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Theorem seqid2 13400
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 Mario Carneiro, 27-May-2014.)
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)
Assertion
Ref Expression
seqid2 (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑁))
Distinct variable groups:   𝑥,𝐹   𝑥,𝐾   𝑥,𝑀   𝑥,𝑁   𝜑,𝑥   𝑥,𝑆   𝑥, +   𝑥,𝑍

Proof of Theorem seqid2
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 seqid2.3 . . 3 (𝜑𝑁 ∈ (ℤ𝐾))
2 eluzfz2 12898 . . 3 (𝑁 ∈ (ℤ𝐾) → 𝑁 ∈ (𝐾...𝑁))
31, 2syl 17 . 2 (𝜑𝑁 ∈ (𝐾...𝑁))
4 eleq1 2899 . . . . . 6 (𝑥 = 𝐾 → (𝑥 ∈ (𝐾...𝑁) ↔ 𝐾 ∈ (𝐾...𝑁)))
5 fveq2 6643 . . . . . . 7 (𝑥 = 𝐾 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝐾))
65eqeq2d 2832 . . . . . 6 (𝑥 = 𝐾 → ((seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥) ↔ (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝐾)))
74, 6imbi12d 348 . . . . 5 (𝑥 = 𝐾 → ((𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥)) ↔ (𝐾 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝐾))))
87imbi2d 344 . . . 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 2899 . . . . . 6 (𝑥 = 𝑛 → (𝑥 ∈ (𝐾...𝑁) ↔ 𝑛 ∈ (𝐾...𝑁)))
10 fveq2 6643 . . . . . . 7 (𝑥 = 𝑛 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑛))
1110eqeq2d 2832 . . . . . 6 (𝑥 = 𝑛 → ((seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥) ↔ (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑛)))
129, 11imbi12d 348 . . . . 5 (𝑥 = 𝑛 → ((𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥)) ↔ (𝑛 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑛))))
1312imbi2d 344 . . . 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 2899 . . . . . 6 (𝑥 = (𝑛 + 1) → (𝑥 ∈ (𝐾...𝑁) ↔ (𝑛 + 1) ∈ (𝐾...𝑁)))
15 fveq2 6643 . . . . . . 7 (𝑥 = (𝑛 + 1) → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘(𝑛 + 1)))
1615eqeq2d 2832 . . . . . 6 (𝑥 = (𝑛 + 1) → ((seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥) ↔ (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘(𝑛 + 1))))
1714, 16imbi12d 348 . . . . 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 344 . . . 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 2899 . . . . . 6 (𝑥 = 𝑁 → (𝑥 ∈ (𝐾...𝑁) ↔ 𝑁 ∈ (𝐾...𝑁)))
20 fveq2 6643 . . . . . . 7 (𝑥 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑁))
2120eqeq2d 2832 . . . . . 6 (𝑥 = 𝑁 → ((seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥) ↔ (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑁)))
2219, 21imbi12d 348 . . . . 5 (𝑥 = 𝑁 → ((𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥)) ↔ (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑁))))
2322imbi2d 344 . . . 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 2822 . . . . 5 (𝐾 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝐾))
25242a1i 12 . . . 4 (𝐾 ∈ ℤ → (𝜑 → (𝐾 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝐾))))
26 peano2fzr 12903 . . . . . . . 8 ((𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁)) → 𝑛 ∈ (𝐾...𝑁))
2726adantl 485 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → 𝑛 ∈ (𝐾...𝑁))
2827expr 460 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝐾)) → ((𝑛 + 1) ∈ (𝐾...𝑁) → 𝑛 ∈ (𝐾...𝑁)))
2928imim1d 82 . . . . 5 ((𝜑𝑛 ∈ (ℤ𝐾)) → ((𝑛 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑛)) → ((𝑛 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑛))))
30 oveq1 7137 . . . . . 6 ((seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑛) → ((seq𝑀( + , 𝐹)‘𝐾) + (𝐹‘(𝑛 + 1))) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
31 fveqeq2 6652 . . . . . . . . . 10 (𝑥 = (𝑛 + 1) → ((𝐹𝑥) = 𝑍 ↔ (𝐹‘(𝑛 + 1)) = 𝑍))
32 seqid2.5 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ ((𝐾 + 1)...𝑁)) → (𝐹𝑥) = 𝑍)
3332ralrimiva 3170 . . . . . . . . . . 11 (𝜑 → ∀𝑥 ∈ ((𝐾 + 1)...𝑁)(𝐹𝑥) = 𝑍)
3433adantr 484 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → ∀𝑥 ∈ ((𝐾 + 1)...𝑁)(𝐹𝑥) = 𝑍)
35 eluzp1p1 12248 . . . . . . . . . . . 12 (𝑛 ∈ (ℤ𝐾) → (𝑛 + 1) ∈ (ℤ‘(𝐾 + 1)))
3635ad2antrl 727 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (𝑛 + 1) ∈ (ℤ‘(𝐾 + 1)))
37 elfzuz3 12888 . . . . . . . . . . . 12 ((𝑛 + 1) ∈ (𝐾...𝑁) → 𝑁 ∈ (ℤ‘(𝑛 + 1)))
3837ad2antll 728 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → 𝑁 ∈ (ℤ‘(𝑛 + 1)))
39 elfzuzb 12885 . . . . . . . . . . 11 ((𝑛 + 1) ∈ ((𝐾 + 1)...𝑁) ↔ ((𝑛 + 1) ∈ (ℤ‘(𝐾 + 1)) ∧ 𝑁 ∈ (ℤ‘(𝑛 + 1))))
4036, 38, 39sylanbrc 586 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (𝑛 + 1) ∈ ((𝐾 + 1)...𝑁))
4131, 34, 40rspcdva 3602 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (𝐹‘(𝑛 + 1)) = 𝑍)
4241oveq2d 7146 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → ((seq𝑀( + , 𝐹)‘𝐾) + (𝐹‘(𝑛 + 1))) = ((seq𝑀( + , 𝐹)‘𝐾) + 𝑍))
43 oveq1 7137 . . . . . . . . . . 11 (𝑥 = (seq𝑀( + , 𝐹)‘𝐾) → (𝑥 + 𝑍) = ((seq𝑀( + , 𝐹)‘𝐾) + 𝑍))
44 id 22 . . . . . . . . . . 11 (𝑥 = (seq𝑀( + , 𝐹)‘𝐾) → 𝑥 = (seq𝑀( + , 𝐹)‘𝐾))
4543, 44eqeq12d 2837 . . . . . . . . . 10 (𝑥 = (seq𝑀( + , 𝐹)‘𝐾) → ((𝑥 + 𝑍) = 𝑥 ↔ ((seq𝑀( + , 𝐹)‘𝐾) + 𝑍) = (seq𝑀( + , 𝐹)‘𝐾)))
46 seqid2.1 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → (𝑥 + 𝑍) = 𝑥)
4746ralrimiva 3170 . . . . . . . . . 10 (𝜑 → ∀𝑥𝑆 (𝑥 + 𝑍) = 𝑥)
48 seqid2.4 . . . . . . . . . 10 (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) ∈ 𝑆)
4945, 47, 48rspcdva 3602 . . . . . . . . 9 (𝜑 → ((seq𝑀( + , 𝐹)‘𝐾) + 𝑍) = (seq𝑀( + , 𝐹)‘𝐾))
5049adantr 484 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → ((seq𝑀( + , 𝐹)‘𝐾) + 𝑍) = (seq𝑀( + , 𝐹)‘𝐾))
5142, 50eqtr2d 2857 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (seq𝑀( + , 𝐹)‘𝐾) = ((seq𝑀( + , 𝐹)‘𝐾) + (𝐹‘(𝑛 + 1))))
52 simprl 770 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → 𝑛 ∈ (ℤ𝐾))
53 seqid2.2 . . . . . . . . . 10 (𝜑𝐾 ∈ (ℤ𝑀))
5453adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → 𝐾 ∈ (ℤ𝑀))
55 uztrn 12239 . . . . . . . . 9 ((𝑛 ∈ (ℤ𝐾) ∧ 𝐾 ∈ (ℤ𝑀)) → 𝑛 ∈ (ℤ𝑀))
5652, 54, 55syl2anc 587 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → 𝑛 ∈ (ℤ𝑀))
57 seqp1 13367 . . . . . . . 8 (𝑛 ∈ (ℤ𝑀) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
5856, 57syl 17 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
5951, 58eqeq12d 2837 . . . . . 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)))))
6030, 59syl5ibr 249 . . . . 5 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → ((seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑛) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘(𝑛 + 1))))
6129, 60animpimp2impd 843 . . . 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))))))
628, 13, 18, 23, 25, 61uzind4 12284 . . 3 (𝑁 ∈ (ℤ𝐾) → (𝜑 → (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑁))))
631, 62mpcom 38 . 2 (𝜑 → (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑁)))
643, 63mpd 15 1 (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wral 3126  cfv 6328  (class class class)co 7130  1c1 10515   + caddc 10517  cz 11959  cuz 12221  ...cfz 12875  seqcseq 13352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-resscn 10571  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-addrcl 10575  ax-mulcl 10576  ax-mulrcl 10577  ax-mulcom 10578  ax-addass 10579  ax-mulass 10580  ax-distr 10581  ax-i2m1 10582  ax-1ne0 10583  ax-1rid 10584  ax-rnegex 10585  ax-rrecex 10586  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589  ax-pre-ltadd 10590  ax-pre-mulgt0 10591
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-er 8264  df-en 8485  df-dom 8486  df-sdom 8487  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-sub 10849  df-neg 10850  df-nn 11616  df-n0 11876  df-z 11960  df-uz 12222  df-fz 12876  df-seq 13353
This theorem is referenced by:  seqcoll  13806  seqcoll2  13807  fsumcvg  15048  fprodcvg  15263  ovolicc1  24098  lgsdilem2  25895
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