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Theorem fprodcom2fi 11653
Description: Interchange order of multiplication. Note that 𝐵(𝑗) and 𝐷(𝑘) are not necessarily constant expressions. (Contributed by Scott Fenton, 1-Feb-2018.) (Proof shortened by JJ, 2-Aug-2021.)
Hypotheses
Ref Expression
fprodcom2.1 (𝜑𝐴 ∈ Fin)
fprodcom2.2 (𝜑𝐶 ∈ Fin)
fprodcom2.3 ((𝜑𝑗𝐴) → 𝐵 ∈ Fin)
fprodcom2fi.d ((𝜑𝑘𝐶) → 𝐷 ∈ Fin)
fprodcom2.4 (𝜑 → ((𝑗𝐴𝑘𝐵) ↔ (𝑘𝐶𝑗𝐷)))
fprodcom2.5 ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐸 ∈ ℂ)
Assertion
Ref Expression
fprodcom2fi (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 fprodcom2fi
Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relxp 4750 . . . . . . . . 9 Rel ({𝑗} × 𝐵)
21rgenw 2545 . . . . . . . 8 𝑗𝐴 Rel ({𝑗} × 𝐵)
3 reliun 4762 . . . . . . . 8 (Rel 𝑗𝐴 ({𝑗} × 𝐵) ↔ ∀𝑗𝐴 Rel ({𝑗} × 𝐵))
42, 3mpbir 146 . . . . . . 7 Rel 𝑗𝐴 ({𝑗} × 𝐵)
5 relcnv 5021 . . . . . . 7 Rel 𝑘𝐶 ({𝑘} × 𝐷)
6 ancom 266 . . . . . . . . . . . 12 ((𝑥 = 𝑗𝑦 = 𝑘) ↔ (𝑦 = 𝑘𝑥 = 𝑗))
7 vex 2755 . . . . . . . . . . . . 13 𝑥 ∈ V
8 vex 2755 . . . . . . . . . . . . 13 𝑦 ∈ V
97, 8opth 4252 . . . . . . . . . . . 12 (⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ↔ (𝑥 = 𝑗𝑦 = 𝑘))
108, 7opth 4252 . . . . . . . . . . . 12 (⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ↔ (𝑦 = 𝑘𝑥 = 𝑗))
116, 9, 103bitr4i 212 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ↔ ⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩)
1211a1i 9 . . . . . . . . . 10 (𝜑 → (⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ↔ ⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩))
13 fprodcom2.4 . . . . . . . . . 10 (𝜑 → ((𝑗𝐴𝑘𝐵) ↔ (𝑘𝐶𝑗𝐷)))
1412, 13anbi12d 473 . . . . . . . . 9 (𝜑 → ((⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗𝐴𝑘𝐵)) ↔ (⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘𝐶𝑗𝐷))))
15142exbidv 1879 . . . . . . . 8 (𝜑 → (∃𝑗𝑘(⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗𝐴𝑘𝐵)) ↔ ∃𝑗𝑘(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘𝐶𝑗𝐷))))
16 eliunxp 4781 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗𝑘(⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗𝐴𝑘𝐵)))
177, 8opelcnv 4824 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷) ↔ ⟨𝑦, 𝑥⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷))
18 eliunxp 4781 . . . . . . . . 9 (⟨𝑦, 𝑥⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷) ↔ ∃𝑘𝑗(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘𝐶𝑗𝐷)))
19 excom 1675 . . . . . . . . 9 (∃𝑘𝑗(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘𝐶𝑗𝐷)) ↔ ∃𝑗𝑘(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘𝐶𝑗𝐷)))
2017, 18, 193bitri 206 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷) ↔ ∃𝑗𝑘(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘𝐶𝑗𝐷)))
2115, 16, 203bitr4g 223 . . . . . . 7 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷)))
224, 5, 21eqrelrdv 4737 . . . . . 6 (𝜑 𝑗𝐴 ({𝑗} × 𝐵) = 𝑘𝐶 ({𝑘} × 𝐷))
23 nfcv 2332 . . . . . . 7 𝑥({𝑗} × 𝐵)
24 nfcv 2332 . . . . . . . 8 𝑗{𝑥}
25 nfcsb1v 3105 . . . . . . . 8 𝑗𝑥 / 𝑗𝐵
2624, 25nfxp 4668 . . . . . . 7 𝑗({𝑥} × 𝑥 / 𝑗𝐵)
27 sneq 3618 . . . . . . . 8 (𝑗 = 𝑥 → {𝑗} = {𝑥})
28 csbeq1a 3081 . . . . . . . 8 (𝑗 = 𝑥𝐵 = 𝑥 / 𝑗𝐵)
2927, 28xpeq12d 4666 . . . . . . 7 (𝑗 = 𝑥 → ({𝑗} × 𝐵) = ({𝑥} × 𝑥 / 𝑗𝐵))
3023, 26, 29cbviun 3938 . . . . . 6 𝑗𝐴 ({𝑗} × 𝐵) = 𝑥𝐴 ({𝑥} × 𝑥 / 𝑗𝐵)
31 nfcv 2332 . . . . . . . 8 𝑦({𝑘} × 𝐷)
32 nfcv 2332 . . . . . . . . 9 𝑘{𝑦}
33 nfcsb1v 3105 . . . . . . . . 9 𝑘𝑦 / 𝑘𝐷
3432, 33nfxp 4668 . . . . . . . 8 𝑘({𝑦} × 𝑦 / 𝑘𝐷)
35 sneq 3618 . . . . . . . . 9 (𝑘 = 𝑦 → {𝑘} = {𝑦})
36 csbeq1a 3081 . . . . . . . . 9 (𝑘 = 𝑦𝐷 = 𝑦 / 𝑘𝐷)
3735, 36xpeq12d 4666 . . . . . . . 8 (𝑘 = 𝑦 → ({𝑘} × 𝐷) = ({𝑦} × 𝑦 / 𝑘𝐷))
3831, 34, 37cbviun 3938 . . . . . . 7 𝑘𝐶 ({𝑘} × 𝐷) = 𝑦𝐶 ({𝑦} × 𝑦 / 𝑘𝐷)
3938cnveqi 4817 . . . . . 6 𝑘𝐶 ({𝑘} × 𝐷) = 𝑦𝐶 ({𝑦} × 𝑦 / 𝑘𝐷)
4022, 30, 393eqtr3g 2245 . . . . 5 (𝜑 𝑥𝐴 ({𝑥} × 𝑥 / 𝑗𝐵) = 𝑦𝐶 ({𝑦} × 𝑦 / 𝑘𝐷))
4140prodeq1d 11591 . . . 4 (𝜑 → ∏𝑧 𝑥𝐴 ({𝑥} × 𝑥 / 𝑗𝐵)(2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸 = ∏𝑧 𝑦𝐶 ({𝑦} × 𝑦 / 𝑘𝐷)(2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸)
428, 7op1std 6167 . . . . . . 7 (𝑤 = ⟨𝑦, 𝑥⟩ → (1st𝑤) = 𝑦)
4342csbeq1d 3079 . . . . . 6 (𝑤 = ⟨𝑦, 𝑥⟩ → (1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸 = 𝑦 / 𝑘(2nd𝑤) / 𝑗𝐸)
448, 7op2ndd 6168 . . . . . . . 8 (𝑤 = ⟨𝑦, 𝑥⟩ → (2nd𝑤) = 𝑥)
4544csbeq1d 3079 . . . . . . 7 (𝑤 = ⟨𝑦, 𝑥⟩ → (2nd𝑤) / 𝑗𝐸 = 𝑥 / 𝑗𝐸)
4645csbeq2dv 3098 . . . . . 6 (𝑤 = ⟨𝑦, 𝑥⟩ → 𝑦 / 𝑘(2nd𝑤) / 𝑗𝐸 = 𝑦 / 𝑘𝑥 / 𝑗𝐸)
4743, 46eqtrd 2222 . . . . 5 (𝑤 = ⟨𝑦, 𝑥⟩ → (1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸 = 𝑦 / 𝑘𝑥 / 𝑗𝐸)
487, 8op2ndd 6168 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) = 𝑦)
4948csbeq1d 3079 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸 = 𝑦 / 𝑘(1st𝑧) / 𝑗𝐸)
507, 8op1std 6167 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → (1st𝑧) = 𝑥)
5150csbeq1d 3079 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (1st𝑧) / 𝑗𝐸 = 𝑥 / 𝑗𝐸)
5251csbeq2dv 3098 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑦 / 𝑘(1st𝑧) / 𝑗𝐸 = 𝑦 / 𝑘𝑥 / 𝑗𝐸)
5349, 52eqtrd 2222 . . . . 5 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸 = 𝑦 / 𝑘𝑥 / 𝑗𝐸)
54 fprodcom2.2 . . . . . 6 (𝜑𝐶 ∈ Fin)
55 snfig 6832 . . . . . . . . 9 (𝑦 ∈ V → {𝑦} ∈ Fin)
5655elv 2756 . . . . . . . 8 {𝑦} ∈ Fin
57 fprodcom2fi.d . . . . . . . . . 10 ((𝜑𝑘𝐶) → 𝐷 ∈ Fin)
5857ralrimiva 2563 . . . . . . . . 9 (𝜑 → ∀𝑘𝐶 𝐷 ∈ Fin)
5933nfel1 2343 . . . . . . . . . 10 𝑘𝑦 / 𝑘𝐷 ∈ Fin
6036eleq1d 2258 . . . . . . . . . 10 (𝑘 = 𝑦 → (𝐷 ∈ Fin ↔ 𝑦 / 𝑘𝐷 ∈ Fin))
6159, 60rspc 2850 . . . . . . . . 9 (𝑦𝐶 → (∀𝑘𝐶 𝐷 ∈ Fin → 𝑦 / 𝑘𝐷 ∈ Fin))
6258, 61mpan9 281 . . . . . . . 8 ((𝜑𝑦𝐶) → 𝑦 / 𝑘𝐷 ∈ Fin)
63 xpfi 6947 . . . . . . . 8 (({𝑦} ∈ Fin ∧ 𝑦 / 𝑘𝐷 ∈ Fin) → ({𝑦} × 𝑦 / 𝑘𝐷) ∈ Fin)
6456, 62, 63sylancr 414 . . . . . . 7 ((𝜑𝑦𝐶) → ({𝑦} × 𝑦 / 𝑘𝐷) ∈ Fin)
6564ralrimiva 2563 . . . . . 6 (𝜑 → ∀𝑦𝐶 ({𝑦} × 𝑦 / 𝑘𝐷) ∈ Fin)
66 disjsnxp 6256 . . . . . . 7 Disj 𝑦𝐶 ({𝑦} × 𝑦 / 𝑘𝐷)
6766a1i 9 . . . . . 6 (𝜑Disj 𝑦𝐶 ({𝑦} × 𝑦 / 𝑘𝐷))
68 iunfidisj 6964 . . . . . 6 ((𝐶 ∈ Fin ∧ ∀𝑦𝐶 ({𝑦} × 𝑦 / 𝑘𝐷) ∈ Fin ∧ Disj 𝑦𝐶 ({𝑦} × 𝑦 / 𝑘𝐷)) → 𝑦𝐶 ({𝑦} × 𝑦 / 𝑘𝐷) ∈ Fin)
6954, 65, 67, 68syl3anc 1249 . . . . 5 (𝜑 𝑦𝐶 ({𝑦} × 𝑦 / 𝑘𝐷) ∈ Fin)
70 reliun 4762 . . . . . . 7 (Rel 𝑦𝐶 ({𝑦} × 𝑦 / 𝑘𝐷) ↔ ∀𝑦𝐶 Rel ({𝑦} × 𝑦 / 𝑘𝐷))
71 relxp 4750 . . . . . . . 8 Rel ({𝑦} × 𝑦 / 𝑘𝐷)
7271a1i 9 . . . . . . 7 (𝑦𝐶 → Rel ({𝑦} × 𝑦 / 𝑘𝐷))
7370, 72mprgbir 2548 . . . . . 6 Rel 𝑦𝐶 ({𝑦} × 𝑦 / 𝑘𝐷)
7473a1i 9 . . . . 5 (𝜑 → Rel 𝑦𝐶 ({𝑦} × 𝑦 / 𝑘𝐷))
75 csbeq1 3075 . . . . . . . 8 (𝑥 = (2nd𝑤) → 𝑥 / 𝑗𝐸 = (2nd𝑤) / 𝑗𝐸)
7675csbeq2dv 3098 . . . . . . 7 (𝑥 = (2nd𝑤) → (1st𝑤) / 𝑘𝑥 / 𝑗𝐸 = (1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸)
7776eleq1d 2258 . . . . . 6 (𝑥 = (2nd𝑤) → ((1st𝑤) / 𝑘𝑥 / 𝑗𝐸 ∈ ℂ ↔ (1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸 ∈ ℂ))
78 csbeq1 3075 . . . . . . . 8 (𝑦 = (1st𝑤) → 𝑦 / 𝑘𝐷 = (1st𝑤) / 𝑘𝐷)
79 csbeq1 3075 . . . . . . . . 9 (𝑦 = (1st𝑤) → 𝑦 / 𝑘𝑥 / 𝑗𝐸 = (1st𝑤) / 𝑘𝑥 / 𝑗𝐸)
8079eleq1d 2258 . . . . . . . 8 (𝑦 = (1st𝑤) → (𝑦 / 𝑘𝑥 / 𝑗𝐸 ∈ ℂ ↔ (1st𝑤) / 𝑘𝑥 / 𝑗𝐸 ∈ ℂ))
8178, 80raleqbidv 2698 . . . . . . 7 (𝑦 = (1st𝑤) → (∀𝑥 𝑦 / 𝑘𝐷𝑦 / 𝑘𝑥 / 𝑗𝐸 ∈ ℂ ↔ ∀𝑥 (1st𝑤) / 𝑘𝐷(1st𝑤) / 𝑘𝑥 / 𝑗𝐸 ∈ ℂ))
82 simpl 109 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐶𝑥𝑦 / 𝑘𝐷)) → 𝜑)
8333, 36opeliunxp2f 6257 . . . . . . . . . . . . . . 15 (⟨𝑦, 𝑥⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷) ↔ (𝑦𝐶𝑥𝑦 / 𝑘𝐷))
8417, 83sylbbr 136 . . . . . . . . . . . . . 14 ((𝑦𝐶𝑥𝑦 / 𝑘𝐷) → ⟨𝑥, 𝑦⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷))
8584adantl 277 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑦𝐶𝑥𝑦 / 𝑘𝐷)) → ⟨𝑥, 𝑦⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷))
8622adantr 276 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑦𝐶𝑥𝑦 / 𝑘𝐷)) → 𝑗𝐴 ({𝑗} × 𝐵) = 𝑘𝐶 ({𝑘} × 𝐷))
8785, 86eleqtrrd 2269 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑦𝐶𝑥𝑦 / 𝑘𝐷)) → ⟨𝑥, 𝑦⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐵))
88 eliun 3905 . . . . . . . . . . . 12 (⟨𝑥, 𝑦⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗𝐴𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵))
8987, 88sylib 122 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦𝐶𝑥𝑦 / 𝑘𝐷)) → ∃𝑗𝐴𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵))
90 simpr 110 . . . . . . . . . . . . . . . 16 ((𝑗𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵)) → ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵))
91 opelxp 4671 . . . . . . . . . . . . . . . 16 (⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵) ↔ (𝑥 ∈ {𝑗} ∧ 𝑦𝐵))
9290, 91sylib 122 . . . . . . . . . . . . . . 15 ((𝑗𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵)) → (𝑥 ∈ {𝑗} ∧ 𝑦𝐵))
9392simpld 112 . . . . . . . . . . . . . 14 ((𝑗𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵)) → 𝑥 ∈ {𝑗})
94 elsni 3625 . . . . . . . . . . . . . 14 (𝑥 ∈ {𝑗} → 𝑥 = 𝑗)
9593, 94syl 14 . . . . . . . . . . . . 13 ((𝑗𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵)) → 𝑥 = 𝑗)
96 simpl 109 . . . . . . . . . . . . 13 ((𝑗𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵)) → 𝑗𝐴)
9795, 96eqeltrd 2266 . . . . . . . . . . . 12 ((𝑗𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵)) → 𝑥𝐴)
9897rexlimiva 2602 . . . . . . . . . . 11 (∃𝑗𝐴𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵) → 𝑥𝐴)
9989, 98syl 14 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐶𝑥𝑦 / 𝑘𝐷)) → 𝑥𝐴)
10025nfcri 2326 . . . . . . . . . . . 12 𝑗 𝑦𝑥 / 𝑗𝐵
10194equcomd 1718 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ {𝑗} → 𝑗 = 𝑥)
102101, 28syl 14 . . . . . . . . . . . . . . . 16 (𝑥 ∈ {𝑗} → 𝐵 = 𝑥 / 𝑗𝐵)
103102eleq2d 2259 . . . . . . . . . . . . . . 15 (𝑥 ∈ {𝑗} → (𝑦𝐵𝑦𝑥 / 𝑗𝐵))
104103biimpa 296 . . . . . . . . . . . . . 14 ((𝑥 ∈ {𝑗} ∧ 𝑦𝐵) → 𝑦𝑥 / 𝑗𝐵)
10591, 104sylbi 121 . . . . . . . . . . . . 13 (⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵) → 𝑦𝑥 / 𝑗𝐵)
106105a1i 9 . . . . . . . . . . . 12 (𝑗𝐴 → (⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵) → 𝑦𝑥 / 𝑗𝐵))
107100, 106rexlimi 2600 . . . . . . . . . . 11 (∃𝑗𝐴𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵) → 𝑦𝑥 / 𝑗𝐵)
10889, 107syl 14 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐶𝑥𝑦 / 𝑘𝐷)) → 𝑦𝑥 / 𝑗𝐵)
109 fprodcom2.5 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐸 ∈ ℂ)
110109ralrimivva 2572 . . . . . . . . . . . . 13 (𝜑 → ∀𝑗𝐴𝑘𝐵 𝐸 ∈ ℂ)
111 nfcsb1v 3105 . . . . . . . . . . . . . . . 16 𝑗𝑥 / 𝑗𝐸
112111nfel1 2343 . . . . . . . . . . . . . . 15 𝑗𝑥 / 𝑗𝐸 ∈ ℂ
11325, 112nfralw 2527 . . . . . . . . . . . . . 14 𝑗𝑘 𝑥 / 𝑗𝐵𝑥 / 𝑗𝐸 ∈ ℂ
114 csbeq1a 3081 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑥𝐸 = 𝑥 / 𝑗𝐸)
115114eleq1d 2258 . . . . . . . . . . . . . . 15 (𝑗 = 𝑥 → (𝐸 ∈ ℂ ↔ 𝑥 / 𝑗𝐸 ∈ ℂ))
11628, 115raleqbidv 2698 . . . . . . . . . . . . . 14 (𝑗 = 𝑥 → (∀𝑘𝐵 𝐸 ∈ ℂ ↔ ∀𝑘 𝑥 / 𝑗𝐵𝑥 / 𝑗𝐸 ∈ ℂ))
117113, 116rspc 2850 . . . . . . . . . . . . 13 (𝑥𝐴 → (∀𝑗𝐴𝑘𝐵 𝐸 ∈ ℂ → ∀𝑘 𝑥 / 𝑗𝐵𝑥 / 𝑗𝐸 ∈ ℂ))
118110, 117mpan9 281 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ∀𝑘 𝑥 / 𝑗𝐵𝑥 / 𝑗𝐸 ∈ ℂ)
119 nfcsb1v 3105 . . . . . . . . . . . . . 14 𝑘𝑦 / 𝑘𝑥 / 𝑗𝐸
120119nfel1 2343 . . . . . . . . . . . . 13 𝑘𝑦 / 𝑘𝑥 / 𝑗𝐸 ∈ ℂ
121 csbeq1a 3081 . . . . . . . . . . . . . 14 (𝑘 = 𝑦𝑥 / 𝑗𝐸 = 𝑦 / 𝑘𝑥 / 𝑗𝐸)
122121eleq1d 2258 . . . . . . . . . . . . 13 (𝑘 = 𝑦 → (𝑥 / 𝑗𝐸 ∈ ℂ ↔ 𝑦 / 𝑘𝑥 / 𝑗𝐸 ∈ ℂ))
123120, 122rspc 2850 . . . . . . . . . . . 12 (𝑦𝑥 / 𝑗𝐵 → (∀𝑘 𝑥 / 𝑗𝐵𝑥 / 𝑗𝐸 ∈ ℂ → 𝑦 / 𝑘𝑥 / 𝑗𝐸 ∈ ℂ))
124118, 123syl5com 29 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝑦𝑥 / 𝑗𝐵𝑦 / 𝑘𝑥 / 𝑗𝐸 ∈ ℂ))
125124impr 379 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝑥 / 𝑗𝐵)) → 𝑦 / 𝑘𝑥 / 𝑗𝐸 ∈ ℂ)
12682, 99, 108, 125syl12anc 1247 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐶𝑥𝑦 / 𝑘𝐷)) → 𝑦 / 𝑘𝑥 / 𝑗𝐸 ∈ ℂ)
127126ralrimivva 2572 . . . . . . . 8 (𝜑 → ∀𝑦𝐶𝑥 𝑦 / 𝑘𝐷𝑦 / 𝑘𝑥 / 𝑗𝐸 ∈ ℂ)
128127adantr 276 . . . . . . 7 ((𝜑𝑤 𝑦𝐶 ({𝑦} × 𝑦 / 𝑘𝐷)) → ∀𝑦𝐶𝑥 𝑦 / 𝑘𝐷𝑦 / 𝑘𝑥 / 𝑗𝐸 ∈ ℂ)
129 simpr 110 . . . . . . . . 9 ((𝜑𝑤 𝑦𝐶 ({𝑦} × 𝑦 / 𝑘𝐷)) → 𝑤 𝑦𝐶 ({𝑦} × 𝑦 / 𝑘𝐷))
130 eliun 3905 . . . . . . . . 9 (𝑤 𝑦𝐶 ({𝑦} × 𝑦 / 𝑘𝐷) ↔ ∃𝑦𝐶 𝑤 ∈ ({𝑦} × 𝑦 / 𝑘𝐷))
131129, 130sylib 122 . . . . . . . 8 ((𝜑𝑤 𝑦𝐶 ({𝑦} × 𝑦 / 𝑘𝐷)) → ∃𝑦𝐶 𝑤 ∈ ({𝑦} × 𝑦 / 𝑘𝐷))
132 xp1st 6184 . . . . . . . . . . . 12 (𝑤 ∈ ({𝑦} × 𝑦 / 𝑘𝐷) → (1st𝑤) ∈ {𝑦})
133132adantl 277 . . . . . . . . . . 11 ((𝑦𝐶𝑤 ∈ ({𝑦} × 𝑦 / 𝑘𝐷)) → (1st𝑤) ∈ {𝑦})
134 elsni 3625 . . . . . . . . . . 11 ((1st𝑤) ∈ {𝑦} → (1st𝑤) = 𝑦)
135133, 134syl 14 . . . . . . . . . 10 ((𝑦𝐶𝑤 ∈ ({𝑦} × 𝑦 / 𝑘𝐷)) → (1st𝑤) = 𝑦)
136 simpl 109 . . . . . . . . . 10 ((𝑦𝐶𝑤 ∈ ({𝑦} × 𝑦 / 𝑘𝐷)) → 𝑦𝐶)
137135, 136eqeltrd 2266 . . . . . . . . 9 ((𝑦𝐶𝑤 ∈ ({𝑦} × 𝑦 / 𝑘𝐷)) → (1st𝑤) ∈ 𝐶)
138137rexlimiva 2602 . . . . . . . 8 (∃𝑦𝐶 𝑤 ∈ ({𝑦} × 𝑦 / 𝑘𝐷) → (1st𝑤) ∈ 𝐶)
139131, 138syl 14 . . . . . . 7 ((𝜑𝑤 𝑦𝐶 ({𝑦} × 𝑦 / 𝑘𝐷)) → (1st𝑤) ∈ 𝐶)
14081, 128, 139rspcdva 2861 . . . . . 6 ((𝜑𝑤 𝑦𝐶 ({𝑦} × 𝑦 / 𝑘𝐷)) → ∀𝑥 (1st𝑤) / 𝑘𝐷(1st𝑤) / 𝑘𝑥 / 𝑗𝐸 ∈ ℂ)
141 xp2nd 6185 . . . . . . . . . 10 (𝑤 ∈ ({𝑦} × 𝑦 / 𝑘𝐷) → (2nd𝑤) ∈ 𝑦 / 𝑘𝐷)
142141adantl 277 . . . . . . . . 9 ((𝑦𝐶𝑤 ∈ ({𝑦} × 𝑦 / 𝑘𝐷)) → (2nd𝑤) ∈ 𝑦 / 𝑘𝐷)
143135csbeq1d 3079 . . . . . . . . 9 ((𝑦𝐶𝑤 ∈ ({𝑦} × 𝑦 / 𝑘𝐷)) → (1st𝑤) / 𝑘𝐷 = 𝑦 / 𝑘𝐷)
144142, 143eleqtrrd 2269 . . . . . . . 8 ((𝑦𝐶𝑤 ∈ ({𝑦} × 𝑦 / 𝑘𝐷)) → (2nd𝑤) ∈ (1st𝑤) / 𝑘𝐷)
145144rexlimiva 2602 . . . . . . 7 (∃𝑦𝐶 𝑤 ∈ ({𝑦} × 𝑦 / 𝑘𝐷) → (2nd𝑤) ∈ (1st𝑤) / 𝑘𝐷)
146131, 145syl 14 . . . . . 6 ((𝜑𝑤 𝑦𝐶 ({𝑦} × 𝑦 / 𝑘𝐷)) → (2nd𝑤) ∈ (1st𝑤) / 𝑘𝐷)
14777, 140, 146rspcdva 2861 . . . . 5 ((𝜑𝑤 𝑦𝐶 ({𝑦} × 𝑦 / 𝑘𝐷)) → (1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸 ∈ ℂ)
14847, 53, 69, 74, 147fprodcnv 11652 . . . 4 (𝜑 → ∏𝑤 𝑦𝐶 ({𝑦} × 𝑦 / 𝑘𝐷)(1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸 = ∏𝑧 𝑦𝐶 ({𝑦} × 𝑦 / 𝑘𝐷)(2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸)
14941, 148eqtr4d 2225 . . 3 (𝜑 → ∏𝑧 𝑥𝐴 ({𝑥} × 𝑥 / 𝑗𝐵)(2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸 = ∏𝑤 𝑦𝐶 ({𝑦} × 𝑦 / 𝑘𝐷)(1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸)
150 fprodcom2.1 . . . 4 (𝜑𝐴 ∈ Fin)
151 fprodcom2.3 . . . . . 6 ((𝜑𝑗𝐴) → 𝐵 ∈ Fin)
152151ralrimiva 2563 . . . . 5 (𝜑 → ∀𝑗𝐴 𝐵 ∈ Fin)
15325nfel1 2343 . . . . . 6 𝑗𝑥 / 𝑗𝐵 ∈ Fin
15428eleq1d 2258 . . . . . 6 (𝑗 = 𝑥 → (𝐵 ∈ Fin ↔ 𝑥 / 𝑗𝐵 ∈ Fin))
155153, 154rspc 2850 . . . . 5 (𝑥𝐴 → (∀𝑗𝐴 𝐵 ∈ Fin → 𝑥 / 𝑗𝐵 ∈ Fin))
156152, 155mpan9 281 . . . 4 ((𝜑𝑥𝐴) → 𝑥 / 𝑗𝐵 ∈ Fin)
15753, 150, 156, 125fprod2d 11650 . . 3 (𝜑 → ∏𝑥𝐴𝑦 𝑥 / 𝑗𝐵𝑦 / 𝑘𝑥 / 𝑗𝐸 = ∏𝑧 𝑥𝐴 ({𝑥} × 𝑥 / 𝑗𝐵)(2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸)
15847, 54, 62, 126fprod2d 11650 . . 3 (𝜑 → ∏𝑦𝐶𝑥 𝑦 / 𝑘𝐷𝑦 / 𝑘𝑥 / 𝑗𝐸 = ∏𝑤 𝑦𝐶 ({𝑦} × 𝑦 / 𝑘𝐷)(1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸)
159149, 157, 1583eqtr4d 2232 . 2 (𝜑 → ∏𝑥𝐴𝑦 𝑥 / 𝑗𝐵𝑦 / 𝑘𝑥 / 𝑗𝐸 = ∏𝑦𝐶𝑥 𝑦 / 𝑘𝐷𝑦 / 𝑘𝑥 / 𝑗𝐸)
160 nfcv 2332 . . 3 𝑥𝑘𝐵 𝐸
161 nfcv 2332 . . . . 5 𝑗𝑦
162161, 111nfcsbw 3108 . . . 4 𝑗𝑦 / 𝑘𝑥 / 𝑗𝐸
16325, 162nfcprod 11582 . . 3 𝑗𝑦 𝑥 / 𝑗𝐵𝑦 / 𝑘𝑥 / 𝑗𝐸
164 nfcv 2332 . . . . 5 𝑦𝐸
165 nfcsb1v 3105 . . . . 5 𝑘𝑦 / 𝑘𝐸
166 csbeq1a 3081 . . . . 5 (𝑘 = 𝑦𝐸 = 𝑦 / 𝑘𝐸)
167164, 165, 166cbvprodi 11587 . . . 4 𝑘𝐵 𝐸 = ∏𝑦𝐵 𝑦 / 𝑘𝐸
168114csbeq2dv 3098 . . . . . 6 (𝑗 = 𝑥𝑦 / 𝑘𝐸 = 𝑦 / 𝑘𝑥 / 𝑗𝐸)
169168adantr 276 . . . . 5 ((𝑗 = 𝑥𝑦𝐵) → 𝑦 / 𝑘𝐸 = 𝑦 / 𝑘𝑥 / 𝑗𝐸)
17028, 169prodeq12dv 11596 . . . 4 (𝑗 = 𝑥 → ∏𝑦𝐵 𝑦 / 𝑘𝐸 = ∏𝑦 𝑥 / 𝑗𝐵𝑦 / 𝑘𝑥 / 𝑗𝐸)
171167, 170eqtrid 2234 . . 3 (𝑗 = 𝑥 → ∏𝑘𝐵 𝐸 = ∏𝑦 𝑥 / 𝑗𝐵𝑦 / 𝑘𝑥 / 𝑗𝐸)
172160, 163, 171cbvprodi 11587 . 2 𝑗𝐴𝑘𝐵 𝐸 = ∏𝑥𝐴𝑦 𝑥 / 𝑗𝐵𝑦 / 𝑘𝑥 / 𝑗𝐸
173 nfcv 2332 . . 3 𝑦𝑗𝐷 𝐸
17433, 119nfcprod 11582 . . 3 𝑘𝑥 𝑦 / 𝑘𝐷𝑦 / 𝑘𝑥 / 𝑗𝐸
175 nfcv 2332 . . . . 5 𝑥𝐸
176175, 111, 114cbvprodi 11587 . . . 4 𝑗𝐷 𝐸 = ∏𝑥𝐷 𝑥 / 𝑗𝐸
177121adantr 276 . . . . 5 ((𝑘 = 𝑦𝑥𝐷) → 𝑥 / 𝑗𝐸 = 𝑦 / 𝑘𝑥 / 𝑗𝐸)
17836, 177prodeq12dv 11596 . . . 4 (𝑘 = 𝑦 → ∏𝑥𝐷 𝑥 / 𝑗𝐸 = ∏𝑥 𝑦 / 𝑘𝐷𝑦 / 𝑘𝑥 / 𝑗𝐸)
179176, 178eqtrid 2234 . . 3 (𝑘 = 𝑦 → ∏𝑗𝐷 𝐸 = ∏𝑥 𝑦 / 𝑘𝐷𝑦 / 𝑘𝑥 / 𝑗𝐸)
180173, 174, 179cbvprodi 11587 . 2 𝑘𝐶𝑗𝐷 𝐸 = ∏𝑦𝐶𝑥 𝑦 / 𝑘𝐷𝑦 / 𝑘𝑥 / 𝑗𝐸
181159, 172, 1803eqtr4g 2247 1 (𝜑 → ∏𝑗𝐴𝑘𝐵 𝐸 = ∏𝑘𝐶𝑗𝐷 𝐸)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wex 1503  wcel 2160  wral 2468  wrex 2469  Vcvv 2752  csb 3072  {csn 3607  cop 3610   ciun 3901  Disj wdisj 3995   × cxp 4639  ccnv 4640  Rel wrel 4646  cfv 5231  1st c1st 6157  2nd c2nd 6158  Fincfn 6758  cc 7828  cprod 11577
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7921  ax-resscn 7922  ax-1cn 7923  ax-1re 7924  ax-icn 7925  ax-addcl 7926  ax-addrcl 7927  ax-mulcl 7928  ax-mulrcl 7929  ax-addcom 7930  ax-mulcom 7931  ax-addass 7932  ax-mulass 7933  ax-distr 7934  ax-i2m1 7935  ax-0lt1 7936  ax-1rid 7937  ax-0id 7938  ax-rnegex 7939  ax-precex 7940  ax-cnre 7941  ax-pre-ltirr 7942  ax-pre-ltwlin 7943  ax-pre-lttrn 7944  ax-pre-apti 7945  ax-pre-ltadd 7946  ax-pre-mulgt0 7947  ax-pre-mulext 7948  ax-arch 7949  ax-caucvg 7950
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-disj 3996  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-isom 5240  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-irdg 6389  df-frec 6410  df-1o 6435  df-oadd 6439  df-er 6553  df-en 6759  df-dom 6760  df-fin 6761  df-pnf 8013  df-mnf 8014  df-xr 8015  df-ltxr 8016  df-le 8017  df-sub 8149  df-neg 8150  df-reap 8551  df-ap 8558  df-div 8649  df-inn 8939  df-2 8997  df-3 8998  df-4 8999  df-n0 9196  df-z 9273  df-uz 9548  df-q 9639  df-rp 9673  df-fz 10028  df-fzo 10162  df-seqfrec 10465  df-exp 10539  df-ihash 10775  df-cj 10870  df-re 10871  df-im 10872  df-rsqrt 11026  df-abs 11027  df-clim 11306  df-proddc 11578
This theorem is referenced by:  fprodcom  11654  fprod0diagfz  11655
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