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Theorem seq3id3 10537
Description: A sequence that consists entirely of "zeroes" sums to "zero". More precisely, a constant sequence with value an element which is a + -idempotent sums (or "+'s") to that element. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Jim Kingdon, 8-Apr-2023.)
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
seq3id3 (𝜑 → (seq𝑀( + , 𝐹)‘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 seq3id3
Dummy variables 𝑘 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iseqid3s.2 . . 3 (𝜑𝑁 ∈ (ℤ𝑀))
2 eluzfz2 10061 . . 3 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
3 fveqeq2 5543 . . . . 5 (𝑤 = 𝑀 → ((seq𝑀( + , 𝐹)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹)‘𝑀) = 𝑍))
43imbi2d 230 . . . 4 (𝑤 = 𝑀 → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = 𝑍)))
5 fveqeq2 5543 . . . . 5 (𝑤 = 𝑘 → ((seq𝑀( + , 𝐹)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍))
65imbi2d 230 . . . 4 (𝑤 = 𝑘 → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹)‘𝑘) = 𝑍)))
7 fveqeq2 5543 . . . . 5 (𝑤 = (𝑘 + 1) → ((seq𝑀( + , 𝐹)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍))
87imbi2d 230 . . . 4 (𝑤 = (𝑘 + 1) → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍)))
9 fveqeq2 5543 . . . . 5 (𝑤 = 𝑁 → ((seq𝑀( + , 𝐹)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹)‘𝑁) = 𝑍))
109imbi2d 230 . . . 4 (𝑤 = 𝑁 → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍)))
11 eluzel2 9562 . . . . . . . 8 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
121, 11syl 14 . . . . . . 7 (𝜑𝑀 ∈ ℤ)
13 iseqid3s.f . . . . . . 7 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
14 iseqid3s.cl . . . . . . 7 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
1512, 13, 14seq3-1 10490 . . . . . 6 (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹𝑀))
16 iseqid3s.3 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) = 𝑍)
1716ralrimiva 2563 . . . . . . 7 (𝜑 → ∀𝑥 ∈ (𝑀...𝑁)(𝐹𝑥) = 𝑍)
18 eluzfz1 10060 . . . . . . . 8 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
19 fveqeq2 5543 . . . . . . . . 9 (𝑥 = 𝑀 → ((𝐹𝑥) = 𝑍 ↔ (𝐹𝑀) = 𝑍))
2019rspcv 2852 . . . . . . . 8 (𝑀 ∈ (𝑀...𝑁) → (∀𝑥 ∈ (𝑀...𝑁)(𝐹𝑥) = 𝑍 → (𝐹𝑀) = 𝑍))
211, 18, 203syl 17 . . . . . . 7 (𝜑 → (∀𝑥 ∈ (𝑀...𝑁)(𝐹𝑥) = 𝑍 → (𝐹𝑀) = 𝑍))
2217, 21mpd 13 . . . . . 6 (𝜑 → (𝐹𝑀) = 𝑍)
2315, 22eqtrd 2222 . . . . 5 (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = 𝑍)
2423a1i 9 . . . 4 (𝑁 ∈ (ℤ𝑀) → (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = 𝑍))
25 elfzouz 10180 . . . . . . . . . . 11 (𝑘 ∈ (𝑀..^𝑁) → 𝑘 ∈ (ℤ𝑀))
2625adantl 277 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ (ℤ𝑀))
2713adantlr 477 . . . . . . . . . 10 (((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
2814adantlr 477 . . . . . . . . . 10 (((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
2926, 27, 28seq3p1 10492 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = ((seq𝑀( + , 𝐹)‘𝑘) + (𝐹‘(𝑘 + 1))))
3029adantr 276 . . . . . . . 8 (((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = ((seq𝑀( + , 𝐹)‘𝑘) + (𝐹‘(𝑘 + 1))))
31 simpr 110 . . . . . . . . 9 (((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → (seq𝑀( + , 𝐹)‘𝑘) = 𝑍)
32 fveqeq2 5543 . . . . . . . . . . 11 (𝑥 = (𝑘 + 1) → ((𝐹𝑥) = 𝑍 ↔ (𝐹‘(𝑘 + 1)) = 𝑍))
3317adantr 276 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ (𝑀...𝑁)(𝐹𝑥) = 𝑍)
34 fzofzp1 10256 . . . . . . . . . . . 12 (𝑘 ∈ (𝑀..^𝑁) → (𝑘 + 1) ∈ (𝑀...𝑁))
3534adantl 277 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ (𝑀...𝑁))
3632, 33, 35rspcdva 2861 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑘 + 1)) = 𝑍)
3736adantr 276 . . . . . . . . 9 (((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → (𝐹‘(𝑘 + 1)) = 𝑍)
3831, 37oveq12d 5913 . . . . . . . 8 (((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → ((seq𝑀( + , 𝐹)‘𝑘) + (𝐹‘(𝑘 + 1))) = (𝑍 + 𝑍))
39 iseqid3s.1 . . . . . . . . 9 (𝜑 → (𝑍 + 𝑍) = 𝑍)
4039ad2antrr 488 . . . . . . . 8 (((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → (𝑍 + 𝑍) = 𝑍)
4130, 38, 403eqtrd 2226 . . . . . . 7 (((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍)
4241ex 115 . . . . . 6 ((𝜑𝑘 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐹)‘𝑘) = 𝑍 → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍))
4342expcom 116 . . . . 5 (𝑘 ∈ (𝑀..^𝑁) → (𝜑 → ((seq𝑀( + , 𝐹)‘𝑘) = 𝑍 → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍)))
4443a2d 26 . . . 4 (𝑘 ∈ (𝑀..^𝑁) → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → (𝜑 → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍)))
454, 6, 8, 10, 24, 44fzind2 10268 . . 3 (𝑁 ∈ (𝑀...𝑁) → (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍))
461, 2, 453syl 17 . 2 (𝜑 → (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍))
4746pm2.43i 49 1 (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2160  wral 2468  cfv 5235  (class class class)co 5895  1c1 7841   + caddc 7843  cz 9282  cuz 9557  ...cfz 10037  ..^cfzo 10171  seqcseq 10475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-addcom 7940  ax-addass 7942  ax-distr 7944  ax-i2m1 7945  ax-0lt1 7946  ax-0id 7948  ax-rnegex 7949  ax-cnre 7951  ax-pre-ltirr 7952  ax-pre-ltwlin 7953  ax-pre-lttrn 7954  ax-pre-ltadd 7956
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5851  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-recs 6329  df-frec 6415  df-pnf 8023  df-mnf 8024  df-xr 8025  df-ltxr 8026  df-le 8027  df-sub 8159  df-neg 8160  df-inn 8949  df-n0 9206  df-z 9283  df-uz 9558  df-fz 10038  df-fzo 10172  df-seqfrec 10476
This theorem is referenced by:  seq3id  10538  ser0  10544  prodf1  11581  mulgnn0z  13086  lgsval2lem  14864
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