ((𝑗𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → 𝑗𝐴)
8684, 85eqeltrd 2688 . . . . . . . . . . . . 13 ((𝑗𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → 𝑚𝐴)
8786rexlimiva 3010 . . . . . . . . . . . 12 (∃𝑗𝐴𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵) → 𝑚𝐴)
8878, 87syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → 𝑚𝐴)
8988expr 641 . . . . . . . . . 10 ((𝜑𝑛𝐶) → (𝑚𝑛 / 𝑘𝐷𝑚𝐴))
9089ssrdv 3574 . . . . . . . . 9 ((𝜑𝑛𝐶) → 𝑛 / 𝑘𝐷𝐴)
91 ssfi 8065 . . . . . . . . 9 ((𝐴 ∈ Fin ∧ 𝑛 / 𝑘𝐷𝐴) → 𝑛 / 𝑘𝐷 ∈ Fin)
9259, 90, 91syl2anc 691 . . . . . . . 8 ((𝜑𝑛𝐶) → 𝑛 / 𝑘𝐷 ∈ Fin)
93 xpfi 8116 . . . . . . . 8 (({𝑛} ∈ Fin ∧ 𝑛 / 𝑘𝐷 ∈ Fin) → ({𝑛} × 𝑛 / 𝑘𝐷) ∈ Fin)
9457, 92, 93sylancr 694 . . . . . . 7 ((𝜑𝑛𝐶) → ({𝑛} × 𝑛 / 𝑘𝐷) ∈ Fin)
9594ralrimiva 2949 . . . . . 6 (𝜑 → ∀𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷) ∈ Fin)
96 iunfi 8137 . . . . . 6 ((𝐶 ∈ Fin ∧ ∀𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷) ∈ Fin) → 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷) ∈ Fin)
9756, 95, 96syl2anc 691 . . . . 5 (𝜑 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷) ∈ Fin)
98 reliun 5162 . . . . . . 7 (Rel 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷) ↔ ∀𝑛𝐶 Rel ({𝑛} × 𝑛 / 𝑘𝐷))
99 relxp 5150 . . . . . . . 8 Rel ({𝑛} × 𝑛 / 𝑘𝐷)
10099a1i 11 . . . . . . 7 (𝑛𝐶 → Rel ({𝑛} × 𝑛 / 𝑘𝐷))
10198, 100mprgbir 2911 . . . . . 6 Rel 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)
102101a1i 11 . . . . 5 (𝜑 → Rel 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷))
103 simpr 476 . . . . . . . 8 ((𝜑𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)) → 𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷))
104 eliun 4460 . . . . . . . 8 (𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷) ↔ ∃𝑛𝐶 𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷))
105103, 104sylib 207 . . . . . . 7 ((𝜑𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)) → ∃𝑛𝐶 𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷))
106 xp2nd 7090 . . . . . . . . . 10 (𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷) → (2nd𝑤) ∈ 𝑛 / 𝑘𝐷)
107106adantl 481 . . . . . . . . 9 ((𝑛𝐶𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷)) → (2nd𝑤) ∈ 𝑛 / 𝑘𝐷)
108 xp1st 7089 . . . . . . . . . . . 12 (𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷) → (1st𝑤) ∈ {𝑛})
109108adantl 481 . . . . . . . . . . 11 ((𝑛𝐶𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷)) → (1st𝑤) ∈ {𝑛})
110 elsni 4142 . . . . . . . . . . 11 ((1st𝑤) ∈ {𝑛} → (1st𝑤) = 𝑛)
111109, 110syl 17 . . . . . . . . . 10 ((𝑛𝐶𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷)) → (1st𝑤) = 𝑛)
112111csbeq1d 3506 . . . . . . . . 9 ((𝑛𝐶𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷)) → (1st𝑤) / 𝑘𝐷 = 𝑛 / 𝑘𝐷)
113107, 112eleqtrrd 2691 . . . . . . . 8 ((𝑛𝐶𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷)) → (2nd𝑤) ∈ (1st𝑤) / 𝑘𝐷)
114113rexlimiva 3010 . . . . . . 7 (∃𝑛𝐶 𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷) → (2nd𝑤) ∈ (1st𝑤) / 𝑘𝐷)
115105, 114syl 17 . . . . . 6 ((𝜑𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)) → (2nd𝑤) ∈ (1st𝑤) / 𝑘𝐷)
116 simpl 472 . . . . . . . . . 10 ((𝑛𝐶𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷)) → 𝑛𝐶)
117111, 116eqeltrd 2688 . . . . . . . . 9 ((𝑛𝐶𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷)) → (1st𝑤) ∈ 𝐶)
118117rexlimiva 3010 . . . . . . . 8 (∃𝑛𝐶 𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷) → (1st𝑤) ∈ 𝐶)
119105, 118syl 17 . . . . . . 7 ((𝜑𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)) → (1st𝑤) ∈ 𝐶)
120 simpl 472 . . . . . . . . . 10 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → 𝜑)
12125nfcri 2745 . . . . . . . . . . . 12 𝑗 𝑛𝑚 / 𝑗𝐵
12283eqcomd 2616 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ {𝑗} → 𝑗 = 𝑚)
123122, 28syl 17 . . . . . . . . . . . . . . . 16 (𝑚 ∈ {𝑗} → 𝐵 = 𝑚 / 𝑗𝐵)
124123eleq2d 2673 . . . . . . . . . . . . . . 15 (𝑚 ∈ {𝑗} → (𝑛𝐵𝑛𝑚 / 𝑗𝐵))
125124biimpa 500 . . . . . . . . . . . . . 14 ((𝑚 ∈ {𝑗} ∧ 𝑛𝐵) → 𝑛𝑚 / 𝑗𝐵)
12680, 125sylbi 206 . . . . . . . . . . . . 13 (⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵) → 𝑛𝑚 / 𝑗𝐵)
127126a1i 11 . . . . . . . . . . . 12 (𝑗𝐴 → (⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵) → 𝑛𝑚 / 𝑗𝐵))
128121, 127rexlimi 3006 . . . . . . . . . . 11 (∃𝑗𝐴𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵) → 𝑛𝑚 / 𝑗𝐵)
12978, 128syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → 𝑛𝑚 / 𝑗𝐵)
130 fprodcom2.5 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐸 ∈ ℂ)
131130ralrimivva 2954 . . . . . . . . . . . . 13 (𝜑 → ∀𝑗𝐴𝑘𝐵 𝐸 ∈ ℂ)
132 nfcsb1v 3515 . . . . . . . . . . . . . . . 16 𝑗𝑚 / 𝑗𝐸
133132nfel1 2765 . . . . . . . . . . . . . . 15 𝑗𝑚 / 𝑗𝐸 ∈ ℂ
13425, 133nfral 2929 . . . . . . . . . . . . . 14 𝑗𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐸 ∈ ℂ
135 csbeq1a 3508 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑚𝐸 = 𝑚 / 𝑗𝐸)
136135eleq1d 2672 . . . . . . . . . . . . . . 15 (𝑗 = 𝑚 → (𝐸 ∈ ℂ ↔ 𝑚 / 𝑗𝐸 ∈ ℂ))
13728, 136raleqbidv 3129 . . . . . . . . . . . . . 14 (𝑗 = 𝑚 → (∀𝑘𝐵 𝐸 ∈ ℂ ↔ ∀𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐸 ∈ ℂ))
138134, 137rspc 3276 . . . . . . . . . . . . 13 (𝑚𝐴 → (∀𝑗𝐴𝑘𝐵 𝐸 ∈ ℂ → ∀𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐸 ∈ ℂ))
139131, 138mpan9 485 . . . . . . . . . . . 12 ((𝜑𝑚𝐴) → ∀𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐸 ∈ ℂ)
140 nfcsb1v 3515 . . . . . . . . . . . . . 14 𝑘𝑛 / 𝑘𝑚 / 𝑗𝐸
141140nfel1 2765 . . . . . . . . . . . . 13 𝑘𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ
142 csbeq1a 3508 . . . . . . . . . . . . . 14 (𝑘 = 𝑛𝑚 / 𝑗𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
143142eleq1d 2672 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → (𝑚 / 𝑗𝐸 ∈ ℂ ↔ 𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ))
144141, 143rspc 3276 . . . . . . . . . . . 12 (𝑛𝑚 / 𝑗𝐵 → (∀𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐸 ∈ ℂ → 𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ))
145139, 144syl5com 31 . . . . . . . . . . 11 ((𝜑𝑚𝐴) → (𝑛𝑚 / 𝑗𝐵𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ))
146145impr 647 . . . . . . . . . 10 ((𝜑 ∧ (𝑚𝐴𝑛𝑚 / 𝑗𝐵)) → 𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ)
147120, 88, 129, 146syl12anc 1316 . . . . . . . . 9 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → 𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ)
148147ralrimivva 2954 . . . . . . . 8 (𝜑 → ∀𝑛𝐶𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ)
149148adantr 480 . . . . . . 7 ((𝜑𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)) → ∀𝑛𝐶𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ)
150 csbeq1 3502 . . . . . . . . 9 (𝑛 = (1st𝑤) → 𝑛 / 𝑘𝐷 = (1st𝑤) / 𝑘𝐷)
151 csbeq1 3502 . . . . . . . . . 10 (𝑛 = (1st𝑤) → 𝑛 / 𝑘𝑚 / 𝑗𝐸 = (1st𝑤) / 𝑘𝑚 / 𝑗𝐸)
152151eleq1d 2672 . . . . . . . . 9 (𝑛 = (1st𝑤) → (𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ ↔ (1st𝑤) / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ))
153150, 152raleqbidv 3129 . . . . . . . 8 (𝑛 = (1st𝑤) → (∀𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ ↔ ∀𝑚 (1st𝑤) / 𝑘𝐷(1st𝑤) / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ))
154153rspcv 3278 . . . . . . 7 ((1st𝑤) ∈ 𝐶 → (∀𝑛𝐶𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ → ∀𝑚 (1st𝑤) / 𝑘𝐷(1st𝑤) / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ))
155119, 149, 154sylc 63 . . . . . 6 ((𝜑𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)) → ∀𝑚 (1st𝑤) / 𝑘𝐷(1st𝑤) / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ)
156 csbeq1 3502 . . . . . . . . 9 (𝑚 = (2nd𝑤) → 𝑚 / 𝑗𝐸 = (2nd𝑤) / 𝑗𝐸)
157156csbeq2dv 3944 . . . . . . . 8 (𝑚 = (2nd𝑤) → (1st𝑤) / 𝑘𝑚 / 𝑗𝐸 = (1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸)
158157eleq1d 2672 . . . . . . 7 (𝑚 = (2nd𝑤) → ((1st𝑤) / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ ↔ (1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸 ∈ ℂ))
159158rspcv 3278 . . . . . 6 ((2nd𝑤) ∈ (1st𝑤) / 𝑘𝐷 → (∀𝑚 (1st𝑤) / 𝑘𝐷(1st𝑤) / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ → (1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸 ∈ ℂ))
160115, 155, 159sylc 63 . . . . 5 ((𝜑𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)) → (1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸 ∈ ℂ)
16149, 55, 97, 102, 160fprodcnv 14552 . . . 4 (𝜑 → ∏𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)(1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸 = ∏𝑧 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)(2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸)
16241, 161eqtr4d 2647 . . 3 (𝜑 → ∏𝑧 𝑚𝐴 ({𝑚} × 𝑚 / 𝑗𝐵)(2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸 = ∏𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)(1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸)
163 fprodcom2.3 . . . . . 6 ((𝜑𝑗𝐴) → 𝐵 ∈ Fin)
164163ralrimiva 2949 . . . . 5 (𝜑 → ∀𝑗𝐴 𝐵 ∈ Fin)
16525nfel1 2765 . . . . . 6 𝑗𝑚 / 𝑗𝐵 ∈ Fin
16628eleq1d 2672 . . . . . 6 (𝑗 = 𝑚 → (𝐵 ∈ Fin ↔ 𝑚 / 𝑗𝐵 ∈ Fin))
167165, 166rspc 3276 . . . . 5 (𝑚𝐴 → (∀𝑗𝐴 𝐵 ∈ Fin → 𝑚 / 𝑗𝐵 ∈ Fin))
168164, 167mpan9 485 . . . 4 ((𝜑𝑚𝐴) → 𝑚 / 𝑗𝐵 ∈ Fin)
16955, 58, 168, 146fprod2d 14550 . . 3 (𝜑 → ∏𝑚𝐴𝑛 𝑚 / 𝑗𝐵𝑛 / 𝑘𝑚 / 𝑗𝐸 = ∏𝑧 𝑚𝐴 ({𝑚} × 𝑚 / 𝑗𝐵)(2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸)
17049, 56, 92, 147fprod2d 14550 . . 3 (𝜑 → ∏𝑛𝐶𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸 = ∏𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)(1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸)
171162, 169, 1703eqtr4d 2654 . 2 (𝜑 → ∏𝑚𝐴𝑛 𝑚 / 𝑗𝐵𝑛 / 𝑘𝑚 / 𝑗𝐸 = ∏𝑛𝐶𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸)
172 nfcv 2751 . . 3 𝑚𝑘𝐵 𝐸
173 nfcv 2751 . . . . 5 𝑗𝑛
174173, 132nfcsb 3517 . . . 4 𝑗𝑛 / 𝑘𝑚 / 𝑗𝐸
17525, 174nfcprod 14480 . . 3 𝑗𝑛 𝑚 / 𝑗𝐵𝑛 / 𝑘𝑚 / 𝑗𝐸
176 nfcv 2751 . . . . 5 𝑛𝐸
177 nfcsb1v 3515 . . . . 5 𝑘𝑛 / 𝑘𝐸
178 csbeq1a 3508 . . . . 5 (𝑘 = 𝑛𝐸 = 𝑛 / 𝑘𝐸)
179176, 177, 178cbvprodi 14486 . . . 4 𝑘𝐵 𝐸 = ∏𝑛𝐵 𝑛 / 𝑘𝐸
180135csbeq2dv 3944 . . . . . 6 (𝑗 = 𝑚𝑛 / 𝑘𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
181180adantr 480 . . . . 5 ((𝑗 = 𝑚𝑛𝐵) → 𝑛 / 𝑘𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
18228, 181prodeq12dv 14495 . . . 4 (𝑗 = 𝑚 → ∏𝑛𝐵 𝑛 / 𝑘𝐸 = ∏𝑛 𝑚 / 𝑗𝐵𝑛 / 𝑘𝑚 / 𝑗𝐸)
183179, 182syl5eq 2656 . . 3 (𝑗 = 𝑚 → ∏𝑘𝐵 𝐸 = ∏𝑛 𝑚 / 𝑗𝐵𝑛 / 𝑘𝑚 / 𝑗𝐸)
184172, 175, 183cbvprodi 14486 . 2 𝑗𝐴𝑘𝐵 𝐸 = ∏𝑚𝐴𝑛 𝑚 / 𝑗𝐵𝑛 / 𝑘𝑚 / 𝑗𝐸
185 nfcv 2751 . . 3 𝑛𝑗𝐷 𝐸
18633, 140nfcprod 14480 . . 3 𝑘𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸
187 nfcv 2751 . . . . 5 𝑚𝐸
188187, 132, 135cbvprodi 14486 . . . 4 𝑗𝐷 𝐸 = ∏𝑚𝐷 𝑚 / 𝑗𝐸
189142adantr 480 . . . . 5 ((𝑘 = 𝑛𝑚𝐷) → 𝑚 / 𝑗𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
19036, 189prodeq12dv 14495 . . . 4 (𝑘 = 𝑛 → ∏𝑚𝐷 𝑚 / 𝑗𝐸 = ∏𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸)
191188, 190syl5eq 2656 . . 3 (𝑘 = 𝑛 → ∏𝑗𝐷 𝐸 = ∏𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸)
192185, 186, 191cbvprodi 14486 . 2 𝑘𝐶𝑗𝐷 𝐸 = ∏𝑛𝐶𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸
193171, 184, 1923eqtr4g 2669 1 (𝜑 → ∏𝑗𝐴𝑘𝐵 𝐸 = ∏𝑘𝐶𝑗𝐷 𝐸)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  ⦋csb 3499   ⊆ wss 3540  {csn 4125  ⟨cop 4131  ∪ ciun 4455   × cxp 5036  ◡ccnv 5037  Rel wrel 5043  ‘cfv 5804  1st c1st 7057  2nd c2nd 7058  Fincfn 7841  ℂcc 9813  ∏cprod 14474 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-prod 14475 This theorem is referenced by: (None)
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