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Theorem summodc 11375
Description: A sum has at most one limit. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 4-May-2023.)
Hypotheses
Ref Expression
isummo.1 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))
isummo.2 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
summodclem2.g 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0))
summodc.3 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0))
Assertion
Ref Expression
summodc (𝜑 → ∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚))))
Distinct variable groups:   𝑘,𝑛,𝐴   𝑛,𝐹   𝜑,𝑘,𝑛   𝐴,𝑓,𝑗,𝑚,𝑘,𝑛   𝐵,𝑛   𝑓,𝐹,𝑘,𝑚   𝜑,𝑓,𝑚,𝑥,𝑘,𝑛   𝑥,𝐴,𝑗   𝐵,𝑓,𝑗,𝑚   𝑗,𝐹,𝑥   𝑛,𝐺,𝑥   𝜑,𝑗,𝑥
Allowed substitution hints:   𝐵(𝑥,𝑘)   𝐺(𝑓,𝑗,𝑘,𝑚)

Proof of Theorem summodc
Dummy variables 𝑎 𝑔 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5511 . . . . . . . . . 10 (𝑚 = 𝑛 → (ℤ𝑚) = (ℤ𝑛))
21sseq2d 3185 . . . . . . . . 9 (𝑚 = 𝑛 → (𝐴 ⊆ (ℤ𝑚) ↔ 𝐴 ⊆ (ℤ𝑛)))
31raleqdv 2678 . . . . . . . . 9 (𝑚 = 𝑛 → (∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ↔ ∀𝑗 ∈ (ℤ𝑛)DECID 𝑗𝐴))
4 seqeq1 10434 . . . . . . . . . 10 (𝑚 = 𝑛 → seq𝑚( + , 𝐹) = seq𝑛( + , 𝐹))
54breq1d 4010 . . . . . . . . 9 (𝑚 = 𝑛 → (seq𝑚( + , 𝐹) ⇝ 𝑦 ↔ seq𝑛( + , 𝐹) ⇝ 𝑦))
62, 3, 53anbi123d 1312 . . . . . . . 8 (𝑚 = 𝑛 → ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ↔ (𝐴 ⊆ (ℤ𝑛) ∧ ∀𝑗 ∈ (ℤ𝑛)DECID 𝑗𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦)))
76cbvrexv 2704 . . . . . . 7 (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ↔ ∃𝑛 ∈ ℤ (𝐴 ⊆ (ℤ𝑛) ∧ ∀𝑗 ∈ (ℤ𝑛)DECID 𝑗𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))
8 reeanv 2646 . . . . . . . . 9 (∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ ∀𝑗 ∈ (ℤ𝑛)DECID 𝑗𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦)) ↔ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ ∃𝑛 ∈ ℤ (𝐴 ⊆ (ℤ𝑛) ∧ ∀𝑗 ∈ (ℤ𝑛)DECID 𝑗𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦)))
9 simprl3 1044 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ ∀𝑗 ∈ (ℤ𝑛)DECID 𝑗𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → seq𝑚( + , 𝐹) ⇝ 𝑥)
10 isummo.1 . . . . . . . . . . . . . 14 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))
11 simpll 527 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ ∀𝑗 ∈ (ℤ𝑛)DECID 𝑗𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → 𝜑)
12 isummo.2 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
1311, 12sylan 283 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ ∀𝑗 ∈ (ℤ𝑛)DECID 𝑗𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) ∧ 𝑘𝐴) → 𝐵 ∈ ℂ)
14 simplrl 535 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ ∀𝑗 ∈ (ℤ𝑛)DECID 𝑗𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → 𝑚 ∈ ℤ)
15 simplrr 536 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ ∀𝑗 ∈ (ℤ𝑛)DECID 𝑗𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → 𝑛 ∈ ℤ)
16 simprl1 1042 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ ∀𝑗 ∈ (ℤ𝑛)DECID 𝑗𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → 𝐴 ⊆ (ℤ𝑚))
17 simprr1 1045 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ ∀𝑗 ∈ (ℤ𝑛)DECID 𝑗𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → 𝐴 ⊆ (ℤ𝑛))
18 eleq1w 2238 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑘 → (𝑗𝐴𝑘𝐴))
1918dcbid 838 . . . . . . . . . . . . . . 15 (𝑗 = 𝑘 → (DECID 𝑗𝐴DECID 𝑘𝐴))
20 simprl2 1043 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ ∀𝑗 ∈ (ℤ𝑛)DECID 𝑗𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)
2120adantr 276 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ ∀𝑗 ∈ (ℤ𝑛)DECID 𝑗𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) ∧ 𝑘 ∈ (ℤ𝑚)) → ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)
22 simpr 110 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ ∀𝑗 ∈ (ℤ𝑛)DECID 𝑗𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) ∧ 𝑘 ∈ (ℤ𝑚)) → 𝑘 ∈ (ℤ𝑚))
2319, 21, 22rspcdva 2846 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ ∀𝑗 ∈ (ℤ𝑛)DECID 𝑗𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) ∧ 𝑘 ∈ (ℤ𝑚)) → DECID 𝑘𝐴)
24 simprr2 1046 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ ∀𝑗 ∈ (ℤ𝑛)DECID 𝑗𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → ∀𝑗 ∈ (ℤ𝑛)DECID 𝑗𝐴)
2524adantr 276 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ ∀𝑗 ∈ (ℤ𝑛)DECID 𝑗𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) ∧ 𝑘 ∈ (ℤ𝑛)) → ∀𝑗 ∈ (ℤ𝑛)DECID 𝑗𝐴)
26 simpr 110 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ ∀𝑗 ∈ (ℤ𝑛)DECID 𝑗𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) ∧ 𝑘 ∈ (ℤ𝑛)) → 𝑘 ∈ (ℤ𝑛))
2719, 25, 26rspcdva 2846 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ ∀𝑗 ∈ (ℤ𝑛)DECID 𝑗𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) ∧ 𝑘 ∈ (ℤ𝑛)) → DECID 𝑘𝐴)
2810, 13, 14, 15, 16, 17, 23, 27sumrbdc 11371 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ ∀𝑗 ∈ (ℤ𝑛)DECID 𝑗𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → (seq𝑚( + , 𝐹) ⇝ 𝑥 ↔ seq𝑛( + , 𝐹) ⇝ 𝑥))
299, 28mpbid 147 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ ∀𝑗 ∈ (ℤ𝑛)DECID 𝑗𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → seq𝑛( + , 𝐹) ⇝ 𝑥)
30 simprr3 1047 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ ∀𝑗 ∈ (ℤ𝑛)DECID 𝑗𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → seq𝑛( + , 𝐹) ⇝ 𝑦)
31 climuni 11285 . . . . . . . . . . . 12 ((seq𝑛( + , 𝐹) ⇝ 𝑥 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦) → 𝑥 = 𝑦)
3229, 30, 31syl2anc 411 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ ∀𝑗 ∈ (ℤ𝑛)DECID 𝑗𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → 𝑥 = 𝑦)
3332exp31 364 . . . . . . . . . 10 (𝜑 → ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ ∀𝑗 ∈ (ℤ𝑛)DECID 𝑗𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦)) → 𝑥 = 𝑦)))
3433rexlimdvv 2601 . . . . . . . . 9 (𝜑 → (∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ ∀𝑗 ∈ (ℤ𝑛)DECID 𝑗𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦)) → 𝑥 = 𝑦))
358, 34biimtrrid 153 . . . . . . . 8 (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ ∃𝑛 ∈ ℤ (𝐴 ⊆ (ℤ𝑛) ∧ ∀𝑗 ∈ (ℤ𝑛)DECID 𝑗𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦)) → 𝑥 = 𝑦))
3635expdimp 259 . . . . . . 7 ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑛 ∈ ℤ (𝐴 ⊆ (ℤ𝑛) ∧ ∀𝑗 ∈ (ℤ𝑛)DECID 𝑗𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦) → 𝑥 = 𝑦))
377, 36biimtrid 152 . . . . . 6 ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) → 𝑥 = 𝑦))
38 summodc.3 . . . . . . 7 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0))
3910, 12, 38summodclem2 11374 . . . . . 6 ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
4037, 39jaod 717 . . . . 5 ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚))) → 𝑥 = 𝑦))
4110, 12, 38summodclem2 11374 . . . . . . . 8 ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑦)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚)) → 𝑦 = 𝑥))
42 equcom 1706 . . . . . . . 8 (𝑦 = 𝑥𝑥 = 𝑦)
4341, 42syl6ib 161 . . . . . . 7 ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑦)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
4443impancom 260 . . . . . 6 ((𝜑 ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚))) → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) → 𝑥 = 𝑦))
45 oveq2 5877 . . . . . . . . . . . 12 (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
46 f1oeq2 5446 . . . . . . . . . . . 12 ((1...𝑚) = (1...𝑛) → (𝑓:(1...𝑚)–1-1-onto𝐴𝑓:(1...𝑛)–1-1-onto𝐴))
4745, 46syl 14 . . . . . . . . . . 11 (𝑚 = 𝑛 → (𝑓:(1...𝑚)–1-1-onto𝐴𝑓:(1...𝑛)–1-1-onto𝐴))
48 fveq2 5511 . . . . . . . . . . . 12 (𝑚 = 𝑛 → (seq1( + , 𝐺)‘𝑚) = (seq1( + , 𝐺)‘𝑛))
4948eqeq2d 2189 . . . . . . . . . . 11 (𝑚 = 𝑛 → (𝑦 = (seq1( + , 𝐺)‘𝑚) ↔ 𝑦 = (seq1( + , 𝐺)‘𝑛)))
5047, 49anbi12d 473 . . . . . . . . . 10 (𝑚 = 𝑛 → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) ↔ (𝑓:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑛))))
5150exbidv 1825 . . . . . . . . 9 (𝑚 = 𝑛 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑛))))
52 f1oeq1 5445 . . . . . . . . . . 11 (𝑓 = 𝑔 → (𝑓:(1...𝑛)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴))
53 breq1 4003 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑎 → (𝑛 ≤ (♯‘𝐴) ↔ 𝑎 ≤ (♯‘𝐴)))
54 fveq2 5511 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑎 → (𝑓𝑛) = (𝑓𝑎))
5554csbeq1d 3064 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑎(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑎) / 𝑘𝐵)
5653, 55ifbieq1d 3556 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑎 → if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0) = if(𝑎 ≤ (♯‘𝐴), (𝑓𝑎) / 𝑘𝐵, 0))
5756cbvmptv 4096 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)) = (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (𝑓𝑎) / 𝑘𝐵, 0))
58 fveq1 5510 . . . . . . . . . . . . . . . . . . 19 (𝑓 = 𝑔 → (𝑓𝑎) = (𝑔𝑎))
5958csbeq1d 3064 . . . . . . . . . . . . . . . . . 18 (𝑓 = 𝑔(𝑓𝑎) / 𝑘𝐵 = (𝑔𝑎) / 𝑘𝐵)
6059ifeq1d 3551 . . . . . . . . . . . . . . . . 17 (𝑓 = 𝑔 → if(𝑎 ≤ (♯‘𝐴), (𝑓𝑎) / 𝑘𝐵, 0) = if(𝑎 ≤ (♯‘𝐴), (𝑔𝑎) / 𝑘𝐵, 0))
6160mpteq2dv 4091 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑔 → (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (𝑓𝑎) / 𝑘𝐵, 0)) = (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (𝑔𝑎) / 𝑘𝐵, 0)))
6257, 61eqtrid 2222 . . . . . . . . . . . . . . 15 (𝑓 = 𝑔 → (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)) = (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (𝑔𝑎) / 𝑘𝐵, 0)))
6338, 62eqtrid 2222 . . . . . . . . . . . . . 14 (𝑓 = 𝑔𝐺 = (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (𝑔𝑎) / 𝑘𝐵, 0)))
6463seqeq3d 10439 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → seq1( + , 𝐺) = seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (𝑔𝑎) / 𝑘𝐵, 0))))
6564fveq1d 5513 . . . . . . . . . . . 12 (𝑓 = 𝑔 → (seq1( + , 𝐺)‘𝑛) = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (𝑔𝑎) / 𝑘𝐵, 0)))‘𝑛))
6665eqeq2d 2189 . . . . . . . . . . 11 (𝑓 = 𝑔 → (𝑦 = (seq1( + , 𝐺)‘𝑛) ↔ 𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (𝑔𝑎) / 𝑘𝐵, 0)))‘𝑛)))
6752, 66anbi12d 473 . . . . . . . . . 10 (𝑓 = 𝑔 → ((𝑓:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑛)) ↔ (𝑔:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (𝑔𝑎) / 𝑘𝐵, 0)))‘𝑛))))
6867cbvexv 1918 . . . . . . . . 9 (∃𝑓(𝑓:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑛)) ↔ ∃𝑔(𝑔:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (𝑔𝑎) / 𝑘𝐵, 0)))‘𝑛)))
6951, 68bitrdi 196 . . . . . . . 8 (𝑚 = 𝑛 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) ↔ ∃𝑔(𝑔:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (𝑔𝑎) / 𝑘𝐵, 0)))‘𝑛))))
7069cbvrexv 2704 . . . . . . 7 (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) ↔ ∃𝑛 ∈ ℕ ∃𝑔(𝑔:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (𝑔𝑎) / 𝑘𝐵, 0)))‘𝑛)))
71 reeanv 2646 . . . . . . . . 9 (∃𝑚 ∈ ℕ ∃𝑛 ∈ ℕ (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ ∃𝑔(𝑔:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (𝑔𝑎) / 𝑘𝐵, 0)))‘𝑛))) ↔ (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ ∃𝑛 ∈ ℕ ∃𝑔(𝑔:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (𝑔𝑎) / 𝑘𝐵, 0)))‘𝑛))))
72 eeanv 1932 . . . . . . . . . . 11 (∃𝑓𝑔((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ (𝑔:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (𝑔𝑎) / 𝑘𝐵, 0)))‘𝑛))) ↔ (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ ∃𝑔(𝑔:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (𝑔𝑎) / 𝑘𝐵, 0)))‘𝑛))))
73 an4 586 . . . . . . . . . . . . 13 (((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ (𝑔:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (𝑔𝑎) / 𝑘𝐵, 0)))‘𝑛))) ↔ ((𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴) ∧ (𝑥 = (seq1( + , 𝐺)‘𝑚) ∧ 𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (𝑔𝑎) / 𝑘𝐵, 0)))‘𝑛))))
74 1zzd 9269 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → 1 ∈ ℤ)
75 simplrr 536 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → 𝑛 ∈ ℕ)
7675nnzd 9363 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → 𝑛 ∈ ℤ)
7774, 76fzfigd 10417 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → (1...𝑛) ∈ Fin)
78 simprr 531 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → 𝑔:(1...𝑛)–1-1-onto𝐴)
7977, 78fihasheqf1od 10753 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → (♯‘(1...𝑛)) = (♯‘𝐴))
8075nnnn0d 9218 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → 𝑛 ∈ ℕ0)
81 hashfz1 10747 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ ℕ0 → (♯‘(1...𝑛)) = 𝑛)
8280, 81syl 14 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → (♯‘(1...𝑛)) = 𝑛)
8379, 82eqtr3d 2212 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → (♯‘𝐴) = 𝑛)
8483breq2d 4012 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → (𝑎 ≤ (♯‘𝐴) ↔ 𝑎𝑛))
8584ifbid 3555 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → if(𝑎 ≤ (♯‘𝐴), (𝑔𝑎) / 𝑘𝐵, 0) = if(𝑎𝑛, (𝑔𝑎) / 𝑘𝐵, 0))
8685mpteq2dv 4091 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (𝑔𝑎) / 𝑘𝐵, 0)) = (𝑎 ∈ ℕ ↦ if(𝑎𝑛, (𝑔𝑎) / 𝑘𝐵, 0)))
8786seqeq3d 10439 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (𝑔𝑎) / 𝑘𝐵, 0))) = seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎𝑛, (𝑔𝑎) / 𝑘𝐵, 0))))
8887fveq1d 5513 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (𝑔𝑎) / 𝑘𝐵, 0)))‘𝑛) = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎𝑛, (𝑔𝑎) / 𝑘𝐵, 0)))‘𝑛))
8988eqeq2d 2189 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → (𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (𝑔𝑎) / 𝑘𝐵, 0)))‘𝑛) ↔ 𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎𝑛, (𝑔𝑎) / 𝑘𝐵, 0)))‘𝑛)))
9089anbi2d 464 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → ((𝑥 = (seq1( + , 𝐺)‘𝑚) ∧ 𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (𝑔𝑎) / 𝑘𝐵, 0)))‘𝑛)) ↔ (𝑥 = (seq1( + , 𝐺)‘𝑚) ∧ 𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎𝑛, (𝑔𝑎) / 𝑘𝐵, 0)))‘𝑛))))
91 simplrl 535 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → 𝑚 ∈ ℕ)
9291nnnn0d 9218 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → 𝑚 ∈ ℕ0)
93 hashfz1 10747 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ ℕ0 → (♯‘(1...𝑚)) = 𝑚)
9492, 93syl 14 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → (♯‘(1...𝑚)) = 𝑚)
9591nnzd 9363 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → 𝑚 ∈ ℤ)
9674, 95fzfigd 10417 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → (1...𝑚) ∈ Fin)
97 simprl 529 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → 𝑓:(1...𝑚)–1-1-onto𝐴)
9896, 97fihasheqf1od 10753 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → (♯‘(1...𝑚)) = (♯‘𝐴))
9994, 98eqtr3d 2212 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → 𝑚 = (♯‘𝐴))
10099fveq2d 5515 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → (seq1( + , 𝐺)‘𝑚) = (seq1( + , 𝐺)‘(♯‘𝐴)))
101 simpll 527 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → 𝜑)
102101, 12sylan 283 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) ∧ 𝑘𝐴) → 𝐵 ∈ ℂ)
10399, 91eqeltrrd 2255 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → (♯‘𝐴) ∈ ℕ)
104103, 75jca 306 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → ((♯‘𝐴) ∈ ℕ ∧ 𝑛 ∈ ℕ))
10599oveq2d 5885 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → (1...𝑚) = (1...(♯‘𝐴)))
106 f1oeq2 5446 . . . . . . . . . . . . . . . . . . . 20 ((1...𝑚) = (1...(♯‘𝐴)) → (𝑓:(1...𝑚)–1-1-onto𝐴𝑓:(1...(♯‘𝐴))–1-1-onto𝐴))
107105, 106syl 14 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → (𝑓:(1...𝑚)–1-1-onto𝐴𝑓:(1...(♯‘𝐴))–1-1-onto𝐴))
10897, 107mpbid 147 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)
109 breq1 4003 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑗 → (𝑛 ≤ (♯‘𝐴) ↔ 𝑗 ≤ (♯‘𝐴)))
110 fveq2 5511 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑗 → (𝑓𝑛) = (𝑓𝑗))
111110csbeq1d 3064 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑗(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑗) / 𝑘𝐵)
112109, 111ifbieq1d 3556 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑗 → if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0) = if(𝑗 ≤ (♯‘𝐴), (𝑓𝑗) / 𝑘𝐵, 0))
113112cbvmptv 4096 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (𝑓𝑗) / 𝑘𝐵, 0))
11438, 113eqtri 2198 . . . . . . . . . . . . . . . . . 18 𝐺 = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (𝑓𝑗) / 𝑘𝐵, 0))
115 breq1 4003 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝑗 → (𝑎𝑛𝑗𝑛))
116 fveq2 5511 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 = 𝑗 → (𝑔𝑎) = (𝑔𝑗))
117116csbeq1d 3064 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝑗(𝑔𝑎) / 𝑘𝐵 = (𝑔𝑗) / 𝑘𝐵)
118115, 117ifbieq1d 3556 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝑗 → if(𝑎𝑛, (𝑔𝑎) / 𝑘𝐵, 0) = if(𝑗𝑛, (𝑔𝑗) / 𝑘𝐵, 0))
119118cbvmptv 4096 . . . . . . . . . . . . . . . . . 18 (𝑎 ∈ ℕ ↦ if(𝑎𝑛, (𝑔𝑎) / 𝑘𝐵, 0)) = (𝑗 ∈ ℕ ↦ if(𝑗𝑛, (𝑔𝑗) / 𝑘𝐵, 0))
12010, 102, 104, 108, 78, 114, 119summodclem3 11372 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → (seq1( + , 𝐺)‘(♯‘𝐴)) = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎𝑛, (𝑔𝑎) / 𝑘𝐵, 0)))‘𝑛))
121100, 120eqtrd 2210 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → (seq1( + , 𝐺)‘𝑚) = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎𝑛, (𝑔𝑎) / 𝑘𝐵, 0)))‘𝑛))
122 eqeq12 2190 . . . . . . . . . . . . . . . 16 ((𝑥 = (seq1( + , 𝐺)‘𝑚) ∧ 𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎𝑛, (𝑔𝑎) / 𝑘𝐵, 0)))‘𝑛)) → (𝑥 = 𝑦 ↔ (seq1( + , 𝐺)‘𝑚) = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎𝑛, (𝑔𝑎) / 𝑘𝐵, 0)))‘𝑛)))
123121, 122syl5ibrcom 157 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → ((𝑥 = (seq1( + , 𝐺)‘𝑚) ∧ 𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎𝑛, (𝑔𝑎) / 𝑘𝐵, 0)))‘𝑛)) → 𝑥 = 𝑦))
12490, 123sylbid 150 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → ((𝑥 = (seq1( + , 𝐺)‘𝑚) ∧ 𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (𝑔𝑎) / 𝑘𝐵, 0)))‘𝑛)) → 𝑥 = 𝑦))
125124expimpd 363 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) → (((𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴) ∧ (𝑥 = (seq1( + , 𝐺)‘𝑚) ∧ 𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (𝑔𝑎) / 𝑘𝐵, 0)))‘𝑛))) → 𝑥 = 𝑦))
12673, 125biimtrid 152 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) → (((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ (𝑔:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (𝑔𝑎) / 𝑘𝐵, 0)))‘𝑛))) → 𝑥 = 𝑦))
127126exlimdvv 1897 . . . . . . . . . . 11 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) → (∃𝑓𝑔((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ (𝑔:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (𝑔𝑎) / 𝑘𝐵, 0)))‘𝑛))) → 𝑥 = 𝑦))
12872, 127biimtrrid 153 . . . . . . . . . 10 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) → ((∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ ∃𝑔(𝑔:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (𝑔𝑎) / 𝑘𝐵, 0)))‘𝑛))) → 𝑥 = 𝑦))
129128rexlimdvva 2602 . . . . . . . . 9 (𝜑 → (∃𝑚 ∈ ℕ ∃𝑛 ∈ ℕ (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ ∃𝑔(𝑔:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (𝑔𝑎) / 𝑘𝐵, 0)))‘𝑛))) → 𝑥 = 𝑦))
13071, 129biimtrrid 153 . . . . . . . 8 (𝜑 → ((∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ ∃𝑛 ∈ ℕ ∃𝑔(𝑔:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (𝑔𝑎) / 𝑘𝐵, 0)))‘𝑛))) → 𝑥 = 𝑦))
131130expdimp 259 . . . . . . 7 ((𝜑 ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚))) → (∃𝑛 ∈ ℕ ∃𝑔(𝑔:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (𝑔𝑎) / 𝑘𝐵, 0)))‘𝑛)) → 𝑥 = 𝑦))
13270, 131biimtrid 152 . . . . . 6 ((𝜑 ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚))) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
13344, 132jaod 717 . . . . 5 ((𝜑 ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚))) → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚))) → 𝑥 = 𝑦))
13440, 133jaodan 797 . . . 4 ((𝜑 ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚)))) → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚))) → 𝑥 = 𝑦))
135134expimpd 363 . . 3 (𝜑 → (((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚))) ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)))) → 𝑥 = 𝑦))
136135alrimivv 1875 . 2 (𝜑 → ∀𝑥𝑦(((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚))) ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)))) → 𝑥 = 𝑦))
137 breq2 4004 . . . . . 6 (𝑥 = 𝑦 → (seq𝑚( + , 𝐹) ⇝ 𝑥 ↔ seq𝑚( + , 𝐹) ⇝ 𝑦))
1381373anbi3d 1318 . . . . 5 (𝑥 = 𝑦 → ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑦)))
139138rexbidv 2478 . . . 4 (𝑥 = 𝑦 → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ↔ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑦)))
140 eqeq1 2184 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 = (seq1( + , 𝐺)‘𝑚) ↔ 𝑦 = (seq1( + , 𝐺)‘𝑚)))
141140anbi2d 464 . . . . . 6 (𝑥 = 𝑦 → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚))))
142141exbidv 1825 . . . . 5 (𝑥 = 𝑦 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚))))
143142rexbidv 2478 . . . 4 (𝑥 = 𝑦 → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚))))
144139, 143orbi12d 793 . . 3 (𝑥 = 𝑦 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚))) ↔ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)))))
145144mo4 2087 . 2 (∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚))) ↔ ∀𝑥𝑦(((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚))) ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)))) → 𝑥 = 𝑦))
146136, 145sylibr 134 1 (𝜑 → ∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wo 708  DECID wdc 834  w3a 978  wal 1351   = wceq 1353  wex 1492  ∃*wmo 2027  wcel 2148  wral 2455  wrex 2456  csb 3057  wss 3129  ifcif 3534   class class class wbr 4000  cmpt 4061  1-1-ontowf1o 5211  cfv 5212  (class class class)co 5869  cc 7800  0cc0 7802  1c1 7803   + caddc 7805  cle 7983  cn 8908  0cn0 9165  cz 9242  cuz 9517  ...cfz 9995  seqcseq 10431  chash 10739  cli 11270
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-frec 6386  df-1o 6411  df-oadd 6415  df-er 6529  df-en 6735  df-dom 6736  df-fin 6737  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fz 9996  df-fzo 10129  df-seqfrec 10432  df-exp 10506  df-ihash 10740  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-clim 11271
This theorem is referenced by:  fsum3  11379
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