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Theorem seq3z 10528
Description: If the operation + has an absorbing element 𝑍 (a.k.a. zero element), then any sequence containing a 𝑍 evaluates to 𝑍. (Contributed by Mario Carneiro, 27-May-2014.) (Revised by Jim Kingdon, 23-Apr-2023.)
Hypotheses
Ref Expression
seq3homo.1 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
seq3homo.2 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
seqz.3 ((𝜑𝑥𝑆) → (𝑍 + 𝑥) = 𝑍)
seqz.4 ((𝜑𝑥𝑆) → (𝑥 + 𝑍) = 𝑍)
seqz.5 (𝜑𝐾 ∈ (𝑀...𝑁))
seqz.7 (𝜑 → (𝐹𝐾) = 𝑍)
Assertion
Ref Expression
seq3z (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍)
Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝑀,𝑦   𝑥,𝑁,𝑦   𝜑,𝑥,𝑦   𝑥,𝐾,𝑦   𝑥, + ,𝑦   𝑥,𝑆,𝑦   𝑥,𝑍,𝑦

Proof of Theorem seq3z
Dummy variables 𝑘 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 seqz.5 . . 3 (𝜑𝐾 ∈ (𝑀...𝑁))
2 elfzuz3 10039 . . 3 (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ𝐾))
31, 2syl 14 . 2 (𝜑𝑁 ∈ (ℤ𝐾))
4 fveqeq2 5538 . . . 4 (𝑤 = 𝐾 → ((seq𝑀( + , 𝐹)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹)‘𝐾) = 𝑍))
54imbi2d 230 . . 3 (𝑤 = 𝐾 → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = 𝑍)))
6 fveqeq2 5538 . . . 4 (𝑤 = 𝑘 → ((seq𝑀( + , 𝐹)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍))
76imbi2d 230 . . 3 (𝑤 = 𝑘 → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹)‘𝑘) = 𝑍)))
8 fveqeq2 5538 . . . 4 (𝑤 = (𝑘 + 1) → ((seq𝑀( + , 𝐹)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍))
98imbi2d 230 . . 3 (𝑤 = (𝑘 + 1) → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍)))
10 fveqeq2 5538 . . . 4 (𝑤 = 𝑁 → ((seq𝑀( + , 𝐹)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹)‘𝑁) = 𝑍))
1110imbi2d 230 . . 3 (𝑤 = 𝑁 → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍)))
12 elfzuz 10038 . . . . . . . . . 10 (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ𝑀))
131, 12syl 14 . . . . . . . . 9 (𝜑𝐾 ∈ (ℤ𝑀))
14 eluzelz 9554 . . . . . . . . 9 (𝐾 ∈ (ℤ𝑀) → 𝐾 ∈ ℤ)
1513, 14syl 14 . . . . . . . 8 (𝜑𝐾 ∈ ℤ)
16 simpr 110 . . . . . . . . . 10 ((𝜑𝑥 ∈ (ℤ𝐾)) → 𝑥 ∈ (ℤ𝐾))
1713adantr 276 . . . . . . . . . 10 ((𝜑𝑥 ∈ (ℤ𝐾)) → 𝐾 ∈ (ℤ𝑀))
18 uztrn 9561 . . . . . . . . . 10 ((𝑥 ∈ (ℤ𝐾) ∧ 𝐾 ∈ (ℤ𝑀)) → 𝑥 ∈ (ℤ𝑀))
1916, 17, 18syl2anc 411 . . . . . . . . 9 ((𝜑𝑥 ∈ (ℤ𝐾)) → 𝑥 ∈ (ℤ𝑀))
20 seq3homo.2 . . . . . . . . 9 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
2119, 20syldan 282 . . . . . . . 8 ((𝜑𝑥 ∈ (ℤ𝐾)) → (𝐹𝑥) ∈ 𝑆)
22 seq3homo.1 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
2315, 21, 22seq3-1 10477 . . . . . . 7 (𝜑 → (seq𝐾( + , 𝐹)‘𝐾) = (𝐹𝐾))
24 seqz.7 . . . . . . 7 (𝜑 → (𝐹𝐾) = 𝑍)
2523, 24eqtrd 2221 . . . . . 6 (𝜑 → (seq𝐾( + , 𝐹)‘𝐾) = 𝑍)
26 seqeq1 10465 . . . . . . . 8 (𝐾 = 𝑀 → seq𝐾( + , 𝐹) = seq𝑀( + , 𝐹))
2726fveq1d 5531 . . . . . . 7 (𝐾 = 𝑀 → (seq𝐾( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝐾))
2827eqeq1d 2197 . . . . . 6 (𝐾 = 𝑀 → ((seq𝐾( + , 𝐹)‘𝐾) = 𝑍 ↔ (seq𝑀( + , 𝐹)‘𝐾) = 𝑍))
2925, 28syl5ibcom 155 . . . . 5 (𝜑 → (𝐾 = 𝑀 → (seq𝑀( + , 𝐹)‘𝐾) = 𝑍))
30 eluzel2 9550 . . . . . . . . . 10 (𝐾 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
3113, 30syl 14 . . . . . . . . 9 (𝜑𝑀 ∈ ℤ)
3231adantr 276 . . . . . . . 8 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → 𝑀 ∈ ℤ)
33 simpr 110 . . . . . . . 8 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → 𝐾 ∈ (ℤ‘(𝑀 + 1)))
3420adantlr 477 . . . . . . . 8 (((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
3522adantlr 477 . . . . . . . 8 (((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
3632, 33, 34, 35seq3m1 10485 . . . . . . 7 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → (seq𝑀( + , 𝐹)‘𝐾) = ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + (𝐹𝐾)))
3724adantr 276 . . . . . . . 8 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → (𝐹𝐾) = 𝑍)
3837oveq2d 5906 . . . . . . 7 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + (𝐹𝐾)) = ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + 𝑍))
39 oveq1 5897 . . . . . . . . 9 (𝑥 = (seq𝑀( + , 𝐹)‘(𝐾 − 1)) → (𝑥 + 𝑍) = ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + 𝑍))
4039eqeq1d 2197 . . . . . . . 8 (𝑥 = (seq𝑀( + , 𝐹)‘(𝐾 − 1)) → ((𝑥 + 𝑍) = 𝑍 ↔ ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + 𝑍) = 𝑍))
41 seqz.4 . . . . . . . . . 10 ((𝜑𝑥𝑆) → (𝑥 + 𝑍) = 𝑍)
4241ralrimiva 2562 . . . . . . . . 9 (𝜑 → ∀𝑥𝑆 (𝑥 + 𝑍) = 𝑍)
4342adantr 276 . . . . . . . 8 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → ∀𝑥𝑆 (𝑥 + 𝑍) = 𝑍)
44 eqid 2188 . . . . . . . . . 10 (ℤ𝑀) = (ℤ𝑀)
4544, 32, 34, 35seqf 10478 . . . . . . . . 9 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → seq𝑀( + , 𝐹):(ℤ𝑀)⟶𝑆)
46 eluzp1m1 9568 . . . . . . . . . 10 ((𝑀 ∈ ℤ ∧ 𝐾 ∈ (ℤ‘(𝑀 + 1))) → (𝐾 − 1) ∈ (ℤ𝑀))
4731, 46sylan 283 . . . . . . . . 9 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → (𝐾 − 1) ∈ (ℤ𝑀))
4845, 47ffvelcdmd 5667 . . . . . . . 8 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → (seq𝑀( + , 𝐹)‘(𝐾 − 1)) ∈ 𝑆)
4940, 43, 48rspcdva 2860 . . . . . . 7 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + 𝑍) = 𝑍)
5036, 38, 493eqtrd 2225 . . . . . 6 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → (seq𝑀( + , 𝐹)‘𝐾) = 𝑍)
5150ex 115 . . . . 5 (𝜑 → (𝐾 ∈ (ℤ‘(𝑀 + 1)) → (seq𝑀( + , 𝐹)‘𝐾) = 𝑍))
52 uzp1 9578 . . . . . 6 (𝐾 ∈ (ℤ𝑀) → (𝐾 = 𝑀𝐾 ∈ (ℤ‘(𝑀 + 1))))
5313, 52syl 14 . . . . 5 (𝜑 → (𝐾 = 𝑀𝐾 ∈ (ℤ‘(𝑀 + 1))))
5429, 51, 53mpjaod 719 . . . 4 (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = 𝑍)
5554a1i 9 . . 3 (𝐾 ∈ ℤ → (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = 𝑍))
56 simpr 110 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ℤ𝐾)) → 𝑘 ∈ (ℤ𝐾))
5713adantr 276 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ℤ𝐾)) → 𝐾 ∈ (ℤ𝑀))
58 uztrn 9561 . . . . . . . . . 10 ((𝑘 ∈ (ℤ𝐾) ∧ 𝐾 ∈ (ℤ𝑀)) → 𝑘 ∈ (ℤ𝑀))
5956, 57, 58syl2anc 411 . . . . . . . . 9 ((𝜑𝑘 ∈ (ℤ𝐾)) → 𝑘 ∈ (ℤ𝑀))
6020adantlr 477 . . . . . . . . 9 (((𝜑𝑘 ∈ (ℤ𝐾)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
6122adantlr 477 . . . . . . . . 9 (((𝜑𝑘 ∈ (ℤ𝐾)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
6259, 60, 61seq3p1 10479 . . . . . . . 8 ((𝜑𝑘 ∈ (ℤ𝐾)) → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = ((seq𝑀( + , 𝐹)‘𝑘) + (𝐹‘(𝑘 + 1))))
6362adantr 276 . . . . . . 7 (((𝜑𝑘 ∈ (ℤ𝐾)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = ((seq𝑀( + , 𝐹)‘𝑘) + (𝐹‘(𝑘 + 1))))
64 simpr 110 . . . . . . . 8 (((𝜑𝑘 ∈ (ℤ𝐾)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → (seq𝑀( + , 𝐹)‘𝑘) = 𝑍)
6564oveq1d 5905 . . . . . . 7 (((𝜑𝑘 ∈ (ℤ𝐾)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → ((seq𝑀( + , 𝐹)‘𝑘) + (𝐹‘(𝑘 + 1))) = (𝑍 + (𝐹‘(𝑘 + 1))))
66 oveq2 5898 . . . . . . . . . 10 (𝑥 = (𝐹‘(𝑘 + 1)) → (𝑍 + 𝑥) = (𝑍 + (𝐹‘(𝑘 + 1))))
6766eqeq1d 2197 . . . . . . . . 9 (𝑥 = (𝐹‘(𝑘 + 1)) → ((𝑍 + 𝑥) = 𝑍 ↔ (𝑍 + (𝐹‘(𝑘 + 1))) = 𝑍))
68 seqz.3 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → (𝑍 + 𝑥) = 𝑍)
6968ralrimiva 2562 . . . . . . . . . 10 (𝜑 → ∀𝑥𝑆 (𝑍 + 𝑥) = 𝑍)
7069adantr 276 . . . . . . . . 9 ((𝜑𝑘 ∈ (ℤ𝐾)) → ∀𝑥𝑆 (𝑍 + 𝑥) = 𝑍)
71 fveq2 5529 . . . . . . . . . . 11 (𝑥 = (𝑘 + 1) → (𝐹𝑥) = (𝐹‘(𝑘 + 1)))
7271eleq1d 2257 . . . . . . . . . 10 (𝑥 = (𝑘 + 1) → ((𝐹𝑥) ∈ 𝑆 ↔ (𝐹‘(𝑘 + 1)) ∈ 𝑆))
7320ralrimiva 2562 . . . . . . . . . . 11 (𝜑 → ∀𝑥 ∈ (ℤ𝑀)(𝐹𝑥) ∈ 𝑆)
7473adantr 276 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ℤ𝐾)) → ∀𝑥 ∈ (ℤ𝑀)(𝐹𝑥) ∈ 𝑆)
75 peano2uz 9600 . . . . . . . . . . 11 (𝑘 ∈ (ℤ𝑀) → (𝑘 + 1) ∈ (ℤ𝑀))
7659, 75syl 14 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ℤ𝐾)) → (𝑘 + 1) ∈ (ℤ𝑀))
7772, 74, 76rspcdva 2860 . . . . . . . . 9 ((𝜑𝑘 ∈ (ℤ𝐾)) → (𝐹‘(𝑘 + 1)) ∈ 𝑆)
7867, 70, 77rspcdva 2860 . . . . . . . 8 ((𝜑𝑘 ∈ (ℤ𝐾)) → (𝑍 + (𝐹‘(𝑘 + 1))) = 𝑍)
7978adantr 276 . . . . . . 7 (((𝜑𝑘 ∈ (ℤ𝐾)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → (𝑍 + (𝐹‘(𝑘 + 1))) = 𝑍)
8063, 65, 793eqtrd 2225 . . . . . 6 (((𝜑𝑘 ∈ (ℤ𝐾)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍)
8180ex 115 . . . . 5 ((𝜑𝑘 ∈ (ℤ𝐾)) → ((seq𝑀( + , 𝐹)‘𝑘) = 𝑍 → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍))
8281expcom 116 . . . 4 (𝑘 ∈ (ℤ𝐾) → (𝜑 → ((seq𝑀( + , 𝐹)‘𝑘) = 𝑍 → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍)))
8382a2d 26 . . 3 (𝑘 ∈ (ℤ𝐾) → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → (𝜑 → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍)))
845, 7, 9, 11, 55, 83uzind4 9605 . 2 (𝑁 ∈ (ℤ𝐾) → (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍))
853, 84mpcom 36 1 (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wo 709   = wceq 1363  wcel 2159  wral 2467  cfv 5230  (class class class)co 5890  1c1 7829   + caddc 7831  cmin 8145  cz 9270  cuz 9545  ...cfz 10025  seqcseq 10462
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-coll 4132  ax-sep 4135  ax-nul 4143  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-setind 4550  ax-iinf 4601  ax-cnex 7919  ax-resscn 7920  ax-1cn 7921  ax-1re 7922  ax-icn 7923  ax-addcl 7924  ax-addrcl 7925  ax-mulcl 7926  ax-addcom 7928  ax-addass 7930  ax-distr 7932  ax-i2m1 7933  ax-0lt1 7934  ax-0id 7936  ax-rnegex 7937  ax-cnre 7939  ax-pre-ltirr 7940  ax-pre-ltwlin 7941  ax-pre-lttrn 7942  ax-pre-ltadd 7944
This theorem depends on definitions:  df-bi 117  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-nel 2455  df-ral 2472  df-rex 2473  df-reu 2474  df-rab 2476  df-v 2753  df-sbc 2977  df-csb 3072  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-nul 3437  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-int 3859  df-iun 3902  df-br 4018  df-opab 4079  df-mpt 4080  df-tr 4116  df-id 4307  df-iord 4380  df-on 4382  df-ilim 4383  df-suc 4385  df-iom 4604  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237  df-fv 5238  df-riota 5846  df-ov 5893  df-oprab 5894  df-mpo 5895  df-1st 6158  df-2nd 6159  df-recs 6323  df-frec 6409  df-pnf 8011  df-mnf 8012  df-xr 8013  df-ltxr 8014  df-le 8015  df-sub 8147  df-neg 8148  df-inn 8937  df-n0 9194  df-z 9271  df-uz 9546  df-fz 10026  df-seqfrec 10463
This theorem is referenced by:  bcval5  10760  lgsne0  14822
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