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Theorem seq3z 10125
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 9644 . . 3 (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ𝐾))
31, 2syl 14 . 2 (𝜑𝑁 ∈ (ℤ𝐾))
4 fveqeq2 5362 . . . 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 5362 . . . 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 5362 . . . 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 5362 . . . 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 9643 . . . . . . . . . 10 (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ𝑀))
131, 12syl 14 . . . . . . . . 9 (𝜑𝐾 ∈ (ℤ𝑀))
14 eluzelz 9185 . . . . . . . . 9 (𝐾 ∈ (ℤ𝑀) → 𝐾 ∈ ℤ)
1513, 14syl 14 . . . . . . . 8 (𝜑𝐾 ∈ ℤ)
16 simpr 109 . . . . . . . . . 10 ((𝜑𝑥 ∈ (ℤ𝐾)) → 𝑥 ∈ (ℤ𝐾))
1713adantr 272 . . . . . . . . . 10 ((𝜑𝑥 ∈ (ℤ𝐾)) → 𝐾 ∈ (ℤ𝑀))
18 uztrn 9192 . . . . . . . . . 10 ((𝑥 ∈ (ℤ𝐾) ∧ 𝐾 ∈ (ℤ𝑀)) → 𝑥 ∈ (ℤ𝑀))
1916, 17, 18syl2anc 406 . . . . . . . . 9 ((𝜑𝑥 ∈ (ℤ𝐾)) → 𝑥 ∈ (ℤ𝑀))
20 seq3homo.2 . . . . . . . . 9 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
2119, 20syldan 278 . . . . . . . 8 ((𝜑𝑥 ∈ (ℤ𝐾)) → (𝐹𝑥) ∈ 𝑆)
22 seq3homo.1 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
2315, 21, 22seq3-1 10074 . . . . . . 7 (𝜑 → (seq𝐾( + , 𝐹)‘𝐾) = (𝐹𝐾))
24 seqz.7 . . . . . . 7 (𝜑 → (𝐹𝐾) = 𝑍)
2523, 24eqtrd 2132 . . . . . 6 (𝜑 → (seq𝐾( + , 𝐹)‘𝐾) = 𝑍)
26 seqeq1 10062 . . . . . . . 8 (𝐾 = 𝑀 → seq𝐾( + , 𝐹) = seq𝑀( + , 𝐹))
2726fveq1d 5355 . . . . . . 7 (𝐾 = 𝑀 → (seq𝐾( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝐾))
2827eqeq1d 2108 . . . . . 6 (𝐾 = 𝑀 → ((seq𝐾( + , 𝐹)‘𝐾) = 𝑍 ↔ (seq𝑀( + , 𝐹)‘𝐾) = 𝑍))
2925, 28syl5ibcom 154 . . . . 5 (𝜑 → (𝐾 = 𝑀 → (seq𝑀( + , 𝐹)‘𝐾) = 𝑍))
30 eluzel2 9181 . . . . . . . . . 10 (𝐾 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
3113, 30syl 14 . . . . . . . . 9 (𝜑𝑀 ∈ ℤ)
3231adantr 272 . . . . . . . 8 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → 𝑀 ∈ ℤ)
33 simpr 109 . . . . . . . 8 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → 𝐾 ∈ (ℤ‘(𝑀 + 1)))
3420adantlr 464 . . . . . . . 8 (((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
3522adantlr 464 . . . . . . . 8 (((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
3632, 33, 34, 35seq3m1 10082 . . . . . . 7 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → (seq𝑀( + , 𝐹)‘𝐾) = ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + (𝐹𝐾)))
3724adantr 272 . . . . . . . 8 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → (𝐹𝐾) = 𝑍)
3837oveq2d 5722 . . . . . . 7 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + (𝐹𝐾)) = ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + 𝑍))
39 oveq1 5713 . . . . . . . . 9 (𝑥 = (seq𝑀( + , 𝐹)‘(𝐾 − 1)) → (𝑥 + 𝑍) = ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + 𝑍))
4039eqeq1d 2108 . . . . . . . 8 (𝑥 = (seq𝑀( + , 𝐹)‘(𝐾 − 1)) → ((𝑥 + 𝑍) = 𝑍 ↔ ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + 𝑍) = 𝑍))
41 seqz.4 . . . . . . . . . 10 ((𝜑𝑥𝑆) → (𝑥 + 𝑍) = 𝑍)
4241ralrimiva 2464 . . . . . . . . 9 (𝜑 → ∀𝑥𝑆 (𝑥 + 𝑍) = 𝑍)
4342adantr 272 . . . . . . . 8 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → ∀𝑥𝑆 (𝑥 + 𝑍) = 𝑍)
44 eqid 2100 . . . . . . . . . 10 (ℤ𝑀) = (ℤ𝑀)
4544, 32, 34, 35seqf 10075 . . . . . . . . 9 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → seq𝑀( + , 𝐹):(ℤ𝑀)⟶𝑆)
46 eluzp1m1 9199 . . . . . . . . . 10 ((𝑀 ∈ ℤ ∧ 𝐾 ∈ (ℤ‘(𝑀 + 1))) → (𝐾 − 1) ∈ (ℤ𝑀))
4731, 46sylan 279 . . . . . . . . 9 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → (𝐾 − 1) ∈ (ℤ𝑀))
4845, 47ffvelrnd 5488 . . . . . . . 8 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → (seq𝑀( + , 𝐹)‘(𝐾 − 1)) ∈ 𝑆)
4940, 43, 48rspcdva 2749 . . . . . . 7 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + 𝑍) = 𝑍)
5036, 38, 493eqtrd 2136 . . . . . 6 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → (seq𝑀( + , 𝐹)‘𝐾) = 𝑍)
5150ex 114 . . . . 5 (𝜑 → (𝐾 ∈ (ℤ‘(𝑀 + 1)) → (seq𝑀( + , 𝐹)‘𝐾) = 𝑍))
52 uzp1 9209 . . . . . 6 (𝐾 ∈ (ℤ𝑀) → (𝐾 = 𝑀𝐾 ∈ (ℤ‘(𝑀 + 1))))
5313, 52syl 14 . . . . 5 (𝜑 → (𝐾 = 𝑀𝐾 ∈ (ℤ‘(𝑀 + 1))))
5429, 51, 53mpjaod 679 . . . 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 272 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ℤ𝐾)) → 𝐾 ∈ (ℤ𝑀))
58 uztrn 9192 . . . . . . . . . 10 ((𝑘 ∈ (ℤ𝐾) ∧ 𝐾 ∈ (ℤ𝑀)) → 𝑘 ∈ (ℤ𝑀))
5956, 57, 58syl2anc 406 . . . . . . . . 9 ((𝜑𝑘 ∈ (ℤ𝐾)) → 𝑘 ∈ (ℤ𝑀))
6020adantlr 464 . . . . . . . . 9 (((𝜑𝑘 ∈ (ℤ𝐾)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
6122adantlr 464 . . . . . . . . 9 (((𝜑𝑘 ∈ (ℤ𝐾)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
6259, 60, 61seq3p1 10076 . . . . . . . 8 ((𝜑𝑘 ∈ (ℤ𝐾)) → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = ((seq𝑀( + , 𝐹)‘𝑘) + (𝐹‘(𝑘 + 1))))
6362adantr 272 . . . . . . 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 5721 . . . . . . 7 (((𝜑𝑘 ∈ (ℤ𝐾)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → ((seq𝑀( + , 𝐹)‘𝑘) + (𝐹‘(𝑘 + 1))) = (𝑍 + (𝐹‘(𝑘 + 1))))
66 oveq2 5714 . . . . . . . . . 10 (𝑥 = (𝐹‘(𝑘 + 1)) → (𝑍 + 𝑥) = (𝑍 + (𝐹‘(𝑘 + 1))))
6766eqeq1d 2108 . . . . . . . . 9 (𝑥 = (𝐹‘(𝑘 + 1)) → ((𝑍 + 𝑥) = 𝑍 ↔ (𝑍 + (𝐹‘(𝑘 + 1))) = 𝑍))
68 seqz.3 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → (𝑍 + 𝑥) = 𝑍)
6968ralrimiva 2464 . . . . . . . . . 10 (𝜑 → ∀𝑥𝑆 (𝑍 + 𝑥) = 𝑍)
7069adantr 272 . . . . . . . . 9 ((𝜑𝑘 ∈ (ℤ𝐾)) → ∀𝑥𝑆 (𝑍 + 𝑥) = 𝑍)
71 fveq2 5353 . . . . . . . . . . 11 (𝑥 = (𝑘 + 1) → (𝐹𝑥) = (𝐹‘(𝑘 + 1)))
7271eleq1d 2168 . . . . . . . . . 10 (𝑥 = (𝑘 + 1) → ((𝐹𝑥) ∈ 𝑆 ↔ (𝐹‘(𝑘 + 1)) ∈ 𝑆))
7320ralrimiva 2464 . . . . . . . . . . 11 (𝜑 → ∀𝑥 ∈ (ℤ𝑀)(𝐹𝑥) ∈ 𝑆)
7473adantr 272 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ℤ𝐾)) → ∀𝑥 ∈ (ℤ𝑀)(𝐹𝑥) ∈ 𝑆)
75 peano2uz 9228 . . . . . . . . . . 11 (𝑘 ∈ (ℤ𝑀) → (𝑘 + 1) ∈ (ℤ𝑀))
7659, 75syl 14 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ℤ𝐾)) → (𝑘 + 1) ∈ (ℤ𝑀))
7772, 74, 76rspcdva 2749 . . . . . . . . 9 ((𝜑𝑘 ∈ (ℤ𝐾)) → (𝐹‘(𝑘 + 1)) ∈ 𝑆)
7867, 70, 77rspcdva 2749 . . . . . . . 8 ((𝜑𝑘 ∈ (ℤ𝐾)) → (𝑍 + (𝐹‘(𝑘 + 1))) = 𝑍)
7978adantr 272 . . . . . . 7 (((𝜑𝑘 ∈ (ℤ𝐾)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → (𝑍 + (𝐹‘(𝑘 + 1))) = 𝑍)
8063, 65, 793eqtrd 2136 . . . . . 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 9233 . 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 670   = wceq 1299  wcel 1448  wral 2375  cfv 5059  (class class class)co 5706  1c1 7501   + caddc 7503  cmin 7804  cz 8906  cuz 9176  ...cfz 9631  seqcseq 10059
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-addcom 7595  ax-addass 7597  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-0id 7603  ax-rnegex 7604  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-ltadd 7611
This theorem depends on definitions:  df-bi 116  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-inn 8579  df-n0 8830  df-z 8907  df-uz 9177  df-fz 9632  df-seqfrec 10060
This theorem is referenced by:  bcval5  10350
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