ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  iseqid3s GIF version

Theorem iseqid3s 9115
Description: A sequence that consists of zeroes up to 𝑁 sums to zero at 𝑁. In this case by "zero" we mean whatever the identity 𝑍 is for the operation +). (Contributed by Jim Kingdon, 18-Aug-2021.)
Hypotheses
Ref Expression
iseqid3s.1 (𝜑 → (𝑍 + 𝑍) = 𝑍)
iseqid3s.2 (𝜑𝑁 ∈ (ℤ𝑀))
iseqid3s.3 ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) = 𝑍)
iseqid3s.z (𝜑𝑍𝑆)
iseqid3s.s (𝜑𝑆𝑉)
iseqid3s.f ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
iseqid3s.cl ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
Assertion
Ref Expression
iseqid3s (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = 𝑍)
Distinct variable groups:   𝑥,𝑦, +   𝑥,𝐹,𝑦   𝑥,𝑀,𝑦   𝜑,𝑥,𝑦   𝑥,𝑍,𝑦   𝑥,𝑁,𝑦   𝑥,𝑆,𝑦
Allowed substitution hints:   𝑉(𝑥,𝑦)

Proof of Theorem iseqid3s
Dummy variables 𝑘 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iseqid3s.2 . . 3 (𝜑𝑁 ∈ (ℤ𝑀))
2 eluzfz2 8854 . . 3 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
3 fveq2 5156 . . . . . 6 (𝑤 = 𝑀 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑀))
43eqeq1d 2048 . . . . 5 (𝑤 = 𝑀 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = 𝑍))
54imbi2d 219 . . . 4 (𝑤 = 𝑀 → ((𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = 𝑍)))
6 fveq2 5156 . . . . . 6 (𝑤 = 𝑘 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑘))
76eqeq1d 2048 . . . . 5 (𝑤 = 𝑘 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = 𝑍))
87imbi2d 219 . . . 4 (𝑤 = 𝑘 → ((𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = 𝑍)))
9 fveq2 5156 . . . . . 6 (𝑤 = (𝑘 + 1) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)))
109eqeq1d 2048 . . . . 5 (𝑤 = (𝑘 + 1) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = 𝑍))
1110imbi2d 219 . . . 4 (𝑤 = (𝑘 + 1) → ((𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = 𝑍)))
12 fveq2 5156 . . . . . 6 (𝑤 = 𝑁 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑁))
1312eqeq1d 2048 . . . . 5 (𝑤 = 𝑁 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = 𝑍))
1413imbi2d 219 . . . 4 (𝑤 = 𝑁 → ((𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = 𝑍)))
15 eluzel2 8441 . . . . . . . 8 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
161, 15syl 14 . . . . . . 7 (𝜑𝑀 ∈ ℤ)
17 iseqid3s.s . . . . . . 7 (𝜑𝑆𝑉)
18 iseqid3s.f . . . . . . 7 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
19 iseqid3s.cl . . . . . . 7 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
2016, 17, 18, 19iseq1 9091 . . . . . 6 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (𝐹𝑀))
21 iseqid3s.3 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) = 𝑍)
2221ralrimiva 2389 . . . . . . 7 (𝜑 → ∀𝑥 ∈ (𝑀...𝑁)(𝐹𝑥) = 𝑍)
23 eluzfz1 8853 . . . . . . . 8 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
24 fveq2 5156 . . . . . . . . . 10 (𝑥 = 𝑀 → (𝐹𝑥) = (𝐹𝑀))
2524eqeq1d 2048 . . . . . . . . 9 (𝑥 = 𝑀 → ((𝐹𝑥) = 𝑍 ↔ (𝐹𝑀) = 𝑍))
2625rspcv 2649 . . . . . . . 8 (𝑀 ∈ (𝑀...𝑁) → (∀𝑥 ∈ (𝑀...𝑁)(𝐹𝑥) = 𝑍 → (𝐹𝑀) = 𝑍))
271, 23, 263syl 17 . . . . . . 7 (𝜑 → (∀𝑥 ∈ (𝑀...𝑁)(𝐹𝑥) = 𝑍 → (𝐹𝑀) = 𝑍))
2822, 27mpd 13 . . . . . 6 (𝜑 → (𝐹𝑀) = 𝑍)
2920, 28eqtrd 2072 . . . . 5 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = 𝑍)
3029a1i 9 . . . 4 (𝑁 ∈ (ℤ𝑀) → (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = 𝑍))
31 elfzouz 8966 . . . . . . . . . . 11 (𝑘 ∈ (𝑀..^𝑁) → 𝑘 ∈ (ℤ𝑀))
3231adantl 262 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ (ℤ𝑀))
3317adantr 261 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑀..^𝑁)) → 𝑆𝑉)
3418adantlr 446 . . . . . . . . . 10 (((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
3519adantlr 446 . . . . . . . . . 10 (((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
3632, 33, 34, 35iseqp1 9094 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) + (𝐹‘(𝑘 + 1))))
3736adantr 261 . . . . . . . 8 (((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = 𝑍) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) + (𝐹‘(𝑘 + 1))))
38 simpr 103 . . . . . . . . 9 (((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = 𝑍) → (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = 𝑍)
39 fzofzp1 9041 . . . . . . . . . . . 12 (𝑘 ∈ (𝑀..^𝑁) → (𝑘 + 1) ∈ (𝑀...𝑁))
4039adantl 262 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ (𝑀...𝑁))
4122adantr 261 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ (𝑀...𝑁)(𝐹𝑥) = 𝑍)
42 fveq2 5156 . . . . . . . . . . . . 13 (𝑥 = (𝑘 + 1) → (𝐹𝑥) = (𝐹‘(𝑘 + 1)))
4342eqeq1d 2048 . . . . . . . . . . . 12 (𝑥 = (𝑘 + 1) → ((𝐹𝑥) = 𝑍 ↔ (𝐹‘(𝑘 + 1)) = 𝑍))
4443rspcv 2649 . . . . . . . . . . 11 ((𝑘 + 1) ∈ (𝑀...𝑁) → (∀𝑥 ∈ (𝑀...𝑁)(𝐹𝑥) = 𝑍 → (𝐹‘(𝑘 + 1)) = 𝑍))
4540, 41, 44sylc 56 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑘 + 1)) = 𝑍)
4645adantr 261 . . . . . . . . 9 (((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = 𝑍) → (𝐹‘(𝑘 + 1)) = 𝑍)
4738, 46oveq12d 5508 . . . . . . . 8 (((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = 𝑍) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) + (𝐹‘(𝑘 + 1))) = (𝑍 + 𝑍))
48 iseqid3s.1 . . . . . . . . 9 (𝜑 → (𝑍 + 𝑍) = 𝑍)
4948ad2antrr 457 . . . . . . . 8 (((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = 𝑍) → (𝑍 + 𝑍) = 𝑍)
5037, 47, 493eqtrd 2076 . . . . . . 7 (((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = 𝑍) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = 𝑍)
5150ex 108 . . . . . 6 ((𝜑𝑘 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) = 𝑍 → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = 𝑍))
5251expcom 109 . . . . 5 (𝑘 ∈ (𝑀..^𝑁) → (𝜑 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) = 𝑍 → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = 𝑍)))
5352a2d 23 . . . 4 (𝑘 ∈ (𝑀..^𝑁) → ((𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = 𝑍) → (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = 𝑍)))
545, 8, 11, 14, 30, 53fzind2 9053 . . 3 (𝑁 ∈ (𝑀...𝑁) → (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = 𝑍))
551, 2, 543syl 17 . 2 (𝜑 → (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = 𝑍))
5655pm2.43i 43 1 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = 𝑍)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97   = wceq 1243  wcel 1393  wral 2303  cfv 4880  (class class class)co 5490  1c1 6862   + caddc 6864  cz 8208  cuz 8436  ...cfz 8832  ..^cfzo 8957  seqcseq 9080
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3869  ax-sep 3872  ax-nul 3880  ax-pow 3924  ax-pr 3941  ax-un 4157  ax-setind 4247  ax-iinf 4289  ax-cnex 6947  ax-resscn 6948  ax-1cn 6949  ax-1re 6950  ax-icn 6951  ax-addcl 6952  ax-addrcl 6953  ax-mulcl 6954  ax-addcom 6956  ax-addass 6958  ax-distr 6960  ax-i2m1 6961  ax-0id 6964  ax-rnegex 6965  ax-cnre 6967  ax-pre-ltirr 6968  ax-pre-ltwlin 6969  ax-pre-lttrn 6970  ax-pre-ltadd 6972
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2308  df-rex 2309  df-reu 2310  df-rab 2312  df-v 2556  df-sbc 2762  df-csb 2850  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-int 3613  df-iun 3656  df-br 3762  df-opab 3816  df-mpt 3817  df-tr 3852  df-eprel 4023  df-id 4027  df-po 4030  df-iso 4031  df-iord 4090  df-on 4092  df-suc 4095  df-iom 4292  df-xp 4329  df-rel 4330  df-cnv 4331  df-co 4332  df-dm 4333  df-rn 4334  df-res 4335  df-ima 4336  df-iota 4845  df-fun 4882  df-fn 4883  df-f 4884  df-f1 4885  df-fo 4886  df-f1o 4887  df-fv 4888  df-riota 5446  df-ov 5493  df-oprab 5494  df-mpt2 5495  df-1st 5745  df-2nd 5746  df-recs 5898  df-irdg 5935  df-frec 5956  df-1o 5979  df-2o 5980  df-oadd 5983  df-omul 5984  df-er 6084  df-ec 6086  df-qs 6090  df-ni 6374  df-pli 6375  df-mi 6376  df-lti 6377  df-plpq 6414  df-mpq 6415  df-enq 6417  df-nqqs 6418  df-plqqs 6419  df-mqqs 6420  df-1nqqs 6421  df-rq 6422  df-ltnqqs 6423  df-enq0 6494  df-nq0 6495  df-0nq0 6496  df-plq0 6497  df-mq0 6498  df-inp 6536  df-i1p 6537  df-iplp 6538  df-iltp 6540  df-enr 6783  df-nr 6784  df-ltr 6787  df-0r 6788  df-1r 6789  df-0 6868  df-1 6869  df-r 6871  df-lt 6874  df-pnf 7033  df-mnf 7034  df-xr 7035  df-ltxr 7036  df-le 7037  df-sub 7155  df-neg 7156  df-inn 7882  df-n0 8145  df-z 8209  df-uz 8437  df-fz 8833  df-fzo 8958  df-iseq 9081
This theorem is referenced by:  iseqid  9116  iser0  9119
  Copyright terms: Public domain W3C validator