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Theorem iseqid2 9600
 Description: The last few partial sums of a sequence that ends with all zeroes (or any element which is a right-identity for +) are all the same. (Contributed by Jim Kingdon, 5-Mar-2022.)
Hypotheses
Ref Expression
iseqid2.1 ((𝜑𝑥𝑆) → (𝑥 + 𝑍) = 𝑥)
iseqid2.2 (𝜑𝐾 ∈ (ℤ𝑀))
iseqid2.3 (𝜑𝑁 ∈ (ℤ𝐾))
iseqid2.4 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) ∈ 𝑆)
iseqid2.5 ((𝜑𝑥 ∈ ((𝐾 + 1)...𝑁)) → (𝐹𝑥) = 𝑍)
iseqid2.s (𝜑𝑆𝑉)
iseqid2.f ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
iseqid2.cl ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
Assertion
Ref Expression
iseqid2 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝑁))
Distinct variable groups:   𝑥, + ,𝑦   𝑥,𝐹,𝑦   𝑥,𝐾,𝑦   𝑥,𝑀,𝑦   𝑥,𝑁,𝑦   𝑥,𝑆,𝑦   𝑥,𝑍   𝜑,𝑥,𝑦
Allowed substitution hints:   𝑉(𝑥,𝑦)   𝑍(𝑦)

Proof of Theorem iseqid2
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 iseqid2.3 . . 3 (𝜑𝑁 ∈ (ℤ𝐾))
2 eluzfz2 9163 . . 3 (𝑁 ∈ (ℤ𝐾) → 𝑁 ∈ (𝐾...𝑁))
31, 2syl 14 . 2 (𝜑𝑁 ∈ (𝐾...𝑁))
4 eleq1 2145 . . . . . 6 (𝑥 = 𝐾 → (𝑥 ∈ (𝐾...𝑁) ↔ 𝐾 ∈ (𝐾...𝑁)))
5 fveq2 5230 . . . . . . 7 (𝑥 = 𝐾 → (seq𝑀( + , 𝐹, 𝑆)‘𝑥) = (seq𝑀( + , 𝐹, 𝑆)‘𝐾))
65eqeq2d 2094 . . . . . 6 (𝑥 = 𝐾 → ((seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝑥) ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝐾)))
74, 6imbi12d 232 . . . . 5 (𝑥 = 𝐾 → ((𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝑥)) ↔ (𝐾 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝐾))))
87imbi2d 228 . . . 4 (𝑥 = 𝐾 → ((𝜑 → (𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝑥))) ↔ (𝜑 → (𝐾 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝐾)))))
9 eleq1 2145 . . . . . 6 (𝑥 = 𝑛 → (𝑥 ∈ (𝐾...𝑁) ↔ 𝑛 ∈ (𝐾...𝑁)))
10 fveq2 5230 . . . . . . 7 (𝑥 = 𝑛 → (seq𝑀( + , 𝐹, 𝑆)‘𝑥) = (seq𝑀( + , 𝐹, 𝑆)‘𝑛))
1110eqeq2d 2094 . . . . . 6 (𝑥 = 𝑛 → ((seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝑥) ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝑛)))
129, 11imbi12d 232 . . . . 5 (𝑥 = 𝑛 → ((𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝑥)) ↔ (𝑛 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝑛))))
1312imbi2d 228 . . . 4 (𝑥 = 𝑛 → ((𝜑 → (𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝑥))) ↔ (𝜑 → (𝑛 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝑛)))))
14 eleq1 2145 . . . . . 6 (𝑥 = (𝑛 + 1) → (𝑥 ∈ (𝐾...𝑁) ↔ (𝑛 + 1) ∈ (𝐾...𝑁)))
15 fveq2 5230 . . . . . . 7 (𝑥 = (𝑛 + 1) → (seq𝑀( + , 𝐹, 𝑆)‘𝑥) = (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)))
1615eqeq2d 2094 . . . . . 6 (𝑥 = (𝑛 + 1) → ((seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝑥) ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))))
1714, 16imbi12d 232 . . . . 5 (𝑥 = (𝑛 + 1) → ((𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝑥)) ↔ ((𝑛 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)))))
1817imbi2d 228 . . . 4 (𝑥 = (𝑛 + 1) → ((𝜑 → (𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝑥))) ↔ (𝜑 → ((𝑛 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))))))
19 eleq1 2145 . . . . . 6 (𝑥 = 𝑁 → (𝑥 ∈ (𝐾...𝑁) ↔ 𝑁 ∈ (𝐾...𝑁)))
20 fveq2 5230 . . . . . . 7 (𝑥 = 𝑁 → (seq𝑀( + , 𝐹, 𝑆)‘𝑥) = (seq𝑀( + , 𝐹, 𝑆)‘𝑁))
2120eqeq2d 2094 . . . . . 6 (𝑥 = 𝑁 → ((seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝑥) ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝑁)))
2219, 21imbi12d 232 . . . . 5 (𝑥 = 𝑁 → ((𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝑥)) ↔ (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝑁))))
2322imbi2d 228 . . . 4 (𝑥 = 𝑁 → ((𝜑 → (𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝑥))) ↔ (𝜑 → (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝑁)))))
24 eqidd 2084 . . . . 5 (𝐾 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝐾))
25242a1i 27 . . . 4 (𝐾 ∈ ℤ → (𝜑 → (𝐾 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝐾))))
26 peano2fzr 9168 . . . . . . . . . 10 ((𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁)) → 𝑛 ∈ (𝐾...𝑁))
2726adantl 271 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → 𝑛 ∈ (𝐾...𝑁))
2827expr 367 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝐾)) → ((𝑛 + 1) ∈ (𝐾...𝑁) → 𝑛 ∈ (𝐾...𝑁)))
2928imim1d 74 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝐾)) → ((𝑛 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝑛)) → ((𝑛 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝑛))))
30 oveq1 5571 . . . . . . . . . 10 ((seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝑛) → ((seq𝑀( + , 𝐹, 𝑆)‘𝐾) + (𝐹‘(𝑛 + 1))) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1))))
31 fveq2 5230 . . . . . . . . . . . . . . 15 (𝑥 = (𝑛 + 1) → (𝐹𝑥) = (𝐹‘(𝑛 + 1)))
3231eqeq1d 2091 . . . . . . . . . . . . . 14 (𝑥 = (𝑛 + 1) → ((𝐹𝑥) = 𝑍 ↔ (𝐹‘(𝑛 + 1)) = 𝑍))
33 iseqid2.5 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ ((𝐾 + 1)...𝑁)) → (𝐹𝑥) = 𝑍)
3433ralrimiva 2439 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑥 ∈ ((𝐾 + 1)...𝑁)(𝐹𝑥) = 𝑍)
3534adantr 270 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → ∀𝑥 ∈ ((𝐾 + 1)...𝑁)(𝐹𝑥) = 𝑍)
36 eluzp1p1 8761 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (ℤ𝐾) → (𝑛 + 1) ∈ (ℤ‘(𝐾 + 1)))
3736ad2antrl 474 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (𝑛 + 1) ∈ (ℤ‘(𝐾 + 1)))
38 elfzuz3 9154 . . . . . . . . . . . . . . . 16 ((𝑛 + 1) ∈ (𝐾...𝑁) → 𝑁 ∈ (ℤ‘(𝑛 + 1)))
3938ad2antll 475 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → 𝑁 ∈ (ℤ‘(𝑛 + 1)))
40 elfzuzb 9151 . . . . . . . . . . . . . . 15 ((𝑛 + 1) ∈ ((𝐾 + 1)...𝑁) ↔ ((𝑛 + 1) ∈ (ℤ‘(𝐾 + 1)) ∧ 𝑁 ∈ (ℤ‘(𝑛 + 1))))
4137, 39, 40sylanbrc 408 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (𝑛 + 1) ∈ ((𝐾 + 1)...𝑁))
4232, 35, 41rspcdva 2716 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (𝐹‘(𝑛 + 1)) = 𝑍)
4342oveq2d 5580 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → ((seq𝑀( + , 𝐹, 𝑆)‘𝐾) + (𝐹‘(𝑛 + 1))) = ((seq𝑀( + , 𝐹, 𝑆)‘𝐾) + 𝑍))
44 oveq1 5571 . . . . . . . . . . . . . . 15 (𝑥 = (seq𝑀( + , 𝐹, 𝑆)‘𝐾) → (𝑥 + 𝑍) = ((seq𝑀( + , 𝐹, 𝑆)‘𝐾) + 𝑍))
45 id 19 . . . . . . . . . . . . . . 15 (𝑥 = (seq𝑀( + , 𝐹, 𝑆)‘𝐾) → 𝑥 = (seq𝑀( + , 𝐹, 𝑆)‘𝐾))
4644, 45eqeq12d 2097 . . . . . . . . . . . . . 14 (𝑥 = (seq𝑀( + , 𝐹, 𝑆)‘𝐾) → ((𝑥 + 𝑍) = 𝑥 ↔ ((seq𝑀( + , 𝐹, 𝑆)‘𝐾) + 𝑍) = (seq𝑀( + , 𝐹, 𝑆)‘𝐾)))
47 iseqid2.1 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑆) → (𝑥 + 𝑍) = 𝑥)
4847ralrimiva 2439 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑥𝑆 (𝑥 + 𝑍) = 𝑥)
49 iseqid2.4 . . . . . . . . . . . . . 14 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) ∈ 𝑆)
5046, 48, 49rspcdva 2716 . . . . . . . . . . . . 13 (𝜑 → ((seq𝑀( + , 𝐹, 𝑆)‘𝐾) + 𝑍) = (seq𝑀( + , 𝐹, 𝑆)‘𝐾))
5150adantr 270 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → ((seq𝑀( + , 𝐹, 𝑆)‘𝐾) + 𝑍) = (seq𝑀( + , 𝐹, 𝑆)‘𝐾))
5243, 51eqtr2d 2116 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = ((seq𝑀( + , 𝐹, 𝑆)‘𝐾) + (𝐹‘(𝑛 + 1))))
53 simprl 498 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → 𝑛 ∈ (ℤ𝐾))
54 iseqid2.2 . . . . . . . . . . . . . 14 (𝜑𝐾 ∈ (ℤ𝑀))
5554adantr 270 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → 𝐾 ∈ (ℤ𝑀))
56 uztrn 8752 . . . . . . . . . . . . 13 ((𝑛 ∈ (ℤ𝐾) ∧ 𝐾 ∈ (ℤ𝑀)) → 𝑛 ∈ (ℤ𝑀))
5753, 55, 56syl2anc 403 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → 𝑛 ∈ (ℤ𝑀))
58 iseqid2.f . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
5958adantlr 461 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
60 iseqid2.cl . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
6160adantlr 461 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
6257, 59, 61iseqp1 9574 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1))))
6352, 62eqeq12d 2097 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → ((seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)) ↔ ((seq𝑀( + , 𝐹, 𝑆)‘𝐾) + (𝐹‘(𝑛 + 1))) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))))
6430, 63syl5ibr 154 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → ((seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝑛) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))))
6564expr 367 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝐾)) → ((𝑛 + 1) ∈ (𝐾...𝑁) → ((seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝑛) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)))))
6665a2d 26 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝐾)) → (((𝑛 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝑛)) → ((𝑛 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)))))
6729, 66syld 44 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝐾)) → ((𝑛 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝑛)) → ((𝑛 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)))))
6867expcom 114 . . . . 5 (𝑛 ∈ (ℤ𝐾) → (𝜑 → ((𝑛 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝑛)) → ((𝑛 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))))))
6968a2d 26 . . . 4 (𝑛 ∈ (ℤ𝐾) → ((𝜑 → (𝑛 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝑛))) → (𝜑 → ((𝑛 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))))))
708, 13, 18, 23, 25, 69uzind4 8793 . . 3 (𝑁 ∈ (ℤ𝐾) → (𝜑 → (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝑁))))
711, 70mpcom 36 . 2 (𝜑 → (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝑁)))
723, 71mpd 13 1 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝑁))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102   = wceq 1285   ∈ wcel 1434  ∀wral 2353  ‘cfv 4953  (class class class)co 5564  1c1 7080   + caddc 7082  ℤcz 8468  ℤ≥cuz 8736  ...cfz 9141  seqcseq 9557 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3914  ax-sep 3917  ax-nul 3925  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309  ax-iinf 4358  ax-cnex 7165  ax-resscn 7166  ax-1cn 7167  ax-1re 7168  ax-icn 7169  ax-addcl 7170  ax-addrcl 7171  ax-mulcl 7172  ax-addcom 7174  ax-addass 7176  ax-distr 7178  ax-i2m1 7179  ax-0lt1 7180  ax-0id 7182  ax-rnegex 7183  ax-cnre 7185  ax-pre-ltirr 7186  ax-pre-ltwlin 7187  ax-pre-lttrn 7188  ax-pre-ltadd 7190 This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2826  df-csb 2919  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-nul 3269  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-int 3658  df-iun 3701  df-br 3807  df-opab 3861  df-mpt 3862  df-tr 3897  df-id 4077  df-iord 4150  df-on 4152  df-ilim 4153  df-suc 4155  df-iom 4361  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fun 4955  df-fn 4956  df-f 4957  df-f1 4958  df-fo 4959  df-f1o 4960  df-fv 4961  df-riota 5520  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-1st 5819  df-2nd 5820  df-recs 5975  df-frec 6061  df-pnf 7253  df-mnf 7254  df-xr 7255  df-ltxr 7256  df-le 7257  df-sub 7384  df-neg 7385  df-inn 8143  df-n0 8392  df-z 8469  df-uz 8737  df-fz 9142  df-iseq 9558 This theorem is referenced by:  fisumcvg  10385
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