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Theorem iseqid3s 9780
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 9340 . . 3 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
3 fveq2 5252 . . . . . 6 (𝑤 = 𝑀 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑀))
43eqeq1d 2091 . . . . 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 5252 . . . . . 6 (𝑤 = 𝑘 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑘))
76eqeq1d 2091 . . . . 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 5252 . . . . . 6 (𝑤 = (𝑘 + 1) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)))
109eqeq1d 2091 . . . . 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 5252 . . . . . 6 (𝑤 = 𝑁 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑁))
1312eqeq1d 2091 . . . . 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 8918 . . . . . . . 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 9751 . . . . . 6 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (𝐹𝑀))
20 iseqid3s.3 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) = 𝑍)
2120ralrimiva 2440 . . . . . . 7 (𝜑 → ∀𝑥 ∈ (𝑀...𝑁)(𝐹𝑥) = 𝑍)
22 eluzfz1 9339 . . . . . . . 8 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
23 fveq2 5252 . . . . . . . . . 10 (𝑥 = 𝑀 → (𝐹𝑥) = (𝐹𝑀))
2423eqeq1d 2091 . . . . . . . . 9 (𝑥 = 𝑀 → ((𝐹𝑥) = 𝑍 ↔ (𝐹𝑀) = 𝑍))
2524rspcv 2708 . . . . . . . 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 2115 . . . . 5 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = 𝑍)
2928a1i 9 . . . 4 (𝑁 ∈ (ℤ𝑀) → (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = 𝑍))
30 elfzouz 9451 . . . . . . . . . . 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 9756 . . . . . . . . 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 5252 . . . . . . . . . . . 12 (𝑥 = (𝑘 + 1) → (𝐹𝑥) = (𝐹‘(𝑘 + 1)))
3837eqeq1d 2091 . . . . . . . . . . 11 (𝑥 = (𝑘 + 1) → ((𝐹𝑥) = 𝑍 ↔ (𝐹‘(𝑘 + 1)) = 𝑍))
3921adantr 270 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ (𝑀...𝑁)(𝐹𝑥) = 𝑍)
40 fzofzp1 9526 . . . . . . . . . . . 12 (𝑘 ∈ (𝑀..^𝑁) → (𝑘 + 1) ∈ (𝑀...𝑁))
4140adantl 271 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ (𝑀...𝑁))
4238, 39, 41rspcdva 2717 . . . . . . . . . 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 5608 . . . . . . . 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 2119 . . . . . . 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 9538 . . 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 2353  cfv 4968  (class class class)co 5590  1c1 7253   + caddc 7255  cz 8645  cuz 8913  ...cfz 9318  ..^cfzo 9442  seqcseq 9739
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 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315  ax-iinf 4365  ax-cnex 7338  ax-resscn 7339  ax-1cn 7340  ax-1re 7341  ax-icn 7342  ax-addcl 7343  ax-addrcl 7344  ax-mulcl 7345  ax-addcom 7347  ax-addass 7349  ax-distr 7351  ax-i2m1 7352  ax-0lt1 7353  ax-0id 7355  ax-rnegex 7356  ax-cnre 7358  ax-pre-ltirr 7359  ax-pre-ltwlin 7360  ax-pre-lttrn 7361  ax-pre-ltadd 7363
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 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4083  df-iord 4156  df-on 4158  df-ilim 4159  df-suc 4161  df-iom 4368  df-xp 4406  df-rel 4407  df-cnv 4408  df-co 4409  df-dm 4410  df-rn 4411  df-res 4412  df-ima 4413  df-iota 4933  df-fun 4970  df-fn 4971  df-f 4972  df-f1 4973  df-fo 4974  df-f1o 4975  df-fv 4976  df-riota 5546  df-ov 5593  df-oprab 5594  df-mpt2 5595  df-1st 5845  df-2nd 5846  df-recs 6001  df-frec 6087  df-pnf 7426  df-mnf 7427  df-xr 7428  df-ltxr 7429  df-le 7430  df-sub 7557  df-neg 7558  df-inn 8316  df-n0 8565  df-z 8646  df-uz 8914  df-fz 9319  df-fzo 9443  df-iseq 9740
This theorem is referenced by:  iseqid  9781  iser0  9786
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