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Theorem seq3z 10296
 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 9815 . . 3 (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ𝐾))
31, 2syl 14 . 2 (𝜑𝑁 ∈ (ℤ𝐾))
4 fveqeq2 5430 . . . 4 (𝑤 = 𝐾 → ((seq𝑀( + , 𝐹)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹)‘𝐾) = 𝑍))
54imbi2d 229 . . 3 (𝑤 = 𝐾 → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = 𝑍)))
6 fveqeq2 5430 . . . 4 (𝑤 = 𝑘 → ((seq𝑀( + , 𝐹)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍))
76imbi2d 229 . . 3 (𝑤 = 𝑘 → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹)‘𝑘) = 𝑍)))
8 fveqeq2 5430 . . . 4 (𝑤 = (𝑘 + 1) → ((seq𝑀( + , 𝐹)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍))
98imbi2d 229 . . 3 (𝑤 = (𝑘 + 1) → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍)))
10 fveqeq2 5430 . . . 4 (𝑤 = 𝑁 → ((seq𝑀( + , 𝐹)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹)‘𝑁) = 𝑍))
1110imbi2d 229 . . 3 (𝑤 = 𝑁 → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍)))
12 elfzuz 9814 . . . . . . . . . 10 (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ𝑀))
131, 12syl 14 . . . . . . . . 9 (𝜑𝐾 ∈ (ℤ𝑀))
14 eluzelz 9347 . . . . . . . . 9 (𝐾 ∈ (ℤ𝑀) → 𝐾 ∈ ℤ)
1513, 14syl 14 . . . . . . . 8 (𝜑𝐾 ∈ ℤ)
16 simpr 109 . . . . . . . . . 10 ((𝜑𝑥 ∈ (ℤ𝐾)) → 𝑥 ∈ (ℤ𝐾))
1713adantr 274 . . . . . . . . . 10 ((𝜑𝑥 ∈ (ℤ𝐾)) → 𝐾 ∈ (ℤ𝑀))
18 uztrn 9354 . . . . . . . . . 10 ((𝑥 ∈ (ℤ𝐾) ∧ 𝐾 ∈ (ℤ𝑀)) → 𝑥 ∈ (ℤ𝑀))
1916, 17, 18syl2anc 408 . . . . . . . . 9 ((𝜑𝑥 ∈ (ℤ𝐾)) → 𝑥 ∈ (ℤ𝑀))
20 seq3homo.2 . . . . . . . . 9 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
2119, 20syldan 280 . . . . . . . 8 ((𝜑𝑥 ∈ (ℤ𝐾)) → (𝐹𝑥) ∈ 𝑆)
22 seq3homo.1 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
2315, 21, 22seq3-1 10245 . . . . . . 7 (𝜑 → (seq𝐾( + , 𝐹)‘𝐾) = (𝐹𝐾))
24 seqz.7 . . . . . . 7 (𝜑 → (𝐹𝐾) = 𝑍)
2523, 24eqtrd 2172 . . . . . 6 (𝜑 → (seq𝐾( + , 𝐹)‘𝐾) = 𝑍)
26 seqeq1 10233 . . . . . . . 8 (𝐾 = 𝑀 → seq𝐾( + , 𝐹) = seq𝑀( + , 𝐹))
2726fveq1d 5423 . . . . . . 7 (𝐾 = 𝑀 → (seq𝐾( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝐾))
2827eqeq1d 2148 . . . . . 6 (𝐾 = 𝑀 → ((seq𝐾( + , 𝐹)‘𝐾) = 𝑍 ↔ (seq𝑀( + , 𝐹)‘𝐾) = 𝑍))
2925, 28syl5ibcom 154 . . . . 5 (𝜑 → (𝐾 = 𝑀 → (seq𝑀( + , 𝐹)‘𝐾) = 𝑍))
30 eluzel2 9343 . . . . . . . . . 10 (𝐾 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
3113, 30syl 14 . . . . . . . . 9 (𝜑𝑀 ∈ ℤ)
3231adantr 274 . . . . . . . 8 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → 𝑀 ∈ ℤ)
33 simpr 109 . . . . . . . 8 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → 𝐾 ∈ (ℤ‘(𝑀 + 1)))
3420adantlr 468 . . . . . . . 8 (((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
3522adantlr 468 . . . . . . . 8 (((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
3632, 33, 34, 35seq3m1 10253 . . . . . . 7 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → (seq𝑀( + , 𝐹)‘𝐾) = ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + (𝐹𝐾)))
3724adantr 274 . . . . . . . 8 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → (𝐹𝐾) = 𝑍)
3837oveq2d 5790 . . . . . . 7 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + (𝐹𝐾)) = ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + 𝑍))
39 oveq1 5781 . . . . . . . . 9 (𝑥 = (seq𝑀( + , 𝐹)‘(𝐾 − 1)) → (𝑥 + 𝑍) = ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + 𝑍))
4039eqeq1d 2148 . . . . . . . 8 (𝑥 = (seq𝑀( + , 𝐹)‘(𝐾 − 1)) → ((𝑥 + 𝑍) = 𝑍 ↔ ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + 𝑍) = 𝑍))
41 seqz.4 . . . . . . . . . 10 ((𝜑𝑥𝑆) → (𝑥 + 𝑍) = 𝑍)
4241ralrimiva 2505 . . . . . . . . 9 (𝜑 → ∀𝑥𝑆 (𝑥 + 𝑍) = 𝑍)
4342adantr 274 . . . . . . . 8 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → ∀𝑥𝑆 (𝑥 + 𝑍) = 𝑍)
44 eqid 2139 . . . . . . . . . 10 (ℤ𝑀) = (ℤ𝑀)
4544, 32, 34, 35seqf 10246 . . . . . . . . 9 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → seq𝑀( + , 𝐹):(ℤ𝑀)⟶𝑆)
46 eluzp1m1 9361 . . . . . . . . . 10 ((𝑀 ∈ ℤ ∧ 𝐾 ∈ (ℤ‘(𝑀 + 1))) → (𝐾 − 1) ∈ (ℤ𝑀))
4731, 46sylan 281 . . . . . . . . 9 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → (𝐾 − 1) ∈ (ℤ𝑀))
4845, 47ffvelrnd 5556 . . . . . . . 8 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → (seq𝑀( + , 𝐹)‘(𝐾 − 1)) ∈ 𝑆)
4940, 43, 48rspcdva 2794 . . . . . . 7 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + 𝑍) = 𝑍)
5036, 38, 493eqtrd 2176 . . . . . 6 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → (seq𝑀( + , 𝐹)‘𝐾) = 𝑍)
5150ex 114 . . . . 5 (𝜑 → (𝐾 ∈ (ℤ‘(𝑀 + 1)) → (seq𝑀( + , 𝐹)‘𝐾) = 𝑍))
52 uzp1 9371 . . . . . 6 (𝐾 ∈ (ℤ𝑀) → (𝐾 = 𝑀𝐾 ∈ (ℤ‘(𝑀 + 1))))
5313, 52syl 14 . . . . 5 (𝜑 → (𝐾 = 𝑀𝐾 ∈ (ℤ‘(𝑀 + 1))))
5429, 51, 53mpjaod 707 . . . 4 (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = 𝑍)
5554a1i 9 . . 3 (𝐾 ∈ ℤ → (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = 𝑍))
56 simpr 109 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ℤ𝐾)) → 𝑘 ∈ (ℤ𝐾))
5713adantr 274 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ℤ𝐾)) → 𝐾 ∈ (ℤ𝑀))
58 uztrn 9354 . . . . . . . . . 10 ((𝑘 ∈ (ℤ𝐾) ∧ 𝐾 ∈ (ℤ𝑀)) → 𝑘 ∈ (ℤ𝑀))
5956, 57, 58syl2anc 408 . . . . . . . . 9 ((𝜑𝑘 ∈ (ℤ𝐾)) → 𝑘 ∈ (ℤ𝑀))
6020adantlr 468 . . . . . . . . 9 (((𝜑𝑘 ∈ (ℤ𝐾)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
6122adantlr 468 . . . . . . . . 9 (((𝜑𝑘 ∈ (ℤ𝐾)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
6259, 60, 61seq3p1 10247 . . . . . . . 8 ((𝜑𝑘 ∈ (ℤ𝐾)) → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = ((seq𝑀( + , 𝐹)‘𝑘) + (𝐹‘(𝑘 + 1))))
6362adantr 274 . . . . . . 7 (((𝜑𝑘 ∈ (ℤ𝐾)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = ((seq𝑀( + , 𝐹)‘𝑘) + (𝐹‘(𝑘 + 1))))
64 simpr 109 . . . . . . . 8 (((𝜑𝑘 ∈ (ℤ𝐾)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → (seq𝑀( + , 𝐹)‘𝑘) = 𝑍)
6564oveq1d 5789 . . . . . . 7 (((𝜑𝑘 ∈ (ℤ𝐾)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → ((seq𝑀( + , 𝐹)‘𝑘) + (𝐹‘(𝑘 + 1))) = (𝑍 + (𝐹‘(𝑘 + 1))))
66 oveq2 5782 . . . . . . . . . 10 (𝑥 = (𝐹‘(𝑘 + 1)) → (𝑍 + 𝑥) = (𝑍 + (𝐹‘(𝑘 + 1))))
6766eqeq1d 2148 . . . . . . . . 9 (𝑥 = (𝐹‘(𝑘 + 1)) → ((𝑍 + 𝑥) = 𝑍 ↔ (𝑍 + (𝐹‘(𝑘 + 1))) = 𝑍))
68 seqz.3 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → (𝑍 + 𝑥) = 𝑍)
6968ralrimiva 2505 . . . . . . . . . 10 (𝜑 → ∀𝑥𝑆 (𝑍 + 𝑥) = 𝑍)
7069adantr 274 . . . . . . . . 9 ((𝜑𝑘 ∈ (ℤ𝐾)) → ∀𝑥𝑆 (𝑍 + 𝑥) = 𝑍)
71 fveq2 5421 . . . . . . . . . . 11 (𝑥 = (𝑘 + 1) → (𝐹𝑥) = (𝐹‘(𝑘 + 1)))
7271eleq1d 2208 . . . . . . . . . 10 (𝑥 = (𝑘 + 1) → ((𝐹𝑥) ∈ 𝑆 ↔ (𝐹‘(𝑘 + 1)) ∈ 𝑆))
7320ralrimiva 2505 . . . . . . . . . . 11 (𝜑 → ∀𝑥 ∈ (ℤ𝑀)(𝐹𝑥) ∈ 𝑆)
7473adantr 274 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ℤ𝐾)) → ∀𝑥 ∈ (ℤ𝑀)(𝐹𝑥) ∈ 𝑆)
75 peano2uz 9390 . . . . . . . . . . 11 (𝑘 ∈ (ℤ𝑀) → (𝑘 + 1) ∈ (ℤ𝑀))
7659, 75syl 14 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ℤ𝐾)) → (𝑘 + 1) ∈ (ℤ𝑀))
7772, 74, 76rspcdva 2794 . . . . . . . . 9 ((𝜑𝑘 ∈ (ℤ𝐾)) → (𝐹‘(𝑘 + 1)) ∈ 𝑆)
7867, 70, 77rspcdva 2794 . . . . . . . 8 ((𝜑𝑘 ∈ (ℤ𝐾)) → (𝑍 + (𝐹‘(𝑘 + 1))) = 𝑍)
7978adantr 274 . . . . . . 7 (((𝜑𝑘 ∈ (ℤ𝐾)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → (𝑍 + (𝐹‘(𝑘 + 1))) = 𝑍)
8063, 65, 793eqtrd 2176 . . . . . 6 (((𝜑𝑘 ∈ (ℤ𝐾)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍)
8180ex 114 . . . . 5 ((𝜑𝑘 ∈ (ℤ𝐾)) → ((seq𝑀( + , 𝐹)‘𝑘) = 𝑍 → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍))
8281expcom 115 . . . 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 9395 . 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 103   ∨ wo 697   = wceq 1331   ∈ wcel 1480  ∀wral 2416  ‘cfv 5123  (class class class)co 5774  1c1 7633   + caddc 7635   − cmin 7945  ℤcz 9066  ℤ≥cuz 9338  ...cfz 9802  seqcseq 10230 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-addcom 7732  ax-addass 7734  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-0id 7740  ax-rnegex 7741  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-ltadd 7748 This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-inn 8733  df-n0 8990  df-z 9067  df-uz 9339  df-fz 9803  df-seqfrec 10231 This theorem is referenced by:  bcval5  10521
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