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Theorem seq3id2 10275
 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 Mario Carneiro, 13-Jul-2013.) (Revised by Jim Kingdon, 12-Nov-2022.)
Hypotheses
Ref Expression
seqid2.1 ((𝜑𝑥𝑆) → (𝑥 + 𝑍) = 𝑥)
seqid2.2 (𝜑𝐾 ∈ (ℤ𝑀))
seqid2.3 (𝜑𝑁 ∈ (ℤ𝐾))
seqid2.4 (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) ∈ 𝑆)
seqid2.5 ((𝜑𝑥 ∈ ((𝐾 + 1)...𝑁)) → (𝐹𝑥) = 𝑍)
seq3id2.f ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
seq3id2.cl ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
Assertion
Ref Expression
seq3id2 (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑁))
Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝐾,𝑦   𝑥,𝑀,𝑦   𝑥,𝑁,𝑦   𝜑,𝑥,𝑦   𝑥,𝑆,𝑦   𝑥, + ,𝑦   𝑥,𝑍
Allowed substitution hint:   𝑍(𝑦)

Proof of Theorem seq3id2
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 seqid2.3 . . 3 (𝜑𝑁 ∈ (ℤ𝐾))
2 eluzfz2 9805 . . 3 (𝑁 ∈ (ℤ𝐾) → 𝑁 ∈ (𝐾...𝑁))
31, 2syl 14 . 2 (𝜑𝑁 ∈ (𝐾...𝑁))
4 eleq1 2200 . . . . . 6 (𝑥 = 𝐾 → (𝑥 ∈ (𝐾...𝑁) ↔ 𝐾 ∈ (𝐾...𝑁)))
5 fveq2 5414 . . . . . . 7 (𝑥 = 𝐾 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝐾))
65eqeq2d 2149 . . . . . 6 (𝑥 = 𝐾 → ((seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥) ↔ (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝐾)))
74, 6imbi12d 233 . . . . 5 (𝑥 = 𝐾 → ((𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥)) ↔ (𝐾 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝐾))))
87imbi2d 229 . . . 4 (𝑥 = 𝐾 → ((𝜑 → (𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥))) ↔ (𝜑 → (𝐾 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝐾)))))
9 eleq1 2200 . . . . . 6 (𝑥 = 𝑛 → (𝑥 ∈ (𝐾...𝑁) ↔ 𝑛 ∈ (𝐾...𝑁)))
10 fveq2 5414 . . . . . . 7 (𝑥 = 𝑛 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑛))
1110eqeq2d 2149 . . . . . 6 (𝑥 = 𝑛 → ((seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥) ↔ (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑛)))
129, 11imbi12d 233 . . . . 5 (𝑥 = 𝑛 → ((𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥)) ↔ (𝑛 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑛))))
1312imbi2d 229 . . . 4 (𝑥 = 𝑛 → ((𝜑 → (𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥))) ↔ (𝜑 → (𝑛 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑛)))))
14 eleq1 2200 . . . . . 6 (𝑥 = (𝑛 + 1) → (𝑥 ∈ (𝐾...𝑁) ↔ (𝑛 + 1) ∈ (𝐾...𝑁)))
15 fveq2 5414 . . . . . . 7 (𝑥 = (𝑛 + 1) → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘(𝑛 + 1)))
1615eqeq2d 2149 . . . . . 6 (𝑥 = (𝑛 + 1) → ((seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥) ↔ (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘(𝑛 + 1))))
1714, 16imbi12d 233 . . . . 5 (𝑥 = (𝑛 + 1) → ((𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥)) ↔ ((𝑛 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘(𝑛 + 1)))))
1817imbi2d 229 . . . 4 (𝑥 = (𝑛 + 1) → ((𝜑 → (𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥))) ↔ (𝜑 → ((𝑛 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘(𝑛 + 1))))))
19 eleq1 2200 . . . . . 6 (𝑥 = 𝑁 → (𝑥 ∈ (𝐾...𝑁) ↔ 𝑁 ∈ (𝐾...𝑁)))
20 fveq2 5414 . . . . . . 7 (𝑥 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑁))
2120eqeq2d 2149 . . . . . 6 (𝑥 = 𝑁 → ((seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥) ↔ (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑁)))
2219, 21imbi12d 233 . . . . 5 (𝑥 = 𝑁 → ((𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥)) ↔ (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑁))))
2322imbi2d 229 . . . 4 (𝑥 = 𝑁 → ((𝜑 → (𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥))) ↔ (𝜑 → (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑁)))))
24 eqidd 2138 . . . . 5 (𝐾 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝐾))
25242a1i 27 . . . 4 (𝐾 ∈ ℤ → (𝜑 → (𝐾 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝐾))))
26 peano2fzr 9810 . . . . . . . 8 ((𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁)) → 𝑛 ∈ (𝐾...𝑁))
2726adantl 275 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → 𝑛 ∈ (𝐾...𝑁))
2827expr 372 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝐾)) → ((𝑛 + 1) ∈ (𝐾...𝑁) → 𝑛 ∈ (𝐾...𝑁)))
2928imim1d 75 . . . . 5 ((𝜑𝑛 ∈ (ℤ𝐾)) → ((𝑛 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑛)) → ((𝑛 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑛))))
30 oveq1 5774 . . . . . 6 ((seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑛) → ((seq𝑀( + , 𝐹)‘𝐾) + (𝐹‘(𝑛 + 1))) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
31 fveqeq2 5423 . . . . . . . . . 10 (𝑥 = (𝑛 + 1) → ((𝐹𝑥) = 𝑍 ↔ (𝐹‘(𝑛 + 1)) = 𝑍))
32 seqid2.5 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ ((𝐾 + 1)...𝑁)) → (𝐹𝑥) = 𝑍)
3332ralrimiva 2503 . . . . . . . . . . 11 (𝜑 → ∀𝑥 ∈ ((𝐾 + 1)...𝑁)(𝐹𝑥) = 𝑍)
3433adantr 274 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → ∀𝑥 ∈ ((𝐾 + 1)...𝑁)(𝐹𝑥) = 𝑍)
35 eluzp1p1 9344 . . . . . . . . . . . 12 (𝑛 ∈ (ℤ𝐾) → (𝑛 + 1) ∈ (ℤ‘(𝐾 + 1)))
3635ad2antrl 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (𝑛 + 1) ∈ (ℤ‘(𝐾 + 1)))
37 elfzuz3 9796 . . . . . . . . . . . 12 ((𝑛 + 1) ∈ (𝐾...𝑁) → 𝑁 ∈ (ℤ‘(𝑛 + 1)))
3837ad2antll 482 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → 𝑁 ∈ (ℤ‘(𝑛 + 1)))
39 elfzuzb 9793 . . . . . . . . . . 11 ((𝑛 + 1) ∈ ((𝐾 + 1)...𝑁) ↔ ((𝑛 + 1) ∈ (ℤ‘(𝐾 + 1)) ∧ 𝑁 ∈ (ℤ‘(𝑛 + 1))))
4036, 38, 39sylanbrc 413 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (𝑛 + 1) ∈ ((𝐾 + 1)...𝑁))
4131, 34, 40rspcdva 2789 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (𝐹‘(𝑛 + 1)) = 𝑍)
4241oveq2d 5783 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → ((seq𝑀( + , 𝐹)‘𝐾) + (𝐹‘(𝑛 + 1))) = ((seq𝑀( + , 𝐹)‘𝐾) + 𝑍))
43 oveq1 5774 . . . . . . . . . . 11 (𝑥 = (seq𝑀( + , 𝐹)‘𝐾) → (𝑥 + 𝑍) = ((seq𝑀( + , 𝐹)‘𝐾) + 𝑍))
44 id 19 . . . . . . . . . . 11 (𝑥 = (seq𝑀( + , 𝐹)‘𝐾) → 𝑥 = (seq𝑀( + , 𝐹)‘𝐾))
4543, 44eqeq12d 2152 . . . . . . . . . 10 (𝑥 = (seq𝑀( + , 𝐹)‘𝐾) → ((𝑥 + 𝑍) = 𝑥 ↔ ((seq𝑀( + , 𝐹)‘𝐾) + 𝑍) = (seq𝑀( + , 𝐹)‘𝐾)))
46 seqid2.1 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → (𝑥 + 𝑍) = 𝑥)
4746ralrimiva 2503 . . . . . . . . . 10 (𝜑 → ∀𝑥𝑆 (𝑥 + 𝑍) = 𝑥)
48 seqid2.4 . . . . . . . . . 10 (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) ∈ 𝑆)
4945, 47, 48rspcdva 2789 . . . . . . . . 9 (𝜑 → ((seq𝑀( + , 𝐹)‘𝐾) + 𝑍) = (seq𝑀( + , 𝐹)‘𝐾))
5049adantr 274 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → ((seq𝑀( + , 𝐹)‘𝐾) + 𝑍) = (seq𝑀( + , 𝐹)‘𝐾))
5142, 50eqtr2d 2171 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (seq𝑀( + , 𝐹)‘𝐾) = ((seq𝑀( + , 𝐹)‘𝐾) + (𝐹‘(𝑛 + 1))))
52 simprl 520 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → 𝑛 ∈ (ℤ𝐾))
53 seqid2.2 . . . . . . . . . 10 (𝜑𝐾 ∈ (ℤ𝑀))
5453adantr 274 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → 𝐾 ∈ (ℤ𝑀))
55 uztrn 9335 . . . . . . . . 9 ((𝑛 ∈ (ℤ𝐾) ∧ 𝐾 ∈ (ℤ𝑀)) → 𝑛 ∈ (ℤ𝑀))
5652, 54, 55syl2anc 408 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → 𝑛 ∈ (ℤ𝑀))
57 seq3id2.f . . . . . . . . 9 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
5857adantlr 468 . . . . . . . 8 (((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
59 seq3id2.cl . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
6059adantlr 468 . . . . . . . 8 (((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
6156, 58, 60seq3p1 10228 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
6251, 61eqeq12d 2152 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → ((seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘(𝑛 + 1)) ↔ ((seq𝑀( + , 𝐹)‘𝐾) + (𝐹‘(𝑛 + 1))) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
6330, 62syl5ibr 155 . . . . 5 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → ((seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑛) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘(𝑛 + 1))))
6429, 63animpimp2impd 548 . . . 4 (𝑛 ∈ (ℤ𝐾) → ((𝜑 → (𝑛 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑛))) → (𝜑 → ((𝑛 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘(𝑛 + 1))))))
658, 13, 18, 23, 25, 64uzind4 9376 . . 3 (𝑁 ∈ (ℤ𝐾) → (𝜑 → (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑁))))
661, 65mpcom 36 . 2 (𝜑 → (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑁)))
673, 66mpd 13 1 (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑁))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480  ∀wral 2414  ‘cfv 5118  (class class class)co 5767  1c1 7614   + caddc 7616  ℤcz 9047  ℤ≥cuz 9319  ...cfz 9783  seqcseq 10211 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-addcom 7713  ax-addass 7715  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-0id 7721  ax-rnegex 7722  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-ltadd 7729 This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-frec 6281  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-inn 8714  df-n0 8971  df-z 9048  df-uz 9320  df-fz 9784  df-seqfrec 10212 This theorem is referenced by:  seq3coll  10578  fsum3cvg  11139  fproddccvg  11334
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