MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  seqid2 Structured version   Visualization version   GIF version

Theorem seqid2 14099
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 13592 . . 3 (𝑁 ∈ (ℤ𝐾) → 𝑁 ∈ (𝐾...𝑁))
31, 2syl 17 . 2 (𝜑𝑁 ∈ (𝐾...𝑁))
4 eleq1 2832 . . . . . 6 (𝑥 = 𝐾 → (𝑥 ∈ (𝐾...𝑁) ↔ 𝐾 ∈ (𝐾...𝑁)))
5 fveq2 6920 . . . . . . 7 (𝑥 = 𝐾 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝐾))
65eqeq2d 2751 . . . . . 6 (𝑥 = 𝐾 → ((seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥) ↔ (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝐾)))
74, 6imbi12d 344 . . . . 5 (𝑥 = 𝐾 → ((𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥)) ↔ (𝐾 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝐾))))
87imbi2d 340 . . . 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 2832 . . . . . 6 (𝑥 = 𝑛 → (𝑥 ∈ (𝐾...𝑁) ↔ 𝑛 ∈ (𝐾...𝑁)))
10 fveq2 6920 . . . . . . 7 (𝑥 = 𝑛 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑛))
1110eqeq2d 2751 . . . . . 6 (𝑥 = 𝑛 → ((seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥) ↔ (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑛)))
129, 11imbi12d 344 . . . . 5 (𝑥 = 𝑛 → ((𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥)) ↔ (𝑛 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑛))))
1312imbi2d 340 . . . 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 2832 . . . . . 6 (𝑥 = (𝑛 + 1) → (𝑥 ∈ (𝐾...𝑁) ↔ (𝑛 + 1) ∈ (𝐾...𝑁)))
15 fveq2 6920 . . . . . . 7 (𝑥 = (𝑛 + 1) → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘(𝑛 + 1)))
1615eqeq2d 2751 . . . . . 6 (𝑥 = (𝑛 + 1) → ((seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥) ↔ (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘(𝑛 + 1))))
1714, 16imbi12d 344 . . . . 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 340 . . . 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 2832 . . . . . 6 (𝑥 = 𝑁 → (𝑥 ∈ (𝐾...𝑁) ↔ 𝑁 ∈ (𝐾...𝑁)))
20 fveq2 6920 . . . . . . 7 (𝑥 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑁))
2120eqeq2d 2751 . . . . . 6 (𝑥 = 𝑁 → ((seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥) ↔ (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑁)))
2219, 21imbi12d 344 . . . . 5 (𝑥 = 𝑁 → ((𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥)) ↔ (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑁))))
2322imbi2d 340 . . . 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 2741 . . . . 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 13597 . . . . . . . 8 ((𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁)) → 𝑛 ∈ (𝐾...𝑁))
2726adantl 481 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → 𝑛 ∈ (𝐾...𝑁))
2827expr 456 . . . . . 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 7455 . . . . . 6 ((seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑛) → ((seq𝑀( + , 𝐹)‘𝐾) + (𝐹‘(𝑛 + 1))) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
31 fveqeq2 6929 . . . . . . . . . 10 (𝑥 = (𝑛 + 1) → ((𝐹𝑥) = 𝑍 ↔ (𝐹‘(𝑛 + 1)) = 𝑍))
32 seqid2.5 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ ((𝐾 + 1)...𝑁)) → (𝐹𝑥) = 𝑍)
3332ralrimiva 3152 . . . . . . . . . . 11 (𝜑 → ∀𝑥 ∈ ((𝐾 + 1)...𝑁)(𝐹𝑥) = 𝑍)
3433adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → ∀𝑥 ∈ ((𝐾 + 1)...𝑁)(𝐹𝑥) = 𝑍)
35 eluzp1p1 12931 . . . . . . . . . . . 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 13581 . . . . . . . . . . . 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 13578 . . . . . . . . . . 11 ((𝑛 + 1) ∈ ((𝐾 + 1)...𝑁) ↔ ((𝑛 + 1) ∈ (ℤ‘(𝐾 + 1)) ∧ 𝑁 ∈ (ℤ‘(𝑛 + 1))))
4036, 38, 39sylanbrc 582 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (𝑛 + 1) ∈ ((𝐾 + 1)...𝑁))
4131, 34, 40rspcdva 3636 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (𝐹‘(𝑛 + 1)) = 𝑍)
4241oveq2d 7464 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → ((seq𝑀( + , 𝐹)‘𝐾) + (𝐹‘(𝑛 + 1))) = ((seq𝑀( + , 𝐹)‘𝐾) + 𝑍))
43 oveq1 7455 . . . . . . . . . . 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 2756 . . . . . . . . . 10 (𝑥 = (seq𝑀( + , 𝐹)‘𝐾) → ((𝑥 + 𝑍) = 𝑥 ↔ ((seq𝑀( + , 𝐹)‘𝐾) + 𝑍) = (seq𝑀( + , 𝐹)‘𝐾)))
46 seqid2.1 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → (𝑥 + 𝑍) = 𝑥)
4746ralrimiva 3152 . . . . . . . . . 10 (𝜑 → ∀𝑥𝑆 (𝑥 + 𝑍) = 𝑥)
48 seqid2.4 . . . . . . . . . 10 (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) ∈ 𝑆)
4945, 47, 48rspcdva 3636 . . . . . . . . 9 (𝜑 → ((seq𝑀( + , 𝐹)‘𝐾) + 𝑍) = (seq𝑀( + , 𝐹)‘𝐾))
5049adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → ((seq𝑀( + , 𝐹)‘𝐾) + 𝑍) = (seq𝑀( + , 𝐹)‘𝐾))
5142, 50eqtr2d 2781 . . . . . . 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 480 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → 𝐾 ∈ (ℤ𝑀))
55 uztrn 12921 . . . . . . . . 9 ((𝑛 ∈ (ℤ𝐾) ∧ 𝐾 ∈ (ℤ𝑀)) → 𝑛 ∈ (ℤ𝑀))
5652, 54, 55syl2anc 583 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → 𝑛 ∈ (ℤ𝑀))
57 seqp1 14067 . . . . . . . 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 2756 . . . . . 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, 59imbitrrid 246 . . . . 5 ((𝜑 ∧ (𝑛 ∈ (ℤ𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → ((seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑛) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘(𝑛 + 1))))
6129, 60animpimp2impd 845 . . . 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 12971 . . 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 395   = wceq 1537  wcel 2108  wral 3067  cfv 6573  (class class class)co 7448  1c1 11185   + caddc 11187  cz 12639  cuz 12903  ...cfz 13567  seqcseq 14052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-n0 12554  df-z 12640  df-uz 12904  df-fz 13568  df-seq 14053
This theorem is referenced by:  seqcoll  14513  seqcoll2  14514  fsumcvg  15760  fprodcvg  15978  ovolicc1  25570  lgsdilem2  27395
  Copyright terms: Public domain W3C validator