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Theorem prodmo 14505
 Description: A product has at most one limit. (Contributed by Scott Fenton, 4-Dec-2017.)
Hypotheses
Ref Expression
prodmo.1 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))
prodmo.2 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
prodmo.3 𝐺 = (𝑗 ∈ ℕ ↦ (𝑓𝑗) / 𝑘𝐵)
Assertion
Ref Expression
prodmo (𝜑 → ∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚))))
Distinct variable groups:   𝐴,𝑘,𝑛   𝑘,𝐹,𝑛   𝜑,𝑘,𝑛   𝐴,𝑓,𝑗,𝑚,𝑥   𝐵,𝑓,𝑗,𝑚   𝑓,𝐹,𝑗,𝑘,𝑚   𝜑,𝑓,𝑥   𝑥,𝐹   𝑗,𝐺,𝑥   𝑗,𝑘,𝑚,𝜑,𝑥   𝑥,𝑛,𝜑   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)   𝐵(𝑥,𝑦,𝑘,𝑛)   𝐹(𝑦)   𝐺(𝑦,𝑓,𝑘,𝑚,𝑛)

Proof of Theorem prodmo
Dummy variables 𝑎 𝑔 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 3simpb 1052 . . . . . . 7 ((𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) → (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥))
21reximi 2994 . . . . . 6 (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) → ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥))
3 3simpb 1052 . . . . . . 7 ((𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧) → (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧))
43reximi 2994 . . . . . 6 (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧) → ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧))
5 fveq2 6103 . . . . . . . . . . . 12 (𝑚 = 𝑤 → (ℤ𝑚) = (ℤ𝑤))
65sseq2d 3596 . . . . . . . . . . 11 (𝑚 = 𝑤 → (𝐴 ⊆ (ℤ𝑚) ↔ 𝐴 ⊆ (ℤ𝑤)))
7 seqeq1 12666 . . . . . . . . . . . 12 (𝑚 = 𝑤 → seq𝑚( · , 𝐹) = seq𝑤( · , 𝐹))
87breq1d 4593 . . . . . . . . . . 11 (𝑚 = 𝑤 → (seq𝑚( · , 𝐹) ⇝ 𝑧 ↔ seq𝑤( · , 𝐹) ⇝ 𝑧))
96, 8anbi12d 743 . . . . . . . . . 10 (𝑚 = 𝑤 → ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧) ↔ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))
109cbvrexv 3148 . . . . . . . . 9 (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧) ↔ ∃𝑤 ∈ ℤ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧))
1110anbi2i 726 . . . . . . . 8 ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧)) ↔ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ ∃𝑤 ∈ ℤ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))
12 reeanv 3086 . . . . . . . 8 (∃𝑚 ∈ ℤ ∃𝑤 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)) ↔ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ ∃𝑤 ∈ ℤ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))
1311, 12bitr4i 266 . . . . . . 7 ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧)) ↔ ∃𝑚 ∈ ℤ ∃𝑤 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))
14 simprlr 799 . . . . . . . . . . . . 13 (((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧))) → seq𝑚( · , 𝐹) ⇝ 𝑥)
1514adantl 481 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → seq𝑚( · , 𝐹) ⇝ 𝑥)
16 prodmo.1 . . . . . . . . . . . . 13 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))
17 prodmo.2 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
1817adantlr 747 . . . . . . . . . . . . 13 (((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) ∧ 𝑘𝐴) → 𝐵 ∈ ℂ)
19 simprll 798 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → 𝑚 ∈ ℤ)
20 simprlr 799 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → 𝑤 ∈ ℤ)
21 simprll 798 . . . . . . . . . . . . . 14 (((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧))) → 𝐴 ⊆ (ℤ𝑚))
2221adantl 481 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → 𝐴 ⊆ (ℤ𝑚))
23 simprrl 800 . . . . . . . . . . . . . 14 (((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧))) → 𝐴 ⊆ (ℤ𝑤))
2423adantl 481 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → 𝐴 ⊆ (ℤ𝑤))
2516, 18, 19, 20, 22, 24prodrb 14501 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → (seq𝑚( · , 𝐹) ⇝ 𝑥 ↔ seq𝑤( · , 𝐹) ⇝ 𝑥))
2615, 25mpbid 221 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → seq𝑤( · , 𝐹) ⇝ 𝑥)
27 simprrr 801 . . . . . . . . . . . 12 (((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧))) → seq𝑤( · , 𝐹) ⇝ 𝑧)
2827adantl 481 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → seq𝑤( · , 𝐹) ⇝ 𝑧)
29 climuni 14131 . . . . . . . . . . 11 ((seq𝑤( · , 𝐹) ⇝ 𝑥 ∧ seq𝑤( · , 𝐹) ⇝ 𝑧) → 𝑥 = 𝑧)
3026, 28, 29syl2anc 691 . . . . . . . . . 10 ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → 𝑥 = 𝑧)
3130expcom 450 . . . . . . . . 9 (((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧))) → (𝜑𝑥 = 𝑧))
3231ex 449 . . . . . . . 8 ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) → (((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)) → (𝜑𝑥 = 𝑧)))
3332rexlimivv 3018 . . . . . . 7 (∃𝑚 ∈ ℤ ∃𝑤 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)) → (𝜑𝑥 = 𝑧))
3413, 33sylbi 206 . . . . . 6 ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧)) → (𝜑𝑥 = 𝑧))
352, 4, 34syl2an 493 . . . . 5 ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧)) → (𝜑𝑥 = 𝑧))
36 prodmo.3 . . . . . . . . . 10 𝐺 = (𝑗 ∈ ℕ ↦ (𝑓𝑗) / 𝑘𝐵)
3716, 17, 36prodmolem2 14504 . . . . . . . . 9 ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚)) → 𝑧 = 𝑥))
38 equcomi 1931 . . . . . . . . 9 (𝑧 = 𝑥𝑥 = 𝑧)
3937, 38syl6 34 . . . . . . . 8 ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚)) → 𝑥 = 𝑧))
4039expimpd 627 . . . . . . 7 (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧) ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚))) → 𝑥 = 𝑧))
4140com12 32 . . . . . 6 ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧) ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚))) → (𝜑𝑥 = 𝑧))
4241ancoms 468 . . . . 5 ((∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚)) ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧)) → (𝜑𝑥 = 𝑧))
4316, 17, 36prodmolem2 14504 . . . . . . 7 ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚)) → 𝑥 = 𝑧))
4443expimpd 627 . . . . . 6 (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚))) → 𝑥 = 𝑧))
4544com12 32 . . . . 5 ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚))) → (𝜑𝑥 = 𝑧))
46 reeanv 3086 . . . . . . . 8 (∃𝑚 ∈ ℕ ∃𝑤 ∈ ℕ (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚)) ∧ ∃𝑔(𝑔:(1...𝑤)–1-1-onto𝐴𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))‘𝑤))) ↔ (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚)) ∧ ∃𝑤 ∈ ℕ ∃𝑔(𝑔:(1...𝑤)–1-1-onto𝐴𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))‘𝑤))))
47 eeanv 2170 . . . . . . . . 9 (∃𝑓𝑔((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚)) ∧ (𝑔:(1...𝑤)–1-1-onto𝐴𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))‘𝑤))) ↔ (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚)) ∧ ∃𝑔(𝑔:(1...𝑤)–1-1-onto𝐴𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))‘𝑤))))
48472rexbii 3024 . . . . . . . 8 (∃𝑚 ∈ ℕ ∃𝑤 ∈ ℕ ∃𝑓𝑔((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚)) ∧ (𝑔:(1...𝑤)–1-1-onto𝐴𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))‘𝑤))) ↔ ∃𝑚 ∈ ℕ ∃𝑤 ∈ ℕ (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚)) ∧ ∃𝑔(𝑔:(1...𝑤)–1-1-onto𝐴𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))‘𝑤))))
49 oveq2 6557 . . . . . . . . . . . . . 14 (𝑚 = 𝑤 → (1...𝑚) = (1...𝑤))
50 f1oeq2 6041 . . . . . . . . . . . . . 14 ((1...𝑚) = (1...𝑤) → (𝑓:(1...𝑚)–1-1-onto𝐴𝑓:(1...𝑤)–1-1-onto𝐴))
5149, 50syl 17 . . . . . . . . . . . . 13 (𝑚 = 𝑤 → (𝑓:(1...𝑚)–1-1-onto𝐴𝑓:(1...𝑤)–1-1-onto𝐴))
52 fveq2 6103 . . . . . . . . . . . . . 14 (𝑚 = 𝑤 → (seq1( · , 𝐺)‘𝑚) = (seq1( · , 𝐺)‘𝑤))
5352eqeq2d 2620 . . . . . . . . . . . . 13 (𝑚 = 𝑤 → (𝑧 = (seq1( · , 𝐺)‘𝑚) ↔ 𝑧 = (seq1( · , 𝐺)‘𝑤)))
5451, 53anbi12d 743 . . . . . . . . . . . 12 (𝑚 = 𝑤 → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚)) ↔ (𝑓:(1...𝑤)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑤))))
5554exbidv 1837 . . . . . . . . . . 11 (𝑚 = 𝑤 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑤)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑤))))
56 f1oeq1 6040 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → (𝑓:(1...𝑤)–1-1-onto𝐴𝑔:(1...𝑤)–1-1-onto𝐴))
57 fveq1 6102 . . . . . . . . . . . . . . . . . . 19 (𝑓 = 𝑔 → (𝑓𝑗) = (𝑔𝑗))
5857csbeq1d 3506 . . . . . . . . . . . . . . . . . 18 (𝑓 = 𝑔(𝑓𝑗) / 𝑘𝐵 = (𝑔𝑗) / 𝑘𝐵)
5958mpteq2dv 4673 . . . . . . . . . . . . . . . . 17 (𝑓 = 𝑔 → (𝑗 ∈ ℕ ↦ (𝑓𝑗) / 𝑘𝐵) = (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))
6036, 59syl5eq 2656 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑔𝐺 = (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))
6160seqeq3d 12671 . . . . . . . . . . . . . . 15 (𝑓 = 𝑔 → seq1( · , 𝐺) = seq1( · , (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵)))
6261fveq1d 6105 . . . . . . . . . . . . . 14 (𝑓 = 𝑔 → (seq1( · , 𝐺)‘𝑤) = (seq1( · , (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))‘𝑤))
6362eqeq2d 2620 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → (𝑧 = (seq1( · , 𝐺)‘𝑤) ↔ 𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))‘𝑤)))
6456, 63anbi12d 743 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ((𝑓:(1...𝑤)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑤)) ↔ (𝑔:(1...𝑤)–1-1-onto𝐴𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))‘𝑤))))
6564cbvexv 2263 . . . . . . . . . . 11 (∃𝑓(𝑓:(1...𝑤)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑤)) ↔ ∃𝑔(𝑔:(1...𝑤)–1-1-onto𝐴𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))‘𝑤)))
6655, 65syl6bb 275 . . . . . . . . . 10 (𝑚 = 𝑤 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚)) ↔ ∃𝑔(𝑔:(1...𝑤)–1-1-onto𝐴𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))‘𝑤))))
6766cbvrexv 3148 . . . . . . . . 9 (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚)) ↔ ∃𝑤 ∈ ℕ ∃𝑔(𝑔:(1...𝑤)–1-1-onto𝐴𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))‘𝑤)))
6867anbi2i 726 . . . . . . . 8 ((∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚)) ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚))) ↔ (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚)) ∧ ∃𝑤 ∈ ℕ ∃𝑔(𝑔:(1...𝑤)–1-1-onto𝐴𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))‘𝑤))))
6946, 48, 683bitr4i 291 . . . . . . 7 (∃𝑚 ∈ ℕ ∃𝑤 ∈ ℕ ∃𝑓𝑔((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚)) ∧ (𝑔:(1...𝑤)–1-1-onto𝐴𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))‘𝑤))) ↔ (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚)) ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚))))
70 an4 861 . . . . . . . . . 10 (((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚)) ∧ (𝑔:(1...𝑤)–1-1-onto𝐴𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))‘𝑤))) ↔ ((𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑤)–1-1-onto𝐴) ∧ (𝑥 = (seq1( · , 𝐺)‘𝑚) ∧ 𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))‘𝑤))))
71 simpll 786 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑤 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑤)–1-1-onto𝐴)) → 𝜑)
7271, 17sylan 487 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑤 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑤)–1-1-onto𝐴)) ∧ 𝑘𝐴) → 𝐵 ∈ ℂ)
73 fveq2 6103 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑎 → (𝑓𝑗) = (𝑓𝑎))
7473csbeq1d 3506 . . . . . . . . . . . . . . 15 (𝑗 = 𝑎(𝑓𝑗) / 𝑘𝐵 = (𝑓𝑎) / 𝑘𝐵)
7574cbvmptv 4678 . . . . . . . . . . . . . 14 (𝑗 ∈ ℕ ↦ (𝑓𝑗) / 𝑘𝐵) = (𝑎 ∈ ℕ ↦ (𝑓𝑎) / 𝑘𝐵)
7636, 75eqtri 2632 . . . . . . . . . . . . 13 𝐺 = (𝑎 ∈ ℕ ↦ (𝑓𝑎) / 𝑘𝐵)
77 fveq2 6103 . . . . . . . . . . . . . . 15 (𝑗 = 𝑎 → (𝑔𝑗) = (𝑔𝑎))
7877csbeq1d 3506 . . . . . . . . . . . . . 14 (𝑗 = 𝑎(𝑔𝑗) / 𝑘𝐵 = (𝑔𝑎) / 𝑘𝐵)
7978cbvmptv 4678 . . . . . . . . . . . . 13 (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵) = (𝑎 ∈ ℕ ↦ (𝑔𝑎) / 𝑘𝐵)
80 simplr 788 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑤 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑤)–1-1-onto𝐴)) → (𝑚 ∈ ℕ ∧ 𝑤 ∈ ℕ))
81 simprl 790 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑤 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑤)–1-1-onto𝐴)) → 𝑓:(1...𝑚)–1-1-onto𝐴)
82 simprr 792 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑤 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑤)–1-1-onto𝐴)) → 𝑔:(1...𝑤)–1-1-onto𝐴)
8316, 72, 76, 79, 80, 81, 82prodmolem3 14502 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑤 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑤)–1-1-onto𝐴)) → (seq1( · , 𝐺)‘𝑚) = (seq1( · , (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))‘𝑤))
84 eqeq12 2623 . . . . . . . . . . . 12 ((𝑥 = (seq1( · , 𝐺)‘𝑚) ∧ 𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))‘𝑤)) → (𝑥 = 𝑧 ↔ (seq1( · , 𝐺)‘𝑚) = (seq1( · , (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))‘𝑤)))
8583, 84syl5ibrcom 236 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑤 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑤)–1-1-onto𝐴)) → ((𝑥 = (seq1( · , 𝐺)‘𝑚) ∧ 𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))‘𝑤)) → 𝑥 = 𝑧))
8685expimpd 627 . . . . . . . . . 10 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → (((𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑤)–1-1-onto𝐴) ∧ (𝑥 = (seq1( · , 𝐺)‘𝑚) ∧ 𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))‘𝑤))) → 𝑥 = 𝑧))
8770, 86syl5bi 231 . . . . . . . . 9 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → (((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚)) ∧ (𝑔:(1...𝑤)–1-1-onto𝐴𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))‘𝑤))) → 𝑥 = 𝑧))
8887exlimdvv 1849 . . . . . . . 8 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → (∃𝑓𝑔((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚)) ∧ (𝑔:(1...𝑤)–1-1-onto𝐴𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))‘𝑤))) → 𝑥 = 𝑧))
8988rexlimdvva 3020 . . . . . . 7 (𝜑 → (∃𝑚 ∈ ℕ ∃𝑤 ∈ ℕ ∃𝑓𝑔((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚)) ∧ (𝑔:(1...𝑤)–1-1-onto𝐴𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))‘𝑤))) → 𝑥 = 𝑧))
9069, 89syl5bir 232 . . . . . 6 (𝜑 → ((∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚)) ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚))) → 𝑥 = 𝑧))
9190com12 32 . . . . 5 ((∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚)) ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚))) → (𝜑𝑥 = 𝑧))
9235, 42, 45, 91ccase 984 . . . 4 (((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚))) ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚)))) → (𝜑𝑥 = 𝑧))
9392com12 32 . . 3 (𝜑 → (((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚))) ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚)))) → 𝑥 = 𝑧))
9493alrimivv 1843 . 2 (𝜑 → ∀𝑥𝑧(((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚))) ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚)))) → 𝑥 = 𝑧))
95 breq2 4587 . . . . . 6 (𝑥 = 𝑧 → (seq𝑚( · , 𝐹) ⇝ 𝑥 ↔ seq𝑚( · , 𝐹) ⇝ 𝑧))
96953anbi3d 1397 . . . . 5 (𝑥 = 𝑧 → ((𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧)))
9796rexbidv 3034 . . . 4 (𝑥 = 𝑧 → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ↔ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧)))
98 eqeq1 2614 . . . . . . 7 (𝑥 = 𝑧 → (𝑥 = (seq1( · , 𝐺)‘𝑚) ↔ 𝑧 = (seq1( · , 𝐺)‘𝑚)))
9998anbi2d 736 . . . . . 6 (𝑥 = 𝑧 → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚))))
10099exbidv 1837 . . . . 5 (𝑥 = 𝑧 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚))))
101100rexbidv 3034 . . . 4 (𝑥 = 𝑧 → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚))))
10297, 101orbi12d 742 . . 3 (𝑥 = 𝑧 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚))) ↔ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚)))))
103102mo4 2505 . 2 (∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚))) ↔ ∀𝑥𝑧(((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚))) ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚)))) → 𝑥 = 𝑧))
10494, 103sylibr 223 1 (𝜑 → ∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃*wmo 2459   ≠ wne 2780  ∃wrex 2897  ⦋csb 3499   ⊆ wss 3540  ifcif 4036   class class class wbr 4583   ↦ cmpt 4643  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  0cc0 9815  1c1 9816   · cmul 9820  ℕcn 10897  ℤcz 11254  ℤ≥cuz 11563  ...cfz 12197  seqcseq 12663   ⇝ cli 14063 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-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 This theorem is referenced by:  fprod  14510
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