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Theorem prodmo 14786
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 1142 . . . . . . 7 ((𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) → (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥))
21reximi 3113 . . . . . 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 1142 . . . . . . 7 ((𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧) → (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧))
43reximi 3113 . . . . . 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 6304 . . . . . . . . . . . 12 (𝑚 = 𝑤 → (ℤ𝑚) = (ℤ𝑤))
65sseq2d 3739 . . . . . . . . . . 11 (𝑚 = 𝑤 → (𝐴 ⊆ (ℤ𝑚) ↔ 𝐴 ⊆ (ℤ𝑤)))
7 seqeq1 12919 . . . . . . . . . . . 12 (𝑚 = 𝑤 → seq𝑚( · , 𝐹) = seq𝑤( · , 𝐹))
87breq1d 4770 . . . . . . . . . . 11 (𝑚 = 𝑤 → (seq𝑚( · , 𝐹) ⇝ 𝑧 ↔ seq𝑤( · , 𝐹) ⇝ 𝑧))
96, 8anbi12d 749 . . . . . . . . . 10 (𝑚 = 𝑤 → ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧) ↔ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))
109cbvrexv 3275 . . . . . . . . 9 (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧) ↔ ∃𝑤 ∈ ℤ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧))
1110anbi2i 732 . . . . . . . 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 3209 . . . . . . . 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 267 . . . . . . 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 822 . . . . . . . . . . . . 13 (((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧))) → seq𝑚( · , 𝐹) ⇝ 𝑥)
1514adantl 473 . . . . . . . . . . . 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 753 . . . . . . . . . . . . 13 (((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) ∧ 𝑘𝐴) → 𝐵 ∈ ℂ)
19 simprll 821 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → 𝑚 ∈ ℤ)
20 simprlr 822 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → 𝑤 ∈ ℤ)
21 simprll 821 . . . . . . . . . . . . . 14 (((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧))) → 𝐴 ⊆ (ℤ𝑚))
2221adantl 473 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → 𝐴 ⊆ (ℤ𝑚))
23 simprrl 823 . . . . . . . . . . . . . 14 (((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧))) → 𝐴 ⊆ (ℤ𝑤))
2423adantl 473 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → 𝐴 ⊆ (ℤ𝑤))
2516, 18, 19, 20, 22, 24prodrb 14782 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → (seq𝑚( · , 𝐹) ⇝ 𝑥 ↔ seq𝑤( · , 𝐹) ⇝ 𝑥))
2615, 25mpbid 222 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → seq𝑤( · , 𝐹) ⇝ 𝑥)
27 simprrr 824 . . . . . . . . . . . 12 (((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧))) → seq𝑤( · , 𝐹) ⇝ 𝑧)
2827adantl 473 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → seq𝑤( · , 𝐹) ⇝ 𝑧)
29 climuni 14403 . . . . . . . . . . 11 ((seq𝑤( · , 𝐹) ⇝ 𝑥 ∧ seq𝑤( · , 𝐹) ⇝ 𝑧) → 𝑥 = 𝑧)
3026, 28, 29syl2anc 696 . . . . . . . . . 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 3138 . . . . . . 7 (∃𝑚 ∈ ℤ ∃𝑤 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)) → (𝜑𝑥 = 𝑧))
3413, 33sylbi 207 . . . . . 6 ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧)) → (𝜑𝑥 = 𝑧))
352, 4, 34syl2an 495 . . . . 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 14785 . . . . . . . . 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 2063 . . . . . . . . 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 630 . . . . . . 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 14785 . . . . . . 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 630 . . . . . 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 3209 . . . . . . . 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 2291 . . . . . . . . 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 3144 . . . . . . . 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 6773 . . . . . . . . . . . . . 14 (𝑚 = 𝑤 → (1...𝑚) = (1...𝑤))
50 f1oeq2 6241 . . . . . . . . . . . . . 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 6304 . . . . . . . . . . . . . 14 (𝑚 = 𝑤 → (seq1( · , 𝐺)‘𝑚) = (seq1( · , 𝐺)‘𝑤))
5352eqeq2d 2734 . . . . . . . . . . . . 13 (𝑚 = 𝑤 → (𝑧 = (seq1( · , 𝐺)‘𝑚) ↔ 𝑧 = (seq1( · , 𝐺)‘𝑤)))
5451, 53anbi12d 749 . . . . . . . . . . . 12 (𝑚 = 𝑤 → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚)) ↔ (𝑓:(1...𝑤)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑤))))
5554exbidv 1963 . . . . . . . . . . 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 6240 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → (𝑓:(1...𝑤)–1-1-onto𝐴𝑔:(1...𝑤)–1-1-onto𝐴))
57 fveq1 6303 . . . . . . . . . . . . . . . . . . 19 (𝑓 = 𝑔 → (𝑓𝑗) = (𝑔𝑗))
5857csbeq1d 3646 . . . . . . . . . . . . . . . . . 18 (𝑓 = 𝑔(𝑓𝑗) / 𝑘𝐵 = (𝑔𝑗) / 𝑘𝐵)
5958mpteq2dv 4853 . . . . . . . . . . . . . . . . 17 (𝑓 = 𝑔 → (𝑗 ∈ ℕ ↦ (𝑓𝑗) / 𝑘𝐵) = (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))
6036, 59syl5eq 2770 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑔𝐺 = (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))
6160seqeq3d 12924 . . . . . . . . . . . . . . 15 (𝑓 = 𝑔 → seq1( · , 𝐺) = seq1( · , (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵)))
6261fveq1d 6306 . . . . . . . . . . . . . 14 (𝑓 = 𝑔 → (seq1( · , 𝐺)‘𝑤) = (seq1( · , (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))‘𝑤))
6362eqeq2d 2734 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → (𝑧 = (seq1( · , 𝐺)‘𝑤) ↔ 𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵))‘𝑤)))
6456, 63anbi12d 749 . . . . . . . . . . . 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 2384 . . . . . . . . . . 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 276 . . . . . . . . . 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 3275 . . . . . . . . 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 732 . . . . . . . 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 292 . . . . . . 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 900 . . . . . . . . . 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 807 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑤 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑤)–1-1-onto𝐴)) → 𝜑)
7271, 17sylan 489 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑤 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑤)–1-1-onto𝐴)) ∧ 𝑘𝐴) → 𝐵 ∈ ℂ)
73 fveq2 6304 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑎 → (𝑓𝑗) = (𝑓𝑎))
7473csbeq1d 3646 . . . . . . . . . . . . . . 15 (𝑗 = 𝑎(𝑓𝑗) / 𝑘𝐵 = (𝑓𝑎) / 𝑘𝐵)
7574cbvmptv 4858 . . . . . . . . . . . . . 14 (𝑗 ∈ ℕ ↦ (𝑓𝑗) / 𝑘𝐵) = (𝑎 ∈ ℕ ↦ (𝑓𝑎) / 𝑘𝐵)
7636, 75eqtri 2746 . . . . . . . . . . . . 13 𝐺 = (𝑎 ∈ ℕ ↦ (𝑓𝑎) / 𝑘𝐵)
77 fveq2 6304 . . . . . . . . . . . . . . 15 (𝑗 = 𝑎 → (𝑔𝑗) = (𝑔𝑎))
7877csbeq1d 3646 . . . . . . . . . . . . . 14 (𝑗 = 𝑎(𝑔𝑗) / 𝑘𝐵 = (𝑔𝑎) / 𝑘𝐵)
7978cbvmptv 4858 . . . . . . . . . . . . 13 (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵) = (𝑎 ∈ ℕ ↦ (𝑔𝑎) / 𝑘𝐵)
80 simplr 809 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑤 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑤)–1-1-onto𝐴)) → (𝑚 ∈ ℕ ∧ 𝑤 ∈ ℕ))
81 simprl 811 . . . . . . . . . . . . 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 813 . . . . . . . . . . . . 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 14783 . . . . . . . . . . . 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 2737 . . . . . . . . . . . 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 237 . . . . . . . . . . 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 630 . . . . . . . . . 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 232 . . . . . . . . 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 1975 . . . . . . . 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 3140 . . . . . . 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 233 . . . . . 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 1024 . . . 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 1969 . 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 4764 . . . . . 6 (𝑥 = 𝑧 → (seq𝑚( · , 𝐹) ⇝ 𝑥 ↔ seq𝑚( · , 𝐹) ⇝ 𝑧))
96953anbi3d 1518 . . . . 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 3154 . . . 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 2728 . . . . . . 7 (𝑥 = 𝑧 → (𝑥 = (seq1( · , 𝐺)‘𝑚) ↔ 𝑧 = (seq1( · , 𝐺)‘𝑚)))
9998anbi2d 742 . . . . . 6 (𝑥 = 𝑧 → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚))))
10099exbidv 1963 . . . . 5 (𝑥 = 𝑧 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚))))
101100rexbidv 3154 . . . 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 748 . . 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 2619 . 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 224 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 196  wo 382  wa 383  w3a 1072  wal 1594   = wceq 1596  wex 1817  wcel 2103  ∃*wmo 2572  wne 2896  wrex 3015  csb 3639  wss 3680  ifcif 4194   class class class wbr 4760  cmpt 4837  1-1-ontowf1o 6000  cfv 6001  (class class class)co 6765  cc 10047  0cc0 10049  1c1 10050   · cmul 10054  cn 11133  cz 11490  cuz 11800  ...cfz 12440  seqcseq 12916  cli 14335
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-inf2 8651  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126  ax-pre-sup 10127
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-se 5178  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-isom 6010  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-1o 7680  df-oadd 7684  df-er 7862  df-en 8073  df-dom 8074  df-sdom 8075  df-fin 8076  df-sup 8464  df-oi 8531  df-card 8878  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-div 10798  df-nn 11134  df-2 11192  df-3 11193  df-n0 11406  df-z 11491  df-uz 11801  df-rp 11947  df-fz 12441  df-fzo 12581  df-seq 12917  df-exp 12976  df-hash 13233  df-cj 13959  df-re 13960  df-im 13961  df-sqrt 14095  df-abs 14096  df-clim 14339
This theorem is referenced by:  fprod  14791
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