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Theorem iseqsplit 9212
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 8894 . . 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 5178 . . . . . . 7 (𝑥 = (𝑀 + 1) → (seq𝐾( + , 𝐹, 𝑆)‘𝑥) = (seq𝐾( + , 𝐹, 𝑆)‘(𝑀 + 1)))
6 fveq2 5178 . . . . . . . 8 (𝑥 = (𝑀 + 1) → (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥) = (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑀 + 1)))
76oveq2d 5528 . . . . . . 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 5178 . . . . . . 7 (𝑥 = 𝑛 → (seq𝐾( + , 𝐹, 𝑆)‘𝑥) = (seq𝐾( + , 𝐹, 𝑆)‘𝑛))
13 fveq2 5178 . . . . . . . 8 (𝑥 = 𝑛 → (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥) = (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛))
1413oveq2d 5528 . . . . . . 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 5178 . . . . . . 7 (𝑥 = (𝑛 + 1) → (seq𝐾( + , 𝐹, 𝑆)‘𝑥) = (seq𝐾( + , 𝐹, 𝑆)‘(𝑛 + 1)))
20 fveq2 5178 . . . . . . . 8 (𝑥 = (𝑛 + 1) → (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥) = (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑛 + 1)))
2120oveq2d 5528 . . . . . . 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 5178 . . . . . . 7 (𝑥 = 𝑁 → (seq𝐾( + , 𝐹, 𝑆)‘𝑥) = (seq𝐾( + , 𝐹, 𝑆)‘𝑁))
27 fveq2 5178 . . . . . . . 8 (𝑥 = 𝑁 → (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥) = (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑁))
2827oveq2d 5528 . . . . . . 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 9199 . . . . . . 7 (𝜑 → (seq𝐾( + , 𝐹, 𝑆)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (𝐹‘(𝑀 + 1))))
37 eluzel2 8476 . . . . . . . . . 10 (𝑁 ∈ (ℤ‘(𝑀 + 1)) → (𝑀 + 1) ∈ ℤ)
381, 37syl 14 . . . . . . . . 9 (𝜑 → (𝑀 + 1) ∈ ℤ)
39 eluzelz 8480 . . . . . . . . . . . . 13 (𝑀 ∈ (ℤ𝐾) → 𝑀 ∈ ℤ)
4032, 39syl 14 . . . . . . . . . . . 12 (𝜑𝑀 ∈ ℤ)
41 peano2uzr 8526 . . . . . . . . . . . 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 8487 . . . . . . . . . . 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 9196 . . . . . . . 8 (𝜑 → (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑀 + 1)) = (𝐹‘(𝑀 + 1)))
4847oveq2d 5528 . . . . . . 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 8899 . . . . . . . . . 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 5519 . . . . . . . . . 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 8524 . . . . . . . . . . . . . . 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 8487 . . . . . . . . . . . . 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 9199 . . . . . . . . . . 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 9199 . . . . . . . . . . . . 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 5528 . . . . . . . . . . . 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 9197 . . . . . . . . . . . . . 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 9197 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛) ∈ 𝑆)
74 fzss1 8924 . . . . . . . . . . . . . . . 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 2940 . . . . . . . . . . . . . . 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 8884 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝐾...𝑁) → 𝑥 ∈ (ℤ𝐾))
8079, 34sylan2 270 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝐾...𝑁)) → (𝐹𝑥) ∈ 𝑆)
8180ralrimiva 2392 . . . . . . . . . . . . . . 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 5178 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑛 + 1) → (𝐹𝑥) = (𝐹‘(𝑛 + 1)))
8483eleq1d 2106 . . . . . . . . . . . . . . 15 (𝑥 = (𝑛 + 1) → ((𝐹𝑥) ∈ 𝑆 ↔ (𝐹‘(𝑛 + 1)) ∈ 𝑆))
8584rspcv 2652 . . . . . . . . . . . . . 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 5659 . . . . . . . . . . . . 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 8529 . . 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 2306  wss 2917  cfv 4902  (class class class)co 5512  1c1 6888   + caddc 6890  cz 8243  cuz 8471  ...cfz 8872  seqcseq 9185
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 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311  ax-cnex 6973  ax-resscn 6974  ax-1cn 6975  ax-1re 6976  ax-icn 6977  ax-addcl 6978  ax-addrcl 6979  ax-mulcl 6980  ax-addcom 6982  ax-addass 6984  ax-distr 6986  ax-i2m1 6987  ax-0id 6990  ax-rnegex 6991  ax-cnre 6993  ax-pre-ltirr 6994  ax-pre-ltwlin 6995  ax-pre-lttrn 6996  ax-pre-ltadd 6998
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 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-frec 5978  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6400  df-pli 6401  df-mi 6402  df-lti 6403  df-plpq 6440  df-mpq 6441  df-enq 6443  df-nqqs 6444  df-plqqs 6445  df-mqqs 6446  df-1nqqs 6447  df-rq 6448  df-ltnqqs 6449  df-enq0 6520  df-nq0 6521  df-0nq0 6522  df-plq0 6523  df-mq0 6524  df-inp 6562  df-i1p 6563  df-iplp 6564  df-iltp 6566  df-enr 6809  df-nr 6810  df-ltr 6813  df-0r 6814  df-1r 6815  df-0 6894  df-1 6895  df-r 6897  df-lt 6900  df-pnf 7060  df-mnf 7061  df-xr 7062  df-ltxr 7063  df-le 7064  df-sub 7182  df-neg 7183  df-inn 7913  df-n0 8180  df-z 8244  df-uz 8472  df-fz 8873  df-iseq 9186
This theorem is referenced by:  iseq1p  9213  clim2iser  9830  clim2iser2  9831
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