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Theorem iseqid3s 9562
 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.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

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 9127 . . 3 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
3 fveq2 5209 . . . . . 6 (𝑤 = 𝑀 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑀))
43eqeq1d 2090 . . . . 5 (𝑤 = 𝑀 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = 𝑍))
54imbi2d 228 . . . 4 (𝑤 = 𝑀 → ((𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = 𝑍)))
6 fveq2 5209 . . . . . 6 (𝑤 = 𝑘 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑘))
76eqeq1d 2090 . . . . 5 (𝑤 = 𝑘 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = 𝑍))
87imbi2d 228 . . . 4 (𝑤 = 𝑘 → ((𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = 𝑍)))
9 fveq2 5209 . . . . . 6 (𝑤 = (𝑘 + 1) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)))
109eqeq1d 2090 . . . . 5 (𝑤 = (𝑘 + 1) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = 𝑍))
1110imbi2d 228 . . . 4 (𝑤 = (𝑘 + 1) → ((𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = 𝑍)))
12 fveq2 5209 . . . . . 6 (𝑤 = 𝑁 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑁))
1312eqeq1d 2090 . . . . 5 (𝑤 = 𝑁 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = 𝑍))
1413imbi2d 228 . . . 4 (𝑤 = 𝑁 → ((𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = 𝑍)))
15 eluzel2 8705 . . . . . . . 8 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
161, 15syl 14 . . . . . . 7 (𝜑𝑀 ∈ ℤ)
17 iseqid3s.f . . . . . . 7 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
18 iseqid3s.cl . . . . . . 7 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
1916, 17, 18iseq1 9533 . . . . . 6 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (𝐹𝑀))
20 iseqid3s.3 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) = 𝑍)
2120ralrimiva 2435 . . . . . . 7 (𝜑 → ∀𝑥 ∈ (𝑀...𝑁)(𝐹𝑥) = 𝑍)
22 eluzfz1 9126 . . . . . . . 8 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
23 fveq2 5209 . . . . . . . . . 10 (𝑥 = 𝑀 → (𝐹𝑥) = (𝐹𝑀))
2423eqeq1d 2090 . . . . . . . . 9 (𝑥 = 𝑀 → ((𝐹𝑥) = 𝑍 ↔ (𝐹𝑀) = 𝑍))
2524rspcv 2698 . . . . . . . 8 (𝑀 ∈ (𝑀...𝑁) → (∀𝑥 ∈ (𝑀...𝑁)(𝐹𝑥) = 𝑍 → (𝐹𝑀) = 𝑍))
261, 22, 253syl 17 . . . . . . 7 (𝜑 → (∀𝑥 ∈ (𝑀...𝑁)(𝐹𝑥) = 𝑍 → (𝐹𝑀) = 𝑍))
2721, 26mpd 13 . . . . . 6 (𝜑 → (𝐹𝑀) = 𝑍)
2819, 27eqtrd 2114 . . . . 5 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = 𝑍)
2928a1i 9 . . . 4 (𝑁 ∈ (ℤ𝑀) → (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = 𝑍))
30 elfzouz 9238 . . . . . . . . . . 11 (𝑘 ∈ (𝑀..^𝑁) → 𝑘 ∈ (ℤ𝑀))
3130adantl 271 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ (ℤ𝑀))
3217adantlr 461 . . . . . . . . . 10 (((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
3318adantlr 461 . . . . . . . . . 10 (((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
3431, 32, 33iseqp1 9538 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) + (𝐹‘(𝑘 + 1))))
3534adantr 270 . . . . . . . 8 (((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = 𝑍) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) + (𝐹‘(𝑘 + 1))))
36 simpr 108 . . . . . . . . 9 (((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = 𝑍) → (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = 𝑍)
37 fveq2 5209 . . . . . . . . . . . 12 (𝑥 = (𝑘 + 1) → (𝐹𝑥) = (𝐹‘(𝑘 + 1)))
3837eqeq1d 2090 . . . . . . . . . . 11 (𝑥 = (𝑘 + 1) → ((𝐹𝑥) = 𝑍 ↔ (𝐹‘(𝑘 + 1)) = 𝑍))
3921adantr 270 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ (𝑀...𝑁)(𝐹𝑥) = 𝑍)
40 fzofzp1 9313 . . . . . . . . . . . 12 (𝑘 ∈ (𝑀..^𝑁) → (𝑘 + 1) ∈ (𝑀...𝑁))
4140adantl 271 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ (𝑀...𝑁))
4238, 39, 41rspcdva 2708 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑘 + 1)) = 𝑍)
4342adantr 270 . . . . . . . . 9 (((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = 𝑍) → (𝐹‘(𝑘 + 1)) = 𝑍)
4436, 43oveq12d 5561 . . . . . . . 8 (((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = 𝑍) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) + (𝐹‘(𝑘 + 1))) = (𝑍 + 𝑍))
45 iseqid3s.1 . . . . . . . . 9 (𝜑 → (𝑍 + 𝑍) = 𝑍)
4645ad2antrr 472 . . . . . . . 8 (((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = 𝑍) → (𝑍 + 𝑍) = 𝑍)
4735, 44, 463eqtrd 2118 . . . . . . 7 (((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = 𝑍) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = 𝑍)
4847ex 113 . . . . . 6 ((𝜑𝑘 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) = 𝑍 → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = 𝑍))
4948expcom 114 . . . . 5 (𝑘 ∈ (𝑀..^𝑁) → (𝜑 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) = 𝑍 → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = 𝑍)))
5049a2d 26 . . . 4 (𝑘 ∈ (𝑀..^𝑁) → ((𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = 𝑍) → (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = 𝑍)))
515, 8, 11, 14, 29, 50fzind2 9325 . . 3 (𝑁 ∈ (𝑀...𝑁) → (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = 𝑍))
521, 2, 513syl 17 . 2 (𝜑 → (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = 𝑍))
5352pm2.43i 48 1 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = 𝑍)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102   = wceq 1285   ∈ wcel 1434  ∀wral 2349  ‘cfv 4932  (class class class)co 5543  1c1 7044   + caddc 7046  ℤcz 8432  ℤ≥cuz 8700  ...cfz 9105  ..^cfzo 9229  seqcseq 9521 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-addcom 7138  ax-addass 7140  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-0id 7146  ax-rnegex 7147  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-ltadd 7154 This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-iord 4129  df-on 4131  df-ilim 4132  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-frec 6040  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-inn 8107  df-n0 8356  df-z 8433  df-uz 8701  df-fz 9106  df-fzo 9230  df-iseq 9522 This theorem is referenced by:  iseqid  9563  iser0  9568
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