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Theorem iseqsplit 9092
Description: Split a sequence into two sequences. (Contributed by Jim Kingdon, 16-Aug-2021.)
Hypotheses
Ref Expression
iseqsplit.1 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
iseqsplit.2 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
iseqsplit.3 (𝜑𝑁 ∈ (ℤ‘(𝑀 + 1)))
iseqsplit.s (𝜑𝑆𝑉)
iseqsplit.4 (𝜑𝑀 ∈ (ℤ𝐾))
iseqsplit.5 ((𝜑𝑥 ∈ (ℤ𝐾)) → (𝐹𝑥) ∈ 𝑆)
Assertion
Ref Expression
iseqsplit (𝜑 → (seq𝐾( + , 𝐹, 𝑆)‘𝑁) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑁)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐹   𝑥,𝐾,𝑦,𝑧   𝑥,𝑀,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥,𝑁,𝑦,𝑧   𝑥, + ,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧
Allowed substitution hints:   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem iseqsplit
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 iseqsplit.3 . . 3 (𝜑𝑁 ∈ (ℤ‘(𝑀 + 1)))
2 eluzfz2 8839 . . 3 (𝑁 ∈ (ℤ‘(𝑀 + 1)) → 𝑁 ∈ ((𝑀 + 1)...𝑁))
31, 2syl 14 . 2 (𝜑𝑁 ∈ ((𝑀 + 1)...𝑁))
4 eleq1 2100 . . . . . 6 (𝑥 = (𝑀 + 1) → (𝑥 ∈ ((𝑀 + 1)...𝑁) ↔ (𝑀 + 1) ∈ ((𝑀 + 1)...𝑁)))
5 fveq2 5141 . . . . . . 7 (𝑥 = (𝑀 + 1) → (seq𝐾( + , 𝐹, 𝑆)‘𝑥) = (seq𝐾( + , 𝐹, 𝑆)‘(𝑀 + 1)))
6 fveq2 5141 . . . . . . . 8 (𝑥 = (𝑀 + 1) → (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥) = (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑀 + 1)))
76oveq2d 5491 . . . . . . 7 (𝑥 = (𝑀 + 1) → ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑀 + 1))))
85, 7eqeq12d 2054 . . . . . 6 (𝑥 = (𝑀 + 1) → ((seq𝐾( + , 𝐹, 𝑆)‘𝑥) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥)) ↔ (seq𝐾( + , 𝐹, 𝑆)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑀 + 1)))))
94, 8imbi12d 223 . . . . 5 (𝑥 = (𝑀 + 1) → ((𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑥) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥))) ↔ ((𝑀 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑀 + 1))))))
109imbi2d 219 . . . 4 (𝑥 = (𝑀 + 1) → ((𝜑 → (𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑥) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥)))) ↔ (𝜑 → ((𝑀 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑀 + 1)))))))
11 eleq1 2100 . . . . . 6 (𝑥 = 𝑛 → (𝑥 ∈ ((𝑀 + 1)...𝑁) ↔ 𝑛 ∈ ((𝑀 + 1)...𝑁)))
12 fveq2 5141 . . . . . . 7 (𝑥 = 𝑛 → (seq𝐾( + , 𝐹, 𝑆)‘𝑥) = (seq𝐾( + , 𝐹, 𝑆)‘𝑛))
13 fveq2 5141 . . . . . . . 8 (𝑥 = 𝑛 → (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥) = (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛))
1413oveq2d 5491 . . . . . . 7 (𝑥 = 𝑛 → ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛)))
1512, 14eqeq12d 2054 . . . . . 6 (𝑥 = 𝑛 → ((seq𝐾( + , 𝐹, 𝑆)‘𝑥) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥)) ↔ (seq𝐾( + , 𝐹, 𝑆)‘𝑛) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛))))
1611, 15imbi12d 223 . . . . 5 (𝑥 = 𝑛 → ((𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑥) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥))) ↔ (𝑛 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑛) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛)))))
1716imbi2d 219 . . . 4 (𝑥 = 𝑛 → ((𝜑 → (𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑥) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥)))) ↔ (𝜑 → (𝑛 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑛) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛))))))
18 eleq1 2100 . . . . . 6 (𝑥 = (𝑛 + 1) → (𝑥 ∈ ((𝑀 + 1)...𝑁) ↔ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)))
19 fveq2 5141 . . . . . . 7 (𝑥 = (𝑛 + 1) → (seq𝐾( + , 𝐹, 𝑆)‘𝑥) = (seq𝐾( + , 𝐹, 𝑆)‘(𝑛 + 1)))
20 fveq2 5141 . . . . . . . 8 (𝑥 = (𝑛 + 1) → (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥) = (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑛 + 1)))
2120oveq2d 5491 . . . . . . 7 (𝑥 = (𝑛 + 1) → ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑛 + 1))))
2219, 21eqeq12d 2054 . . . . . 6 (𝑥 = (𝑛 + 1) → ((seq𝐾( + , 𝐹, 𝑆)‘𝑥) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥)) ↔ (seq𝐾( + , 𝐹, 𝑆)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑛 + 1)))))
2318, 22imbi12d 223 . . . . 5 (𝑥 = (𝑛 + 1) → ((𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑥) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥))) ↔ ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑛 + 1))))))
2423imbi2d 219 . . . 4 (𝑥 = (𝑛 + 1) → ((𝜑 → (𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑥) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥)))) ↔ (𝜑 → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑛 + 1)))))))
25 eleq1 2100 . . . . . 6 (𝑥 = 𝑁 → (𝑥 ∈ ((𝑀 + 1)...𝑁) ↔ 𝑁 ∈ ((𝑀 + 1)...𝑁)))
26 fveq2 5141 . . . . . . 7 (𝑥 = 𝑁 → (seq𝐾( + , 𝐹, 𝑆)‘𝑥) = (seq𝐾( + , 𝐹, 𝑆)‘𝑁))
27 fveq2 5141 . . . . . . . 8 (𝑥 = 𝑁 → (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥) = (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑁))
2827oveq2d 5491 . . . . . . 7 (𝑥 = 𝑁 → ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑁)))
2926, 28eqeq12d 2054 . . . . . 6 (𝑥 = 𝑁 → ((seq𝐾( + , 𝐹, 𝑆)‘𝑥) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥)) ↔ (seq𝐾( + , 𝐹, 𝑆)‘𝑁) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑁))))
3025, 29imbi12d 223 . . . . 5 (𝑥 = 𝑁 → ((𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑥) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥))) ↔ (𝑁 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑁) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑁)))))
3130imbi2d 219 . . . 4 (𝑥 = 𝑁 → ((𝜑 → (𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑥) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥)))) ↔ (𝜑 → (𝑁 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑁) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑁))))))
32 iseqsplit.4 . . . . . . . 8 (𝜑𝑀 ∈ (ℤ𝐾))
33 iseqsplit.s . . . . . . . 8 (𝜑𝑆𝑉)
34 iseqsplit.5 . . . . . . . 8 ((𝜑𝑥 ∈ (ℤ𝐾)) → (𝐹𝑥) ∈ 𝑆)
35 iseqsplit.1 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
3632, 33, 34, 35iseqp1 9079 . . . . . . 7 (𝜑 → (seq𝐾( + , 𝐹, 𝑆)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (𝐹‘(𝑀 + 1))))
37 eluzel2 8426 . . . . . . . . . 10 (𝑁 ∈ (ℤ‘(𝑀 + 1)) → (𝑀 + 1) ∈ ℤ)
381, 37syl 14 . . . . . . . . 9 (𝜑 → (𝑀 + 1) ∈ ℤ)
39 eluzelz 8430 . . . . . . . . . . . . 13 (𝑀 ∈ (ℤ𝐾) → 𝑀 ∈ ℤ)
4032, 39syl 14 . . . . . . . . . . . 12 (𝜑𝑀 ∈ ℤ)
41 peano2uzr 8476 . . . . . . . . . . . 12 ((𝑀 ∈ ℤ ∧ 𝑥 ∈ (ℤ‘(𝑀 + 1))) → 𝑥 ∈ (ℤ𝑀))
4240, 41sylan 267 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 𝑥 ∈ (ℤ𝑀))
4332adantr 261 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 𝑀 ∈ (ℤ𝐾))
44 uztrn 8437 . . . . . . . . . . 11 ((𝑥 ∈ (ℤ𝑀) ∧ 𝑀 ∈ (ℤ𝐾)) → 𝑥 ∈ (ℤ𝐾))
4542, 43, 44syl2anc 391 . . . . . . . . . 10 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 𝑥 ∈ (ℤ𝐾))
4645, 34syldan 266 . . . . . . . . 9 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → (𝐹𝑥) ∈ 𝑆)
4738, 33, 46, 35iseq1 9076 . . . . . . . 8 (𝜑 → (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑀 + 1)) = (𝐹‘(𝑀 + 1)))
4847oveq2d 5491 . . . . . . 7 (𝜑 → ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑀 + 1))) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (𝐹‘(𝑀 + 1))))
4936, 48eqtr4d 2075 . . . . . 6 (𝜑 → (seq𝐾( + , 𝐹, 𝑆)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑀 + 1))))
5049a1d 22 . . . . 5 (𝜑 → ((𝑀 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑀 + 1)))))
5150a1i 9 . . . 4 ((𝑀 + 1) ∈ ℤ → (𝜑 → ((𝑀 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑀 + 1))))))
52 peano2fzr 8844 . . . . . . . . . 10 ((𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)) → 𝑛 ∈ ((𝑀 + 1)...𝑁))
5352adantl 262 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝑛 ∈ ((𝑀 + 1)...𝑁))
5453expr 357 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ‘(𝑀 + 1))) → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → 𝑛 ∈ ((𝑀 + 1)...𝑁)))
5554imim1d 69 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ‘(𝑀 + 1))) → ((𝑛 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑛) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛))) → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑛) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛)))))
56 oveq1 5482 . . . . . . . . . 10 ((seq𝐾( + , 𝐹, 𝑆)‘𝑛) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛)) → ((seq𝐾( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1))) = (((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛)) + (𝐹‘(𝑛 + 1))))
57 simprl 483 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝑛 ∈ (ℤ‘(𝑀 + 1)))
58 peano2uz 8474 . . . . . . . . . . . . . . 15 (𝑀 ∈ (ℤ𝐾) → (𝑀 + 1) ∈ (ℤ𝐾))
5932, 58syl 14 . . . . . . . . . . . . . 14 (𝜑 → (𝑀 + 1) ∈ (ℤ𝐾))
6059adantr 261 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (𝑀 + 1) ∈ (ℤ𝐾))
61 uztrn 8437 . . . . . . . . . . . . 13 ((𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑀 + 1) ∈ (ℤ𝐾)) → 𝑛 ∈ (ℤ𝐾))
6257, 60, 61syl2anc 391 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝑛 ∈ (ℤ𝐾))
6333adantr 261 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝑆𝑉)
6434adantlr 446 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) ∧ 𝑥 ∈ (ℤ𝐾)) → (𝐹𝑥) ∈ 𝑆)
6535adantlr 446 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
6662, 63, 64, 65iseqp1 9079 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (seq𝐾( + , 𝐹, 𝑆)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1))))
6746adantlr 446 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) ∧ 𝑥 ∈ (ℤ‘(𝑀 + 1))) → (𝐹𝑥) ∈ 𝑆)
6857, 63, 67, 65iseqp1 9079 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑛 + 1)) = ((seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1))))
6968oveq2d 5491 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑛 + 1))) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + ((seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))))
70 simpl 102 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝜑)
7132, 33, 34, 35iseqcl 9077 . . . . . . . . . . . . . 14 (𝜑 → (seq𝐾( + , 𝐹, 𝑆)‘𝑀) ∈ 𝑆)
7271adantr 261 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (seq𝐾( + , 𝐹, 𝑆)‘𝑀) ∈ 𝑆)
7357, 63, 67, 65iseqcl 9077 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛) ∈ 𝑆)
74 fzss1 8869 . . . . . . . . . . . . . . . 16 ((𝑀 + 1) ∈ (ℤ𝐾) → ((𝑀 + 1)...𝑁) ⊆ (𝐾...𝑁))
7532, 58, 743syl 17 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑀 + 1)...𝑁) ⊆ (𝐾...𝑁))
76 simpr 103 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)) → (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))
77 ssel2 2937 . . . . . . . . . . . . . . 15 ((((𝑀 + 1)...𝑁) ⊆ (𝐾...𝑁) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)) → (𝑛 + 1) ∈ (𝐾...𝑁))
7875, 76, 77syl2an 273 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (𝑛 + 1) ∈ (𝐾...𝑁))
79 elfzuz 8829 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝐾...𝑁) → 𝑥 ∈ (ℤ𝐾))
8079, 34sylan2 270 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝐾...𝑁)) → (𝐹𝑥) ∈ 𝑆)
8180ralrimiva 2389 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑥 ∈ (𝐾...𝑁)(𝐹𝑥) ∈ 𝑆)
8281adantr 261 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ∀𝑥 ∈ (𝐾...𝑁)(𝐹𝑥) ∈ 𝑆)
83 fveq2 5141 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑛 + 1) → (𝐹𝑥) = (𝐹‘(𝑛 + 1)))
8483eleq1d 2106 . . . . . . . . . . . . . . 15 (𝑥 = (𝑛 + 1) → ((𝐹𝑥) ∈ 𝑆 ↔ (𝐹‘(𝑛 + 1)) ∈ 𝑆))
8584rspcv 2649 . . . . . . . . . . . . . 14 ((𝑛 + 1) ∈ (𝐾...𝑁) → (∀𝑥 ∈ (𝐾...𝑁)(𝐹𝑥) ∈ 𝑆 → (𝐹‘(𝑛 + 1)) ∈ 𝑆))
8678, 82, 85sylc 56 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (𝐹‘(𝑛 + 1)) ∈ 𝑆)
87 iseqsplit.2 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
8887caovassg 5622 . . . . . . . . . . . . 13 ((𝜑 ∧ ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) ∈ 𝑆 ∧ (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛) ∈ 𝑆 ∧ (𝐹‘(𝑛 + 1)) ∈ 𝑆)) → (((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛)) + (𝐹‘(𝑛 + 1))) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + ((seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))))
8970, 72, 73, 86, 88syl13anc 1137 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛)) + (𝐹‘(𝑛 + 1))) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + ((seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))))
9069, 89eqtr4d 2075 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑛 + 1))) = (((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛)) + (𝐹‘(𝑛 + 1))))
9166, 90eqeq12d 2054 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ((seq𝐾( + , 𝐹, 𝑆)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑛 + 1))) ↔ ((seq𝐾( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1))) = (((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛)) + (𝐹‘(𝑛 + 1)))))
9256, 91syl5ibr 145 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ((seq𝐾( + , 𝐹, 𝑆)‘𝑛) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛)) → (seq𝐾( + , 𝐹, 𝑆)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑛 + 1)))))
9392expr 357 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ‘(𝑀 + 1))) → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → ((seq𝐾( + , 𝐹, 𝑆)‘𝑛) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛)) → (seq𝐾( + , 𝐹, 𝑆)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑛 + 1))))))
9493a2d 23 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ‘(𝑀 + 1))) → (((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑛) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛))) → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑛 + 1))))))
9555, 94syld 40 . . . . . 6 ((𝜑𝑛 ∈ (ℤ‘(𝑀 + 1))) → ((𝑛 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑛) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛))) → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑛 + 1))))))
9695expcom 109 . . . . 5 (𝑛 ∈ (ℤ‘(𝑀 + 1)) → (𝜑 → ((𝑛 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑛) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛))) → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑛 + 1)))))))
9796a2d 23 . . . 4 (𝑛 ∈ (ℤ‘(𝑀 + 1)) → ((𝜑 → (𝑛 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑛) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛)))) → (𝜑 → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑛 + 1)))))))
9810, 17, 24, 31, 51, 97uzind4 8479 . . 3 (𝑁 ∈ (ℤ‘(𝑀 + 1)) → (𝜑 → (𝑁 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑁) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑁)))))
991, 98mpcom 32 . 2 (𝜑 → (𝑁 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑁) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑁))))
1003, 99mpd 13 1 (𝜑 → (seq𝐾( + , 𝐹, 𝑆)‘𝑁) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑁)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  w3a 885   = wceq 1243  wcel 1393  wral 2303  wss 2914  cfv 4865  (class class class)co 5475  1c1 6847   + caddc 6849  cz 8193  cuz 8421  ...cfz 8817  seqcseq 9065
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3869  ax-sep 3872  ax-nul 3880  ax-pow 3924  ax-pr 3941  ax-un 4142  ax-setind 4232  ax-iinf 4274  ax-cnex 6932  ax-resscn 6933  ax-1cn 6934  ax-1re 6935  ax-icn 6936  ax-addcl 6937  ax-addrcl 6938  ax-mulcl 6939  ax-addcom 6941  ax-addass 6943  ax-distr 6945  ax-i2m1 6946  ax-0id 6949  ax-rnegex 6950  ax-cnre 6952  ax-pre-ltirr 6953  ax-pre-ltwlin 6954  ax-pre-lttrn 6955  ax-pre-ltadd 6957
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2308  df-rex 2309  df-reu 2310  df-rab 2312  df-v 2556  df-sbc 2762  df-csb 2850  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-int 3613  df-iun 3656  df-br 3762  df-opab 3816  df-mpt 3817  df-tr 3852  df-eprel 4023  df-id 4027  df-po 4030  df-iso 4031  df-iord 4075  df-on 4077  df-suc 4080  df-iom 4277  df-xp 4314  df-rel 4315  df-cnv 4316  df-co 4317  df-dm 4318  df-rn 4319  df-res 4320  df-ima 4321  df-iota 4830  df-fun 4867  df-fn 4868  df-f 4869  df-f1 4870  df-fo 4871  df-f1o 4872  df-fv 4873  df-riota 5431  df-ov 5478  df-oprab 5479  df-mpt2 5480  df-1st 5730  df-2nd 5731  df-recs 5883  df-irdg 5920  df-frec 5941  df-1o 5964  df-2o 5965  df-oadd 5968  df-omul 5969  df-er 6069  df-ec 6071  df-qs 6075  df-ni 6359  df-pli 6360  df-mi 6361  df-lti 6362  df-plpq 6399  df-mpq 6400  df-enq 6402  df-nqqs 6403  df-plqqs 6404  df-mqqs 6405  df-1nqqs 6406  df-rq 6407  df-ltnqqs 6408  df-enq0 6479  df-nq0 6480  df-0nq0 6481  df-plq0 6482  df-mq0 6483  df-inp 6521  df-i1p 6522  df-iplp 6523  df-iltp 6525  df-enr 6768  df-nr 6769  df-ltr 6772  df-0r 6773  df-1r 6774  df-0 6853  df-1 6854  df-r 6856  df-lt 6859  df-pnf 7018  df-mnf 7019  df-xr 7020  df-ltxr 7021  df-le 7022  df-sub 7140  df-neg 7141  df-inn 7867  df-n0 8130  df-z 8194  df-uz 8422  df-fz 8818  df-iseq 9066
This theorem is referenced by:  iseq1p  9093  clim2iser  9710  clim2iser2  9711
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