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Theorem iseqsplit 9773
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 9341 . . 3 (𝑁 ∈ (ℤ‘(𝑀 + 1)) → 𝑁 ∈ ((𝑀 + 1)...𝑁))
31, 2syl 14 . 2 (𝜑𝑁 ∈ ((𝑀 + 1)...𝑁))
4 eleq1 2145 . . . . . 6 (𝑥 = (𝑀 + 1) → (𝑥 ∈ ((𝑀 + 1)...𝑁) ↔ (𝑀 + 1) ∈ ((𝑀 + 1)...𝑁)))
5 fveq2 5253 . . . . . . 7 (𝑥 = (𝑀 + 1) → (seq𝐾( + , 𝐹, 𝑆)‘𝑥) = (seq𝐾( + , 𝐹, 𝑆)‘(𝑀 + 1)))
6 fveq2 5253 . . . . . . . 8 (𝑥 = (𝑀 + 1) → (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥) = (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑀 + 1)))
76oveq2d 5607 . . . . . . 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 2097 . . . . . 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 232 . . . . 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 228 . . . 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 2145 . . . . . 6 (𝑥 = 𝑛 → (𝑥 ∈ ((𝑀 + 1)...𝑁) ↔ 𝑛 ∈ ((𝑀 + 1)...𝑁)))
12 fveq2 5253 . . . . . . 7 (𝑥 = 𝑛 → (seq𝐾( + , 𝐹, 𝑆)‘𝑥) = (seq𝐾( + , 𝐹, 𝑆)‘𝑛))
13 fveq2 5253 . . . . . . . 8 (𝑥 = 𝑛 → (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥) = (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛))
1413oveq2d 5607 . . . . . . 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 2097 . . . . . 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 232 . . . . 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 228 . . . 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 2145 . . . . . 6 (𝑥 = (𝑛 + 1) → (𝑥 ∈ ((𝑀 + 1)...𝑁) ↔ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)))
19 fveq2 5253 . . . . . . 7 (𝑥 = (𝑛 + 1) → (seq𝐾( + , 𝐹, 𝑆)‘𝑥) = (seq𝐾( + , 𝐹, 𝑆)‘(𝑛 + 1)))
20 fveq2 5253 . . . . . . . 8 (𝑥 = (𝑛 + 1) → (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥) = (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑛 + 1)))
2120oveq2d 5607 . . . . . . 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 2097 . . . . . 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 232 . . . . 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 228 . . . 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 2145 . . . . . 6 (𝑥 = 𝑁 → (𝑥 ∈ ((𝑀 + 1)...𝑁) ↔ 𝑁 ∈ ((𝑀 + 1)...𝑁)))
26 fveq2 5253 . . . . . . 7 (𝑥 = 𝑁 → (seq𝐾( + , 𝐹, 𝑆)‘𝑥) = (seq𝐾( + , 𝐹, 𝑆)‘𝑁))
27 fveq2 5253 . . . . . . . 8 (𝑥 = 𝑁 → (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥) = (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑁))
2827oveq2d 5607 . . . . . . 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 2097 . . . . . 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 232 . . . . 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 228 . . . 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 . . . . . . 7 (𝜑𝑀 ∈ (ℤ𝐾))
33 iseqsplit.5 . . . . . . 7 ((𝜑𝑥 ∈ (ℤ𝐾)) → (𝐹𝑥) ∈ 𝑆)
34 iseqsplit.1 . . . . . . 7 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
3532, 33, 34iseqp1 9757 . . . . . 6 (𝜑 → (seq𝐾( + , 𝐹, 𝑆)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (𝐹‘(𝑀 + 1))))
36 eluzel2 8919 . . . . . . . . 9 (𝑁 ∈ (ℤ‘(𝑀 + 1)) → (𝑀 + 1) ∈ ℤ)
371, 36syl 14 . . . . . . . 8 (𝜑 → (𝑀 + 1) ∈ ℤ)
38 eluzelz 8923 . . . . . . . . . . . 12 (𝑀 ∈ (ℤ𝐾) → 𝑀 ∈ ℤ)
3932, 38syl 14 . . . . . . . . . . 11 (𝜑𝑀 ∈ ℤ)
40 peano2uzr 8968 . . . . . . . . . . 11 ((𝑀 ∈ ℤ ∧ 𝑥 ∈ (ℤ‘(𝑀 + 1))) → 𝑥 ∈ (ℤ𝑀))
4139, 40sylan 277 . . . . . . . . . 10 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 𝑥 ∈ (ℤ𝑀))
4232adantr 270 . . . . . . . . . 10 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 𝑀 ∈ (ℤ𝐾))
43 uztrn 8930 . . . . . . . . . 10 ((𝑥 ∈ (ℤ𝑀) ∧ 𝑀 ∈ (ℤ𝐾)) → 𝑥 ∈ (ℤ𝐾))
4441, 42, 43syl2anc 403 . . . . . . . . 9 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 𝑥 ∈ (ℤ𝐾))
4544, 33syldan 276 . . . . . . . 8 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → (𝐹𝑥) ∈ 𝑆)
4637, 45, 34iseq1 9752 . . . . . . 7 (𝜑 → (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑀 + 1)) = (𝐹‘(𝑀 + 1)))
4746oveq2d 5607 . . . . . 6 (𝜑 → ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑀 + 1))) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (𝐹‘(𝑀 + 1))))
4835, 47eqtr4d 2118 . . . . 5 (𝜑 → (seq𝐾( + , 𝐹, 𝑆)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑀 + 1))))
4948a1i13 24 . . . 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))))))
50 peano2fzr 9346 . . . . . . . . . 10 ((𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)) → 𝑛 ∈ ((𝑀 + 1)...𝑁))
5150adantl 271 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝑛 ∈ ((𝑀 + 1)...𝑁))
5251expr 367 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ‘(𝑀 + 1))) → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → 𝑛 ∈ ((𝑀 + 1)...𝑁)))
5352imim1d 74 . . . . . . 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)))))
54 oveq1 5598 . . . . . . . . . 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))))
55 simprl 498 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝑛 ∈ (ℤ‘(𝑀 + 1)))
56 peano2uz 8966 . . . . . . . . . . . . . . 15 (𝑀 ∈ (ℤ𝐾) → (𝑀 + 1) ∈ (ℤ𝐾))
5732, 56syl 14 . . . . . . . . . . . . . 14 (𝜑 → (𝑀 + 1) ∈ (ℤ𝐾))
5857adantr 270 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (𝑀 + 1) ∈ (ℤ𝐾))
59 uztrn 8930 . . . . . . . . . . . . 13 ((𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑀 + 1) ∈ (ℤ𝐾)) → 𝑛 ∈ (ℤ𝐾))
6055, 58, 59syl2anc 403 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝑛 ∈ (ℤ𝐾))
6133adantlr 461 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) ∧ 𝑥 ∈ (ℤ𝐾)) → (𝐹𝑥) ∈ 𝑆)
6234adantlr 461 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
6360, 61, 62iseqp1 9757 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (seq𝐾( + , 𝐹, 𝑆)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1))))
6445adantlr 461 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) ∧ 𝑥 ∈ (ℤ‘(𝑀 + 1))) → (𝐹𝑥) ∈ 𝑆)
6555, 64, 62iseqp1 9757 . . . . . . . . . . . . 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))))
6665oveq2d 5607 . . . . . . . . . . . 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)))))
67 simpl 107 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝜑)
6832, 33, 34iseqcl 9756 . . . . . . . . . . . . . 14 (𝜑 → (seq𝐾( + , 𝐹, 𝑆)‘𝑀) ∈ 𝑆)
6968adantr 270 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (seq𝐾( + , 𝐹, 𝑆)‘𝑀) ∈ 𝑆)
7055, 64, 62iseqcl 9756 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛) ∈ 𝑆)
71 fveq2 5253 . . . . . . . . . . . . . . 15 (𝑥 = (𝑛 + 1) → (𝐹𝑥) = (𝐹‘(𝑛 + 1)))
7271eleq1d 2151 . . . . . . . . . . . . . 14 (𝑥 = (𝑛 + 1) → ((𝐹𝑥) ∈ 𝑆 ↔ (𝐹‘(𝑛 + 1)) ∈ 𝑆))
73 elfzuz 9331 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝐾...𝑁) → 𝑥 ∈ (ℤ𝐾))
7473, 33sylan2 280 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝐾...𝑁)) → (𝐹𝑥) ∈ 𝑆)
7574ralrimiva 2440 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑥 ∈ (𝐾...𝑁)(𝐹𝑥) ∈ 𝑆)
7675adantr 270 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ∀𝑥 ∈ (𝐾...𝑁)(𝐹𝑥) ∈ 𝑆)
77 fzss1 9371 . . . . . . . . . . . . . . . 16 ((𝑀 + 1) ∈ (ℤ𝐾) → ((𝑀 + 1)...𝑁) ⊆ (𝐾...𝑁))
7832, 56, 773syl 17 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑀 + 1)...𝑁) ⊆ (𝐾...𝑁))
79 simpr 108 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)) → (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))
80 ssel2 3005 . . . . . . . . . . . . . . 15 ((((𝑀 + 1)...𝑁) ⊆ (𝐾...𝑁) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)) → (𝑛 + 1) ∈ (𝐾...𝑁))
8178, 79, 80syl2an 283 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (𝑛 + 1) ∈ (𝐾...𝑁))
8272, 76, 81rspcdva 2717 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (𝐹‘(𝑛 + 1)) ∈ 𝑆)
83 iseqsplit.2 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
8483caovassg 5738 . . . . . . . . . . . . 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)))))
8567, 69, 70, 82, 84syl13anc 1172 . . . . . . . . . . . 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)))))
8666, 85eqtr4d 2118 . . . . . . . . . . 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))))
8763, 86eqeq12d 2097 . . . . . . . . . 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)))))
8854, 87syl5ibr 154 . . . . . . . . 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)))))
8988expr 367 . . . . . . . 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))))))
9089a2d 26 . . . . . . 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))))))
9153, 90syld 44 . . . . . 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))))))
9291expcom 114 . . . . 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)))))))
9392a2d 26 . . . 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)))))))
9410, 17, 24, 31, 49, 93uzind4 8971 . . 3 (𝑁 ∈ (ℤ‘(𝑀 + 1)) → (𝜑 → (𝑁 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑁) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑁)))))
951, 94mpcom 36 . 2 (𝜑 → (𝑁 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑁) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑁))))
963, 95mpd 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 102  w3a 920   = wceq 1285  wcel 1434  wral 2353  wss 2984  cfv 4969  (class class class)co 5591  1c1 7254   + caddc 7256  cz 8646  cuz 8914  ...cfz 9319  seqcseq 9740
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 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366  ax-cnex 7339  ax-resscn 7340  ax-1cn 7341  ax-1re 7342  ax-icn 7343  ax-addcl 7344  ax-addrcl 7345  ax-mulcl 7346  ax-addcom 7348  ax-addass 7350  ax-distr 7352  ax-i2m1 7353  ax-0lt1 7354  ax-0id 7356  ax-rnegex 7357  ax-cnre 7359  ax-pre-ltirr 7360  ax-pre-ltwlin 7361  ax-pre-lttrn 7362  ax-pre-ltadd 7364
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 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4084  df-iord 4157  df-on 4159  df-ilim 4160  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-riota 5547  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-frec 6088  df-pnf 7427  df-mnf 7428  df-xr 7429  df-ltxr 7430  df-le 7431  df-sub 7558  df-neg 7559  df-inn 8317  df-n0 8566  df-z 8647  df-uz 8915  df-fz 9320  df-iseq 9741
This theorem is referenced by:  iseq1p  9774  ibcval5  10006  clim2iser  10549  clim2iser2  10550
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