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Theorem fprodcom2OLD 14934
Description: Obsolete proof of fprodcom2 14933 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 5283 . . . . . . . . 9 Rel ({𝑗} × 𝐵)
21rgenw 3062 . . . . . . . 8 𝑗𝐴 Rel ({𝑗} × 𝐵)
3 reliun 5395 . . . . . . . 8 (Rel 𝑗𝐴 ({𝑗} × 𝐵) ↔ ∀𝑗𝐴 Rel ({𝑗} × 𝐵))
42, 3mpbir 221 . . . . . . 7 Rel 𝑗𝐴 ({𝑗} × 𝐵)
5 relcnv 5661 . . . . . . 7 Rel 𝑘𝐶 ({𝑘} × 𝐷)
6 ancom 465 . . . . . . . . . . . 12 ((𝑥 = 𝑗𝑦 = 𝑘) ↔ (𝑦 = 𝑘𝑥 = 𝑗))
7 vex 3343 . . . . . . . . . . . . 13 𝑥 ∈ V
8 vex 3343 . . . . . . . . . . . . 13 𝑦 ∈ V
97, 8opth 5093 . . . . . . . . . . . 12 (⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ↔ (𝑥 = 𝑗𝑦 = 𝑘))
108, 7opth 5093 . . . . . . . . . . . 12 (⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ↔ (𝑦 = 𝑘𝑥 = 𝑗))
116, 9, 103bitr4i 292 . . . . . . . . . . 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 749 . . . . . . . . 9 (𝜑 → ((⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗𝐴𝑘𝐵)) ↔ (⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘𝐶𝑗𝐷))))
15142exbidv 2001 . . . . . . . 8 (𝜑 → (∃𝑗𝑘(⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗𝐴𝑘𝐵)) ↔ ∃𝑗𝑘(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘𝐶𝑗𝐷))))
16 eliunxp 5415 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗𝑘(⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗𝐴𝑘𝐵)))
177, 8opelcnv 5459 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷) ↔ ⟨𝑦, 𝑥⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷))
18 eliunxp 5415 . . . . . . . . 9 (⟨𝑦, 𝑥⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷) ↔ ∃𝑘𝑗(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘𝐶𝑗𝐷)))
19 excom 2191 . . . . . . . . 9 (∃𝑘𝑗(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘𝐶𝑗𝐷)) ↔ ∃𝑗𝑘(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘𝐶𝑗𝐷)))
2017, 18, 193bitri 286 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷) ↔ ∃𝑗𝑘(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘𝐶𝑗𝐷)))
2115, 16, 203bitr4g 303 . . . . . . 7 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷)))
224, 5, 21eqrelrdv 5373 . . . . . 6 (𝜑 𝑗𝐴 ({𝑗} × 𝐵) = 𝑘𝐶 ({𝑘} × 𝐷))
23 nfcv 2902 . . . . . . 7 𝑚({𝑗} × 𝐵)
24 nfcv 2902 . . . . . . . 8 𝑗{𝑚}
25 nfcsb1v 3690 . . . . . . . 8 𝑗𝑚 / 𝑗𝐵
2624, 25nfxp 5299 . . . . . . 7 𝑗({𝑚} × 𝑚 / 𝑗𝐵)
27 sneq 4331 . . . . . . . 8 (𝑗 = 𝑚 → {𝑗} = {𝑚})
28 csbeq1a 3683 . . . . . . . 8 (𝑗 = 𝑚𝐵 = 𝑚 / 𝑗𝐵)
2927, 28xpeq12d 5297 . . . . . . 7 (𝑗 = 𝑚 → ({𝑗} × 𝐵) = ({𝑚} × 𝑚 / 𝑗𝐵))
3023, 26, 29cbviun 4709 . . . . . 6 𝑗𝐴 ({𝑗} × 𝐵) = 𝑚𝐴 ({𝑚} × 𝑚 / 𝑗𝐵)
31 nfcv 2902 . . . . . . . 8 𝑛({𝑘} × 𝐷)
32 nfcv 2902 . . . . . . . . 9 𝑘{𝑛}
33 nfcsb1v 3690 . . . . . . . . 9 𝑘𝑛 / 𝑘𝐷
3432, 33nfxp 5299 . . . . . . . 8 𝑘({𝑛} × 𝑛 / 𝑘𝐷)
35 sneq 4331 . . . . . . . . 9 (𝑘 = 𝑛 → {𝑘} = {𝑛})
36 csbeq1a 3683 . . . . . . . . 9 (𝑘 = 𝑛𝐷 = 𝑛 / 𝑘𝐷)
3735, 36xpeq12d 5297 . . . . . . . 8 (𝑘 = 𝑛 → ({𝑘} × 𝐷) = ({𝑛} × 𝑛 / 𝑘𝐷))
3831, 34, 37cbviun 4709 . . . . . . 7 𝑘𝐶 ({𝑘} × 𝐷) = 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)
3938cnveqi 5452 . . . . . 6 𝑘𝐶 ({𝑘} × 𝐷) = 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)
4022, 30, 393eqtr3g 2817 . . . . 5 (𝜑 𝑚𝐴 ({𝑚} × 𝑚 / 𝑗𝐵) = 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷))
4140prodeq1d 14870 . . . 4 (𝜑 → ∏𝑧 𝑚𝐴 ({𝑚} × 𝑚 / 𝑗𝐵)(2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸 = ∏𝑧 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)(2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸)
42 vex 3343 . . . . . . . 8 𝑛 ∈ V
43 vex 3343 . . . . . . . 8 𝑚 ∈ V
4442, 43op1std 7344 . . . . . . 7 (𝑤 = ⟨𝑛, 𝑚⟩ → (1st𝑤) = 𝑛)
4544csbeq1d 3681 . . . . . 6 (𝑤 = ⟨𝑛, 𝑚⟩ → (1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸 = 𝑛 / 𝑘(2nd𝑤) / 𝑗𝐸)
4642, 43op2ndd 7345 . . . . . . . 8 (𝑤 = ⟨𝑛, 𝑚⟩ → (2nd𝑤) = 𝑚)
4746csbeq1d 3681 . . . . . . 7 (𝑤 = ⟨𝑛, 𝑚⟩ → (2nd𝑤) / 𝑗𝐸 = 𝑚 / 𝑗𝐸)
4847csbeq2dv 4135 . . . . . 6 (𝑤 = ⟨𝑛, 𝑚⟩ → 𝑛 / 𝑘(2nd𝑤) / 𝑗𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
4945, 48eqtrd 2794 . . . . 5 (𝑤 = ⟨𝑛, 𝑚⟩ → (1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
5043, 42op2ndd 7345 . . . . . . 7 (𝑧 = ⟨𝑚, 𝑛⟩ → (2nd𝑧) = 𝑛)
5150csbeq1d 3681 . . . . . 6 (𝑧 = ⟨𝑚, 𝑛⟩ → (2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸 = 𝑛 / 𝑘(1st𝑧) / 𝑗𝐸)
5243, 42op1std 7344 . . . . . . . 8 (𝑧 = ⟨𝑚, 𝑛⟩ → (1st𝑧) = 𝑚)
5352csbeq1d 3681 . . . . . . 7 (𝑧 = ⟨𝑚, 𝑛⟩ → (1st𝑧) / 𝑗𝐸 = 𝑚 / 𝑗𝐸)
5453csbeq2dv 4135 . . . . . 6 (𝑧 = ⟨𝑚, 𝑛⟩ → 𝑛 / 𝑘(1st𝑧) / 𝑗𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
5551, 54eqtrd 2794 . . . . 5 (𝑧 = ⟨𝑚, 𝑛⟩ → (2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
56 fprodcom2.2 . . . . . 6 (𝜑𝐶 ∈ Fin)
57 snfi 8205 . . . . . . . 8 {𝑛} ∈ Fin
58 fprodcom2.1 . . . . . . . . . 10 (𝜑𝐴 ∈ Fin)
5958adantr 472 . . . . . . . . 9 ((𝜑𝑛𝐶) → 𝐴 ∈ Fin)
6033nfcri 2896 . . . . . . . . . . . . . . . . . 18 𝑘 𝑚𝑛 / 𝑘𝐷
61 id 22 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑛𝑘 = 𝑛)
62 vsnid 4354 . . . . . . . . . . . . . . . . . . . . . 22 𝑘 ∈ {𝑘}
6361, 62syl6eqelr 2848 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑛𝑛 ∈ {𝑘})
6463biantrurd 530 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑛 → (𝑚𝐷 ↔ (𝑛 ∈ {𝑘} ∧ 𝑚𝐷)))
65 opelxp 5303 . . . . . . . . . . . . . . . . . . . 20 (⟨𝑛, 𝑚⟩ ∈ ({𝑘} × 𝐷) ↔ (𝑛 ∈ {𝑘} ∧ 𝑚𝐷))
6664, 65syl6rbbr 279 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑛 → (⟨𝑛, 𝑚⟩ ∈ ({𝑘} × 𝐷) ↔ 𝑚𝐷))
6736eleq2d 2825 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑛 → (𝑚𝐷𝑚𝑛 / 𝑘𝐷))
6866, 67bitrd 268 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑛 → (⟨𝑛, 𝑚⟩ ∈ ({𝑘} × 𝐷) ↔ 𝑚𝑛 / 𝑘𝐷))
6960, 68rspce 3444 . . . . . . . . . . . . . . . . 17 ((𝑛𝐶𝑚𝑛 / 𝑘𝐷) → ∃𝑘𝐶𝑛, 𝑚⟩ ∈ ({𝑘} × 𝐷))
70 eliun 4676 . . . . . . . . . . . . . . . . 17 (⟨𝑛, 𝑚⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷) ↔ ∃𝑘𝐶𝑛, 𝑚⟩ ∈ ({𝑘} × 𝐷))
7169, 70sylibr 224 . . . . . . . . . . . . . . . 16 ((𝑛𝐶𝑚𝑛 / 𝑘𝐷) → ⟨𝑛, 𝑚⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷))
7243, 42opelcnv 5459 . . . . . . . . . . . . . . . 16 (⟨𝑚, 𝑛⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷) ↔ ⟨𝑛, 𝑚⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷))
7371, 72sylibr 224 . . . . . . . . . . . . . . 15 ((𝑛𝐶𝑚𝑛 / 𝑘𝐷) → ⟨𝑚, 𝑛⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷))
7473adantl 473 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → ⟨𝑚, 𝑛⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷))
7522adantr 472 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → 𝑗𝐴 ({𝑗} × 𝐵) = 𝑘𝐶 ({𝑘} × 𝐷))
7674, 75eleqtrrd 2842 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → ⟨𝑚, 𝑛⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐵))
77 eliun 4676 . . . . . . . . . . . . 13 (⟨𝑚, 𝑛⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗𝐴𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵))
7876, 77sylib 208 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → ∃𝑗𝐴𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵))
79 simpr 479 . . . . . . . . . . . . . . . . 17 ((𝑗𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵))
80 opelxp 5303 . . . . . . . . . . . . . . . . 17 (⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵) ↔ (𝑚 ∈ {𝑗} ∧ 𝑛𝐵))
8179, 80sylib 208 . . . . . . . . . . . . . . . 16 ((𝑗𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → (𝑚 ∈ {𝑗} ∧ 𝑛𝐵))
8281simpld 477 . . . . . . . . . . . . . . 15 ((𝑗𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → 𝑚 ∈ {𝑗})
83 elsni 4338 . . . . . . . . . . . . . . 15 (𝑚 ∈ {𝑗} → 𝑚 = 𝑗)
8482, 83syl 17 . . . . . . . . . . . . . 14 ((𝑗𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → 𝑚 = 𝑗)
85 simpl 474 . . . . . . . . . . . . . 14 ((𝑗𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → 𝑗𝐴)
8684, 85eqeltrd 2839 . . . . . . . . . . . . 13 ((𝑗𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → 𝑚𝐴)
8786rexlimiva 3166 . . . . . . . . . . . 12 (∃𝑗𝐴𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵) → 𝑚𝐴)
8878, 87syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → 𝑚𝐴)
8988expr 644 . . . . . . . . . 10 ((𝜑𝑛𝐶) → (𝑚𝑛 / 𝑘𝐷𝑚𝐴))
9089ssrdv 3750 . . . . . . . . 9 ((𝜑𝑛𝐶) → 𝑛 / 𝑘𝐷𝐴)
91 ssfi 8347 . . . . . . . . 9 ((𝐴 ∈ Fin ∧ 𝑛 / 𝑘𝐷𝐴) → 𝑛 / 𝑘𝐷 ∈ Fin)
9259, 90, 91syl2anc 696 . . . . . . . 8 ((𝜑𝑛𝐶) → 𝑛 / 𝑘𝐷 ∈ Fin)
93 xpfi 8398 . . . . . . . 8 (({𝑛} ∈ Fin ∧ 𝑛 / 𝑘𝐷 ∈ Fin) → ({𝑛} × 𝑛 / 𝑘𝐷) ∈ Fin)
9457, 92, 93sylancr 698 . . . . . . 7 ((𝜑𝑛𝐶) → ({𝑛} × 𝑛 / 𝑘𝐷) ∈ Fin)
9594ralrimiva 3104 . . . . . 6 (𝜑 → ∀𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷) ∈ Fin)
96 iunfi 8421 . . . . . 6 ((𝐶 ∈ Fin ∧ ∀𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷) ∈ Fin) → 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷) ∈ Fin)
9756, 95, 96syl2anc 696 . . . . 5 (𝜑 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷) ∈ Fin)
98 reliun 5395 . . . . . . 7 (Rel 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷) ↔ ∀𝑛𝐶 Rel ({𝑛} × 𝑛 / 𝑘𝐷))
99 relxp 5283 . . . . . . . 8 Rel ({𝑛} × 𝑛 / 𝑘𝐷)
10099a1i 11 . . . . . . 7 (𝑛𝐶 → Rel ({𝑛} × 𝑛 / 𝑘𝐷))
10198, 100mprgbir 3065 . . . . . 6 Rel 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)
102101a1i 11 . . . . 5 (𝜑 → Rel 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷))
103 simpr 479 . . . . . . . 8 ((𝜑𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)) → 𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷))
104 eliun 4676 . . . . . . . 8 (𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷) ↔ ∃𝑛𝐶 𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷))
105103, 104sylib 208 . . . . . . 7 ((𝜑𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)) → ∃𝑛𝐶 𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷))
106 xp2nd 7367 . . . . . . . . . 10 (𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷) → (2nd𝑤) ∈ 𝑛 / 𝑘𝐷)
107106adantl 473 . . . . . . . . 9 ((𝑛𝐶𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷)) → (2nd𝑤) ∈ 𝑛 / 𝑘𝐷)
108 xp1st 7366 . . . . . . . . . . . 12 (𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷) → (1st𝑤) ∈ {𝑛})
109108adantl 473 . . . . . . . . . . 11 ((𝑛𝐶𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷)) → (1st𝑤) ∈ {𝑛})
110 elsni 4338 . . . . . . . . . . 11 ((1st𝑤) ∈ {𝑛} → (1st𝑤) = 𝑛)
111109, 110syl 17 . . . . . . . . . 10 ((𝑛𝐶𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷)) → (1st𝑤) = 𝑛)
112111csbeq1d 3681 . . . . . . . . 9 ((𝑛𝐶𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷)) → (1st𝑤) / 𝑘𝐷 = 𝑛 / 𝑘𝐷)
113107, 112eleqtrrd 2842 . . . . . . . 8 ((𝑛𝐶𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷)) → (2nd𝑤) ∈ (1st𝑤) / 𝑘𝐷)
114113rexlimiva 3166 . . . . . . 7 (∃𝑛𝐶 𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷) → (2nd𝑤) ∈ (1st𝑤) / 𝑘𝐷)
115105, 114syl 17 . . . . . 6 ((𝜑𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)) → (2nd𝑤) ∈ (1st𝑤) / 𝑘𝐷)
116 simpl 474 . . . . . . . . . 10 ((𝑛𝐶𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷)) → 𝑛𝐶)
117111, 116eqeltrd 2839 . . . . . . . . 9 ((𝑛𝐶𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷)) → (1st𝑤) ∈ 𝐶)
118117rexlimiva 3166 . . . . . . . 8 (∃𝑛𝐶 𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷) → (1st𝑤) ∈ 𝐶)
119105, 118syl 17 . . . . . . 7 ((𝜑𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)) → (1st𝑤) ∈ 𝐶)
120 simpl 474 . . . . . . . . . 10 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → 𝜑)
12125nfcri 2896 . . . . . . . . . . . 12 𝑗 𝑛𝑚 / 𝑗𝐵
12283eqcomd 2766 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ {𝑗} → 𝑗 = 𝑚)
123122, 28syl 17 . . . . . . . . . . . . . . . 16 (𝑚 ∈ {𝑗} → 𝐵 = 𝑚 / 𝑗𝐵)
124123eleq2d 2825 . . . . . . . . . . . . . . 15 (𝑚 ∈ {𝑗} → (𝑛𝐵𝑛𝑚 / 𝑗𝐵))
125124biimpa 502 . . . . . . . . . . . . . 14 ((𝑚 ∈ {𝑗} ∧ 𝑛𝐵) → 𝑛𝑚 / 𝑗𝐵)
12680, 125sylbi 207 . . . . . . . . . . . . 13 (⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵) → 𝑛𝑚 / 𝑗𝐵)
127126a1i 11 . . . . . . . . . . . 12 (𝑗𝐴 → (⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵) → 𝑛𝑚 / 𝑗𝐵))
128121, 127rexlimi 3162 . . . . . . . . . . 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 3109 . . . . . . . . . . . . 13 (𝜑 → ∀𝑗𝐴𝑘𝐵 𝐸 ∈ ℂ)
132 nfcsb1v 3690 . . . . . . . . . . . . . . . 16 𝑗𝑚 / 𝑗𝐸
133132nfel1 2917 . . . . . . . . . . . . . . 15 𝑗𝑚 / 𝑗𝐸 ∈ ℂ
13425, 133nfral 3083 . . . . . . . . . . . . . 14 𝑗𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐸 ∈ ℂ
135 csbeq1a 3683 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑚𝐸 = 𝑚 / 𝑗𝐸)
136135eleq1d 2824 . . . . . . . . . . . . . . 15 (𝑗 = 𝑚 → (𝐸 ∈ ℂ ↔ 𝑚 / 𝑗𝐸 ∈ ℂ))
13728, 136raleqbidv 3291 . . . . . . . . . . . . . 14 (𝑗 = 𝑚 → (∀𝑘𝐵 𝐸 ∈ ℂ ↔ ∀𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐸 ∈ ℂ))
138134, 137rspc 3443 . . . . . . . . . . . . 13 (𝑚𝐴 → (∀𝑗𝐴𝑘𝐵 𝐸 ∈ ℂ → ∀𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐸 ∈ ℂ))
139131, 138mpan9 487 . . . . . . . . . . . 12 ((𝜑𝑚𝐴) → ∀𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐸 ∈ ℂ)
140 nfcsb1v 3690 . . . . . . . . . . . . . 14 𝑘𝑛 / 𝑘𝑚 / 𝑗𝐸
141140nfel1 2917 . . . . . . . . . . . . 13 𝑘𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ
142 csbeq1a 3683 . . . . . . . . . . . . . 14 (𝑘 = 𝑛𝑚 / 𝑗𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
143142eleq1d 2824 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → (𝑚 / 𝑗𝐸 ∈ ℂ ↔ 𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ))
144141, 143rspc 3443 . . . . . . . . . . . 12 (𝑛𝑚 / 𝑗𝐵 → (∀𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐸 ∈ ℂ → 𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ))
145139, 144syl5com 31 . . . . . . . . . . 11 ((𝜑𝑚𝐴) → (𝑛𝑚 / 𝑗𝐵𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ))
146145impr 650 . . . . . . . . . 10 ((𝜑 ∧ (𝑚𝐴𝑛𝑚 / 𝑗𝐵)) → 𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ)
147120, 88, 129, 146syl12anc 1475 . . . . . . . . 9 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → 𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ)
148147ralrimivva 3109 . . . . . . . 8 (𝜑 → ∀𝑛𝐶𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ)
149148adantr 472 . . . . . . 7 ((𝜑𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)) → ∀𝑛𝐶𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ)
150 csbeq1 3677 . . . . . . . . 9 (𝑛 = (1st𝑤) → 𝑛 / 𝑘𝐷 = (1st𝑤) / 𝑘𝐷)
151 csbeq1 3677 . . . . . . . . . 10 (𝑛 = (1st𝑤) → 𝑛 / 𝑘𝑚 / 𝑗𝐸 = (1st𝑤) / 𝑘𝑚 / 𝑗𝐸)
152151eleq1d 2824 . . . . . . . . 9 (𝑛 = (1st𝑤) → (𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ ↔ (1st𝑤) / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ))
153150, 152raleqbidv 3291 . . . . . . . 8 (𝑛 = (1st𝑤) → (∀𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ ↔ ∀𝑚 (1st𝑤) / 𝑘𝐷(1st𝑤) / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ))
154153rspcv 3445 . . . . . . 7 ((1st𝑤) ∈ 𝐶 → (∀𝑛𝐶𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ → ∀𝑚 (1st𝑤) / 𝑘𝐷(1st𝑤) / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ))
155119, 149, 154sylc 65 . . . . . 6 ((𝜑𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)) → ∀𝑚 (1st𝑤) / 𝑘𝐷(1st𝑤) / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ)
156 csbeq1 3677 . . . . . . . . 9 (𝑚 = (2nd𝑤) → 𝑚 / 𝑗𝐸 = (2nd𝑤) / 𝑗𝐸)
157156csbeq2dv 4135 . . . . . . . 8 (𝑚 = (2nd𝑤) → (1st𝑤) / 𝑘𝑚 / 𝑗𝐸 = (1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸)
158157eleq1d 2824 . . . . . . 7 (𝑚 = (2nd𝑤) → ((1st𝑤) / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ ↔ (1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸 ∈ ℂ))
159158rspcv 3445 . . . . . 6 ((2nd𝑤) ∈ (1st𝑤) / 𝑘𝐷 → (∀𝑚 (1st𝑤) / 𝑘𝐷(1st𝑤) / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ → (1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸 ∈ ℂ))
160115, 155, 159sylc 65 . . . . 5 ((𝜑𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)) → (1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸 ∈ ℂ)
16149, 55, 97, 102, 160fprodcnv 14932 . . . 4 (𝜑 → ∏𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)(1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸 = ∏𝑧 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)(2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸)
16241, 161eqtr4d 2797 . . 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 3104 . . . . 5 (𝜑 → ∀𝑗𝐴 𝐵 ∈ Fin)
16525nfel1 2917 . . . . . 6 𝑗𝑚 / 𝑗𝐵 ∈ Fin
16628eleq1d 2824 . . . . . 6 (𝑗 = 𝑚 → (𝐵 ∈ Fin ↔ 𝑚 / 𝑗𝐵 ∈ Fin))
167165, 166rspc 3443 . . . . 5 (𝑚𝐴 → (∀𝑗𝐴 𝐵 ∈ Fin → 𝑚 / 𝑗𝐵 ∈ Fin))
168164, 167mpan9 487 . . . 4 ((𝜑𝑚𝐴) → 𝑚 / 𝑗𝐵 ∈ Fin)
16955, 58, 168, 146fprod2d 14930 . . 3 (𝜑 → ∏𝑚𝐴𝑛 𝑚 / 𝑗𝐵𝑛 / 𝑘𝑚 / 𝑗𝐸 = ∏𝑧 𝑚𝐴 ({𝑚} × 𝑚 / 𝑗𝐵)(2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸)
17049, 56, 92, 147fprod2d 14930 . . 3 (𝜑 → ∏𝑛𝐶𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸 = ∏𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)(1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸)
171162, 169, 1703eqtr4d 2804 . 2 (𝜑 → ∏𝑚𝐴𝑛 𝑚 / 𝑗𝐵𝑛 / 𝑘𝑚 / 𝑗𝐸 = ∏𝑛𝐶𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸)
172 nfcv 2902 . . 3 𝑚𝑘𝐵 𝐸
173 nfcv 2902 . . . . 5 𝑗𝑛
174173, 132nfcsb 3692 . . . 4 𝑗𝑛 / 𝑘𝑚 / 𝑗𝐸
17525, 174nfcprod 14860 . . 3 𝑗𝑛 𝑚 / 𝑗𝐵𝑛 / 𝑘𝑚 / 𝑗𝐸
176 nfcv 2902 . . . . 5 𝑛𝐸
177 nfcsb1v 3690 . . . . 5 𝑘𝑛 / 𝑘𝐸
178 csbeq1a 3683 . . . . 5 (𝑘 = 𝑛𝐸 = 𝑛 / 𝑘𝐸)
179176, 177, 178cbvprodi 14866 . . . 4 𝑘𝐵 𝐸 = ∏𝑛𝐵 𝑛 / 𝑘𝐸
180135csbeq2dv 4135 . . . . . 6 (𝑗 = 𝑚𝑛 / 𝑘𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
181180adantr 472 . . . . 5 ((𝑗 = 𝑚𝑛𝐵) → 𝑛 / 𝑘𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
18228, 181prodeq12dv 14875 . . . 4 (𝑗 = 𝑚 → ∏𝑛𝐵 𝑛 / 𝑘𝐸 = ∏𝑛 𝑚 / 𝑗𝐵𝑛 / 𝑘𝑚 / 𝑗𝐸)
183179, 182syl5eq 2806 . . 3 (𝑗 = 𝑚 → ∏𝑘𝐵 𝐸 = ∏𝑛 𝑚 / 𝑗𝐵𝑛 / 𝑘𝑚 / 𝑗𝐸)
184172, 175, 183cbvprodi 14866 . 2 𝑗𝐴𝑘𝐵 𝐸 = ∏𝑚𝐴𝑛 𝑚 / 𝑗𝐵𝑛 / 𝑘𝑚 / 𝑗𝐸
185 nfcv 2902 . . 3 𝑛𝑗𝐷 𝐸
18633, 140nfcprod 14860 . . 3 𝑘𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸
187 nfcv 2902 . . . . 5 𝑚𝐸
188187, 132, 135cbvprodi 14866 . . . 4 𝑗𝐷 𝐸 = ∏𝑚𝐷 𝑚 / 𝑗𝐸
189142adantr 472 . . . . 5 ((𝑘 = 𝑛𝑚𝐷) → 𝑚 / 𝑗𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
19036, 189prodeq12dv 14875 . . . 4 (𝑘 = 𝑛 → ∏𝑚𝐷 𝑚 / 𝑗𝐸 = ∏𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸)
191188, 190syl5eq 2806 . . 3 (𝑘 = 𝑛 → ∏𝑗𝐷 𝐸 = ∏𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸)
192185, 186, 191cbvprodi 14866 . 2 𝑘𝐶𝑗𝐷 𝐸 = ∏𝑛𝐶𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸
193171, 184, 1923eqtr4g 2819 1 (𝜑 → ∏𝑗𝐴𝑘𝐵 𝐸 = ∏𝑘𝐶𝑗𝐷 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wex 1853  wcel 2139  wral 3050  wrex 3051  csb 3674  wss 3715  {csn 4321  cop 4327   ciun 4672   × cxp 5264  ccnv 5265  Rel wrel 5271  cfv 6049  1st c1st 7332  2nd c2nd 7333  Fincfn 8123  cc 10146  cprod 14854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-inf2 8713  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225  ax-pre-sup 10226
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-sup 8515  df-oi 8582  df-card 8975  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-div 10897  df-nn 11233  df-2 11291  df-3 11292  df-n0 11505  df-z 11590  df-uz 11900  df-rp 12046  df-fz 12540  df-fzo 12680  df-seq 13016  df-exp 13075  df-hash 13332  df-cj 14058  df-re 14059  df-im 14060  df-sqrt 14194  df-abs 14195  df-clim 14438  df-prod 14855
This theorem is referenced by: (None)
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