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Theorem prodmo 15286
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 1146 . . . . . . 7 ((𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) → (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥))
21reximi 3209 . . . . . 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 1146 . . . . . . 7 ((𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧) → (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧))
43reximi 3209 . . . . . 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 6649 . . . . . . . . . . . 12 (𝑚 = 𝑤 → (ℤ𝑚) = (ℤ𝑤))
65sseq2d 3950 . . . . . . . . . . 11 (𝑚 = 𝑤 → (𝐴 ⊆ (ℤ𝑚) ↔ 𝐴 ⊆ (ℤ𝑤)))
7 seqeq1 13371 . . . . . . . . . . . 12 (𝑚 = 𝑤 → seq𝑚( · , 𝐹) = seq𝑤( · , 𝐹))
87breq1d 5043 . . . . . . . . . . 11 (𝑚 = 𝑤 → (seq𝑚( · , 𝐹) ⇝ 𝑧 ↔ seq𝑤( · , 𝐹) ⇝ 𝑧))
96, 8anbi12d 633 . . . . . . . . . 10 (𝑚 = 𝑤 → ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧) ↔ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))
109cbvrexvw 3400 . . . . . . . . 9 (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧) ↔ ∃𝑤 ∈ ℤ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧))
1110anbi2i 625 . . . . . . . 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 3323 . . . . . . . 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 281 . . . . . . 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 779 . . . . . . . . . . . . 13 (((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧))) → seq𝑚( · , 𝐹) ⇝ 𝑥)
1514adantl 485 . . . . . . . . . . . 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 714 . . . . . . . . . . . . 13 (((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) ∧ 𝑘𝐴) → 𝐵 ∈ ℂ)
19 simprll 778 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → 𝑚 ∈ ℤ)
20 simprlr 779 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → 𝑤 ∈ ℤ)
21 simprll 778 . . . . . . . . . . . . . 14 (((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧))) → 𝐴 ⊆ (ℤ𝑚))
2221adantl 485 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → 𝐴 ⊆ (ℤ𝑚))
23 simprrl 780 . . . . . . . . . . . . . 14 (((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧))) → 𝐴 ⊆ (ℤ𝑤))
2423adantl 485 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → 𝐴 ⊆ (ℤ𝑤))
2516, 18, 19, 20, 22, 24prodrb 15282 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → (seq𝑚( · , 𝐹) ⇝ 𝑥 ↔ seq𝑤( · , 𝐹) ⇝ 𝑥))
2615, 25mpbid 235 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → seq𝑤( · , 𝐹) ⇝ 𝑥)
27 simprrr 781 . . . . . . . . . . . 12 (((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧))) → seq𝑤( · , 𝐹) ⇝ 𝑧)
2827adantl 485 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → seq𝑤( · , 𝐹) ⇝ 𝑧)
29 climuni 14905 . . . . . . . . . . 11 ((seq𝑤( · , 𝐹) ⇝ 𝑥 ∧ seq𝑤( · , 𝐹) ⇝ 𝑧) → 𝑥 = 𝑧)
3026, 28, 29syl2anc 587 . . . . . . . . . 10 ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → 𝑥 = 𝑧)
3130expcom 417 . . . . . . . . 9 (((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧))) → (𝜑𝑥 = 𝑧))
3231ex 416 . . . . . . . 8 ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) → (((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)) → (𝜑𝑥 = 𝑧)))
3332rexlimivv 3254 . . . . . . 7 (∃𝑚 ∈ ℤ ∃𝑤 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)) → (𝜑𝑥 = 𝑧))
3413, 33sylbi 220 . . . . . 6 ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧)) → (𝜑𝑥 = 𝑧))
352, 4, 34syl2an 598 . . . . 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 15285 . . . . . . . . 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 2024 . . . . . . . . 9 (𝑧 = 𝑥𝑥 = 𝑧)
3937, 38syl6 35 . . . . . . . 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 457 . . . . . . 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 462 . . . . 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 15285 . . . . . . 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 457 . . . . . 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 3323 . . . . . . . 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 exdistrv 1956 . . . . . . . . 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 3214 . . . . . . . 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 7147 . . . . . . . . . . . . . 14 (𝑚 = 𝑤 → (1...𝑚) = (1...𝑤))
5049f1oeq2d 6590 . . . . . . . . . . . . 13 (𝑚 = 𝑤 → (𝑓:(1...𝑚)–1-1-onto𝐴𝑓:(1...𝑤)–1-1-onto𝐴))
51 fveq2 6649 . . . . . . . . . . . . . 14 (𝑚 = 𝑤 → (seq1( · , 𝐺)‘𝑚) = (seq1( · , 𝐺)‘𝑤))
5251eqeq2d 2812 . . . . . . . . . . . . 13 (𝑚 = 𝑤 → (𝑧 = (seq1( · , 𝐺)‘𝑚) ↔ 𝑧 = (seq1( · , 𝐺)‘𝑤)))
5350, 52anbi12d 633 . . . . . . . . . . . 12 (𝑚 = 𝑤 → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚)) ↔ (𝑓:(1...𝑤)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑤))))
5453exbidv 1922 . . . . . . . . . . 11 (𝑚 = 𝑤 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑤)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑤))))
55 f1oeq1 6583 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → (𝑓:(1...𝑤)–1-1-onto𝐴𝑔:(1...𝑤)–1-1-onto𝐴))
56 fveq1 6648 . . . . . . . . . . . . . . . . . . 19 (𝑓 = 𝑔 → (𝑓𝑗) = (𝑔𝑗))
5756csbeq1d 3835 . . . . . . . . . . . . . . . . . 18 (𝑓 = 𝑔(𝑓𝑗) / 𝑘𝐵 = (𝑔𝑗) / 𝑘𝐵)
5857mpteq2dv 5129 . . . . . . . . . . . . . . . . 17 (𝑓 = 𝑔 → (𝑗 ∈ ℕ ↦ (𝑓𝑗) / 𝑘𝐵) = (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))
5936, 58syl5eq 2848 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑔𝐺 = (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))
6059seqeq3d 13376 . . . . . . . . . . . . . . 15 (𝑓 = 𝑔 → seq1( · , 𝐺) = seq1( · , (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵)))
6160fveq1d 6651 . . . . . . . . . . . . . 14 (𝑓 = 𝑔 → (seq1( · , 𝐺)‘𝑤) = (seq1( · , (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))‘𝑤))
6261eqeq2d 2812 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → (𝑧 = (seq1( · , 𝐺)‘𝑤) ↔ 𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))‘𝑤)))
6355, 62anbi12d 633 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ((𝑓:(1...𝑤)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑤)) ↔ (𝑔:(1...𝑤)–1-1-onto𝐴𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))‘𝑤))))
6463cbvexvw 2044 . . . . . . . . . . 11 (∃𝑓(𝑓:(1...𝑤)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑤)) ↔ ∃𝑔(𝑔:(1...𝑤)–1-1-onto𝐴𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))‘𝑤)))
6554, 64syl6bb 290 . . . . . . . . . 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))))
6665cbvrexvw 3400 . . . . . . . . 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)))
6766anbi2i 625 . . . . . . . 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))))
6846, 48, 673bitr4i 306 . . . . . . 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))))
69 an4 655 . . . . . . . . . 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))))
7017ad4ant14 751 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑤 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑤)–1-1-onto𝐴)) ∧ 𝑘𝐴) → 𝐵 ∈ ℂ)
71 fveq2 6649 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑎 → (𝑓𝑗) = (𝑓𝑎))
7271csbeq1d 3835 . . . . . . . . . . . . . . 15 (𝑗 = 𝑎(𝑓𝑗) / 𝑘𝐵 = (𝑓𝑎) / 𝑘𝐵)
7372cbvmptv 5136 . . . . . . . . . . . . . 14 (𝑗 ∈ ℕ ↦ (𝑓𝑗) / 𝑘𝐵) = (𝑎 ∈ ℕ ↦ (𝑓𝑎) / 𝑘𝐵)
7436, 73eqtri 2824 . . . . . . . . . . . . 13 𝐺 = (𝑎 ∈ ℕ ↦ (𝑓𝑎) / 𝑘𝐵)
75 fveq2 6649 . . . . . . . . . . . . . . 15 (𝑗 = 𝑎 → (𝑔𝑗) = (𝑔𝑎))
7675csbeq1d 3835 . . . . . . . . . . . . . 14 (𝑗 = 𝑎(𝑔𝑗) / 𝑘𝐵 = (𝑔𝑎) / 𝑘𝐵)
7776cbvmptv 5136 . . . . . . . . . . . . 13 (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵) = (𝑎 ∈ ℕ ↦ (𝑔𝑎) / 𝑘𝐵)
78 simplr 768 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑤 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑤)–1-1-onto𝐴)) → (𝑚 ∈ ℕ ∧ 𝑤 ∈ ℕ))
79 simprl 770 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑤 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑤)–1-1-onto𝐴)) → 𝑓:(1...𝑚)–1-1-onto𝐴)
80 simprr 772 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑤 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑤)–1-1-onto𝐴)) → 𝑔:(1...𝑤)–1-1-onto𝐴)
8116, 70, 74, 77, 78, 79, 80prodmolem3 15283 . . . . . . . . . . . 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))
82 eqeq12 2815 . . . . . . . . . . . 12 ((𝑥 = (seq1( · , 𝐺)‘𝑚) ∧ 𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))‘𝑤)) → (𝑥 = 𝑧 ↔ (seq1( · , 𝐺)‘𝑚) = (seq1( · , (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))‘𝑤)))
8381, 82syl5ibrcom 250 . . . . . . . . . . 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))
8483expimpd 457 . . . . . . . . . 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))
8569, 84syl5bi 245 . . . . . . . . 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))
8685exlimdvv 1935 . . . . . . . 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))
8786rexlimdvva 3256 . . . . . . 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))
8868, 87syl5bir 246 . . . . . 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))
8988com12 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))
9035, 42, 45, 89ccase 1033 . . . 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))
9190com12 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))
9291alrimivv 1929 . 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))
93 breq2 5037 . . . . . 6 (𝑥 = 𝑧 → (seq𝑚( · , 𝐹) ⇝ 𝑥 ↔ seq𝑚( · , 𝐹) ⇝ 𝑧))
94933anbi3d 1439 . . . . 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)))
9594rexbidv 3259 . . . 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)))
96 eqeq1 2805 . . . . . . 7 (𝑥 = 𝑧 → (𝑥 = (seq1( · , 𝐺)‘𝑚) ↔ 𝑧 = (seq1( · , 𝐺)‘𝑚)))
9796anbi2d 631 . . . . . 6 (𝑥 = 𝑧 → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚))))
9897exbidv 1922 . . . . 5 (𝑥 = 𝑧 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚))))
9998rexbidv 3259 . . . 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))))
10095, 99orbi12d 916 . . 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)))))
101100mo4 2628 . 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))
10292, 101sylibr 237 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  wa 399  wo 844  w3a 1084  wal 1536   = wceq 1538  wex 1781  wcel 2112  ∃*wmo 2599  wne 2990  wrex 3110  csb 3831  wss 3884  ifcif 4428   class class class wbr 5033  cmpt 5113  1-1-ontowf1o 6327  cfv 6328  (class class class)co 7139  cc 10528  0cc0 10530  1c1 10531   · cmul 10535  cn 11629  cz 11973  cuz 12235  ...cfz 12889  seqcseq 13368  cli 14837
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-sup 8894  df-oi 8962  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-rp 12382  df-fz 12890  df-fzo 13033  df-seq 13369  df-exp 13430  df-hash 13691  df-cj 14454  df-re 14455  df-im 14456  df-sqrt 14590  df-abs 14591  df-clim 14841
This theorem is referenced by:  fprod  15291
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