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Theorem seqsplit 13609
Description: Split a sequence into two sequences. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)
Hypotheses
Ref Expression
seqsplit.1 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
seqsplit.2 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
seqsplit.3 (𝜑𝑁 ∈ (ℤ‘(𝑀 + 1)))
seqsplit.4 (𝜑𝑀 ∈ (ℤ𝐾))
seqsplit.5 ((𝜑𝑥 ∈ (𝐾...𝑁)) → (𝐹𝑥) ∈ 𝑆)
Assertion
Ref Expression
seqsplit (𝜑 → (seq𝐾( + , 𝐹)‘𝑁) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘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

Proof of Theorem seqsplit
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 seqsplit.3 . . 3 (𝜑𝑁 ∈ (ℤ‘(𝑀 + 1)))
2 eluzfz2 13120 . . 3 (𝑁 ∈ (ℤ‘(𝑀 + 1)) → 𝑁 ∈ ((𝑀 + 1)...𝑁))
31, 2syl 17 . 2 (𝜑𝑁 ∈ ((𝑀 + 1)...𝑁))
4 eleq1 2825 . . . . . 6 (𝑥 = (𝑀 + 1) → (𝑥 ∈ ((𝑀 + 1)...𝑁) ↔ (𝑀 + 1) ∈ ((𝑀 + 1)...𝑁)))
5 fveq2 6717 . . . . . . 7 (𝑥 = (𝑀 + 1) → (seq𝐾( + , 𝐹)‘𝑥) = (seq𝐾( + , 𝐹)‘(𝑀 + 1)))
6 fveq2 6717 . . . . . . . 8 (𝑥 = (𝑀 + 1) → (seq(𝑀 + 1)( + , 𝐹)‘𝑥) = (seq(𝑀 + 1)( + , 𝐹)‘(𝑀 + 1)))
76oveq2d 7229 . . . . . . 7 (𝑥 = (𝑀 + 1) → ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑀 + 1))))
85, 7eqeq12d 2753 . . . . . 6 (𝑥 = (𝑀 + 1) → ((seq𝐾( + , 𝐹)‘𝑥) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)) ↔ (seq𝐾( + , 𝐹)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑀 + 1)))))
94, 8imbi12d 348 . . . . 5 (𝑥 = (𝑀 + 1) → ((𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑥) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥))) ↔ ((𝑀 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑀 + 1))))))
109imbi2d 344 . . . 4 (𝑥 = (𝑀 + 1) → ((𝜑 → (𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑥) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)))) ↔ (𝜑 → ((𝑀 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑀 + 1)))))))
11 eleq1 2825 . . . . . 6 (𝑥 = 𝑛 → (𝑥 ∈ ((𝑀 + 1)...𝑁) ↔ 𝑛 ∈ ((𝑀 + 1)...𝑁)))
12 fveq2 6717 . . . . . . 7 (𝑥 = 𝑛 → (seq𝐾( + , 𝐹)‘𝑥) = (seq𝐾( + , 𝐹)‘𝑛))
13 fveq2 6717 . . . . . . . 8 (𝑥 = 𝑛 → (seq(𝑀 + 1)( + , 𝐹)‘𝑥) = (seq(𝑀 + 1)( + , 𝐹)‘𝑛))
1413oveq2d 7229 . . . . . . 7 (𝑥 = 𝑛 → ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛)))
1512, 14eqeq12d 2753 . . . . . 6 (𝑥 = 𝑛 → ((seq𝐾( + , 𝐹)‘𝑥) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)) ↔ (seq𝐾( + , 𝐹)‘𝑛) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛))))
1611, 15imbi12d 348 . . . . 5 (𝑥 = 𝑛 → ((𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑥) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥))) ↔ (𝑛 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑛) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛)))))
1716imbi2d 344 . . . 4 (𝑥 = 𝑛 → ((𝜑 → (𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑥) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)))) ↔ (𝜑 → (𝑛 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑛) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛))))))
18 eleq1 2825 . . . . . 6 (𝑥 = (𝑛 + 1) → (𝑥 ∈ ((𝑀 + 1)...𝑁) ↔ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)))
19 fveq2 6717 . . . . . . 7 (𝑥 = (𝑛 + 1) → (seq𝐾( + , 𝐹)‘𝑥) = (seq𝐾( + , 𝐹)‘(𝑛 + 1)))
20 fveq2 6717 . . . . . . . 8 (𝑥 = (𝑛 + 1) → (seq(𝑀 + 1)( + , 𝐹)‘𝑥) = (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1)))
2120oveq2d 7229 . . . . . . 7 (𝑥 = (𝑛 + 1) → ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1))))
2219, 21eqeq12d 2753 . . . . . 6 (𝑥 = (𝑛 + 1) → ((seq𝐾( + , 𝐹)‘𝑥) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)) ↔ (seq𝐾( + , 𝐹)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1)))))
2318, 22imbi12d 348 . . . . 5 (𝑥 = (𝑛 + 1) → ((𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑥) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥))) ↔ ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1))))))
2423imbi2d 344 . . . 4 (𝑥 = (𝑛 + 1) → ((𝜑 → (𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑥) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)))) ↔ (𝜑 → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1)))))))
25 eleq1 2825 . . . . . 6 (𝑥 = 𝑁 → (𝑥 ∈ ((𝑀 + 1)...𝑁) ↔ 𝑁 ∈ ((𝑀 + 1)...𝑁)))
26 fveq2 6717 . . . . . . 7 (𝑥 = 𝑁 → (seq𝐾( + , 𝐹)‘𝑥) = (seq𝐾( + , 𝐹)‘𝑁))
27 fveq2 6717 . . . . . . . 8 (𝑥 = 𝑁 → (seq(𝑀 + 1)( + , 𝐹)‘𝑥) = (seq(𝑀 + 1)( + , 𝐹)‘𝑁))
2827oveq2d 7229 . . . . . . 7 (𝑥 = 𝑁 → ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁)))
2926, 28eqeq12d 2753 . . . . . 6 (𝑥 = 𝑁 → ((seq𝐾( + , 𝐹)‘𝑥) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)) ↔ (seq𝐾( + , 𝐹)‘𝑁) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁))))
3025, 29imbi12d 348 . . . . 5 (𝑥 = 𝑁 → ((𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑥) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥))) ↔ (𝑁 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑁) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁)))))
3130imbi2d 344 . . . 4 (𝑥 = 𝑁 → ((𝜑 → (𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑥) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)))) ↔ (𝜑 → (𝑁 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑁) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁))))))
32 seqsplit.4 . . . . . . 7 (𝜑𝑀 ∈ (ℤ𝐾))
33 seqp1 13589 . . . . . . 7 (𝑀 ∈ (ℤ𝐾) → (seq𝐾( + , 𝐹)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (𝐹‘(𝑀 + 1))))
3432, 33syl 17 . . . . . 6 (𝜑 → (seq𝐾( + , 𝐹)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (𝐹‘(𝑀 + 1))))
35 eluzel2 12443 . . . . . . . 8 (𝑁 ∈ (ℤ‘(𝑀 + 1)) → (𝑀 + 1) ∈ ℤ)
36 seq1 13587 . . . . . . . 8 ((𝑀 + 1) ∈ ℤ → (seq(𝑀 + 1)( + , 𝐹)‘(𝑀 + 1)) = (𝐹‘(𝑀 + 1)))
371, 35, 363syl 18 . . . . . . 7 (𝜑 → (seq(𝑀 + 1)( + , 𝐹)‘(𝑀 + 1)) = (𝐹‘(𝑀 + 1)))
3837oveq2d 7229 . . . . . 6 (𝜑 → ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑀 + 1))) = ((seq𝐾( + , 𝐹)‘𝑀) + (𝐹‘(𝑀 + 1))))
3934, 38eqtr4d 2780 . . . . 5 (𝜑 → (seq𝐾( + , 𝐹)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑀 + 1))))
4039a1i13 27 . . . 4 ((𝑀 + 1) ∈ ℤ → (𝜑 → ((𝑀 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑀 + 1))))))
41 peano2fzr 13125 . . . . . . . 8 ((𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)) → 𝑛 ∈ ((𝑀 + 1)...𝑁))
4241adantl 485 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝑛 ∈ ((𝑀 + 1)...𝑁))
4342expr 460 . . . . . 6 ((𝜑𝑛 ∈ (ℤ‘(𝑀 + 1))) → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → 𝑛 ∈ ((𝑀 + 1)...𝑁)))
4443imim1d 82 . . . . 5 ((𝜑𝑛 ∈ (ℤ‘(𝑀 + 1))) → ((𝑛 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑛) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛))) → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑛) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛)))))
45 oveq1 7220 . . . . . 6 ((seq𝐾( + , 𝐹)‘𝑛) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛)) → ((seq𝐾( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))) = (((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛)) + (𝐹‘(𝑛 + 1))))
46 simprl 771 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝑛 ∈ (ℤ‘(𝑀 + 1)))
47 peano2uz 12497 . . . . . . . . . . 11 (𝑀 ∈ (ℤ𝐾) → (𝑀 + 1) ∈ (ℤ𝐾))
4832, 47syl 17 . . . . . . . . . 10 (𝜑 → (𝑀 + 1) ∈ (ℤ𝐾))
4948adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (𝑀 + 1) ∈ (ℤ𝐾))
50 uztrn 12456 . . . . . . . . 9 ((𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑀 + 1) ∈ (ℤ𝐾)) → 𝑛 ∈ (ℤ𝐾))
5146, 49, 50syl2anc 587 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝑛 ∈ (ℤ𝐾))
52 seqp1 13589 . . . . . . . 8 (𝑛 ∈ (ℤ𝐾) → (seq𝐾( + , 𝐹)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
5351, 52syl 17 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (seq𝐾( + , 𝐹)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
54 seqp1 13589 . . . . . . . . . 10 (𝑛 ∈ (ℤ‘(𝑀 + 1)) → (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1)) = ((seq(𝑀 + 1)( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
5546, 54syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1)) = ((seq(𝑀 + 1)( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
5655oveq2d 7229 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1))) = ((seq𝐾( + , 𝐹)‘𝑀) + ((seq(𝑀 + 1)( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
57 simpl 486 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝜑)
58 eluzelz 12448 . . . . . . . . . . . . . . . 16 (𝑀 ∈ (ℤ𝐾) → 𝑀 ∈ ℤ)
5932, 58syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑀 ∈ ℤ)
60 peano2uzr 12499 . . . . . . . . . . . . . . 15 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ‘(𝑀 + 1))) → 𝑁 ∈ (ℤ𝑀))
6159, 1, 60syl2anc 587 . . . . . . . . . . . . . 14 (𝜑𝑁 ∈ (ℤ𝑀))
62 fzss2 13152 . . . . . . . . . . . . . 14 (𝑁 ∈ (ℤ𝑀) → (𝐾...𝑀) ⊆ (𝐾...𝑁))
6361, 62syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝐾...𝑀) ⊆ (𝐾...𝑁))
6463sselda 3901 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐾...𝑀)) → 𝑥 ∈ (𝐾...𝑁))
65 seqsplit.5 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐾...𝑁)) → (𝐹𝑥) ∈ 𝑆)
6664, 65syldan 594 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐾...𝑀)) → (𝐹𝑥) ∈ 𝑆)
67 seqsplit.1 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
6832, 66, 67seqcl 13596 . . . . . . . . . 10 (𝜑 → (seq𝐾( + , 𝐹)‘𝑀) ∈ 𝑆)
6968adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (seq𝐾( + , 𝐹)‘𝑀) ∈ 𝑆)
70 elfzuz3 13109 . . . . . . . . . . . . . 14 (𝑛 ∈ ((𝑀 + 1)...𝑁) → 𝑁 ∈ (ℤ𝑛))
71 fzss2 13152 . . . . . . . . . . . . . 14 (𝑁 ∈ (ℤ𝑛) → ((𝑀 + 1)...𝑛) ⊆ ((𝑀 + 1)...𝑁))
7242, 70, 713syl 18 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ((𝑀 + 1)...𝑛) ⊆ ((𝑀 + 1)...𝑁))
73 fzss1 13151 . . . . . . . . . . . . . . 15 ((𝑀 + 1) ∈ (ℤ𝐾) → ((𝑀 + 1)...𝑁) ⊆ (𝐾...𝑁))
7432, 47, 733syl 18 . . . . . . . . . . . . . 14 (𝜑 → ((𝑀 + 1)...𝑁) ⊆ (𝐾...𝑁))
7574adantr 484 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ((𝑀 + 1)...𝑁) ⊆ (𝐾...𝑁))
7672, 75sstrd 3911 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ((𝑀 + 1)...𝑛) ⊆ (𝐾...𝑁))
7776sselda 3901 . . . . . . . . . . 11 (((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) ∧ 𝑥 ∈ ((𝑀 + 1)...𝑛)) → 𝑥 ∈ (𝐾...𝑁))
7865adantlr 715 . . . . . . . . . . 11 (((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) ∧ 𝑥 ∈ (𝐾...𝑁)) → (𝐹𝑥) ∈ 𝑆)
7977, 78syldan 594 . . . . . . . . . 10 (((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) ∧ 𝑥 ∈ ((𝑀 + 1)...𝑛)) → (𝐹𝑥) ∈ 𝑆)
8067adantlr 715 . . . . . . . . . 10 (((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
8146, 79, 80seqcl 13596 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (seq(𝑀 + 1)( + , 𝐹)‘𝑛) ∈ 𝑆)
82 fveq2 6717 . . . . . . . . . . 11 (𝑥 = (𝑛 + 1) → (𝐹𝑥) = (𝐹‘(𝑛 + 1)))
8382eleq1d 2822 . . . . . . . . . 10 (𝑥 = (𝑛 + 1) → ((𝐹𝑥) ∈ 𝑆 ↔ (𝐹‘(𝑛 + 1)) ∈ 𝑆))
8465ralrimiva 3105 . . . . . . . . . . 11 (𝜑 → ∀𝑥 ∈ (𝐾...𝑁)(𝐹𝑥) ∈ 𝑆)
8584adantr 484 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ∀𝑥 ∈ (𝐾...𝑁)(𝐹𝑥) ∈ 𝑆)
86 simpr 488 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)) → (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))
87 ssel2 3895 . . . . . . . . . . 11 ((((𝑀 + 1)...𝑁) ⊆ (𝐾...𝑁) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)) → (𝑛 + 1) ∈ (𝐾...𝑁))
8874, 86, 87syl2an 599 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (𝑛 + 1) ∈ (𝐾...𝑁))
8983, 85, 88rspcdva 3539 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (𝐹‘(𝑛 + 1)) ∈ 𝑆)
90 seqsplit.2 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
9190caovassg 7406 . . . . . . . . 9 ((𝜑 ∧ ((seq𝐾( + , 𝐹)‘𝑀) ∈ 𝑆 ∧ (seq(𝑀 + 1)( + , 𝐹)‘𝑛) ∈ 𝑆 ∧ (𝐹‘(𝑛 + 1)) ∈ 𝑆)) → (((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛)) + (𝐹‘(𝑛 + 1))) = ((seq𝐾( + , 𝐹)‘𝑀) + ((seq(𝑀 + 1)( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
9257, 69, 81, 89, 91syl13anc 1374 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛)) + (𝐹‘(𝑛 + 1))) = ((seq𝐾( + , 𝐹)‘𝑀) + ((seq(𝑀 + 1)( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
9356, 92eqtr4d 2780 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1))) = (((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛)) + (𝐹‘(𝑛 + 1))))
9453, 93eqeq12d 2753 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ((seq𝐾( + , 𝐹)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1))) ↔ ((seq𝐾( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))) = (((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛)) + (𝐹‘(𝑛 + 1)))))
9545, 94syl5ibr 249 . . . . 5 ((𝜑 ∧ (𝑛 ∈ (ℤ‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ((seq𝐾( + , 𝐹)‘𝑛) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛)) → (seq𝐾( + , 𝐹)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1)))))
9644, 95animpimp2impd 846 . . . 4 (𝑛 ∈ (ℤ‘(𝑀 + 1)) → ((𝜑 → (𝑛 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑛) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛)))) → (𝜑 → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1)))))))
9710, 17, 24, 31, 40, 96uzind4 12502 . . 3 (𝑁 ∈ (ℤ‘(𝑀 + 1)) → (𝜑 → (𝑁 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑁) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁)))))
981, 97mpcom 38 . 2 (𝜑 → (𝑁 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑁) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁))))
993, 98mpd 15 1 (𝜑 → (seq𝐾( + , 𝐹)‘𝑁) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1089   = wceq 1543  wcel 2110  wral 3061  wss 3866  cfv 6380  (class class class)co 7213  1c1 10730   + caddc 10732  cz 12176  cuz 12438  ...cfz 13095  seqcseq 13574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-n0 12091  df-z 12177  df-uz 12439  df-fz 13096  df-seq 13575
This theorem is referenced by:  seq1p  13610  seqf1olem2  13616  bcval5  13884  clim2ser  15218  clim2ser2  15219  isumsplit  15404  clim2div  15453  gsumsgrpccat  18266  gsumccatOLD  18267  mulgnndir  18520  mblfinlem2  35552  fmul01lt1lem1  42800  fmul01lt1lem2  42801
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