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Theorem iseqz 9783
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.)
Hypotheses
Ref Expression
iseqhomo.1 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
iseqhomo.2 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
iseqhomo.s (𝜑𝑆𝑉)
iseqz.3 ((𝜑𝑥𝑆) → (𝑍 + 𝑥) = 𝑍)
iseqz.4 ((𝜑𝑥𝑆) → (𝑥 + 𝑍) = 𝑍)
iseqz.5 (𝜑𝐾 ∈ (𝑀...𝑁))
iseqz.6 (𝜑𝑁𝑉)
iseqz.7 (𝜑 → (𝐹𝐾) = 𝑍)
Assertion
Ref Expression
iseqz (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘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
Allowed substitution hints:   𝑉(𝑥,𝑦)

Proof of Theorem iseqz
Dummy variables 𝑘 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iseqz.5 . . 3 (𝜑𝐾 ∈ (𝑀...𝑁))
2 elfzuz3 9330 . . 3 (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ𝐾))
31, 2syl 14 . 2 (𝜑𝑁 ∈ (ℤ𝐾))
4 fveq2 5251 . . . . 5 (𝑤 = 𝐾 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝐾))
54eqeq1d 2091 . . . 4 (𝑤 = 𝐾 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = 𝑍))
65imbi2d 228 . . 3 (𝑤 = 𝐾 → ((𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = 𝑍)))
7 fveq2 5251 . . . . 5 (𝑤 = 𝑘 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑘))
87eqeq1d 2091 . . . 4 (𝑤 = 𝑘 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = 𝑍))
98imbi2d 228 . . 3 (𝑤 = 𝑘 → ((𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = 𝑍)))
10 fveq2 5251 . . . . 5 (𝑤 = (𝑘 + 1) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)))
1110eqeq1d 2091 . . . 4 (𝑤 = (𝑘 + 1) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = 𝑍))
1211imbi2d 228 . . 3 (𝑤 = (𝑘 + 1) → ((𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = 𝑍)))
13 fveq2 5251 . . . . 5 (𝑤 = 𝑁 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑁))
1413eqeq1d 2091 . . . 4 (𝑤 = 𝑁 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = 𝑍))
1514imbi2d 228 . . 3 (𝑤 = 𝑁 → ((𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = 𝑍)))
16 elfzuz 9329 . . . . . . . . . 10 (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ𝑀))
171, 16syl 14 . . . . . . . . 9 (𝜑𝐾 ∈ (ℤ𝑀))
18 eluzelz 8921 . . . . . . . . 9 (𝐾 ∈ (ℤ𝑀) → 𝐾 ∈ ℤ)
1917, 18syl 14 . . . . . . . 8 (𝜑𝐾 ∈ ℤ)
20 simpr 108 . . . . . . . . . 10 ((𝜑𝑥 ∈ (ℤ𝐾)) → 𝑥 ∈ (ℤ𝐾))
2117adantr 270 . . . . . . . . . 10 ((𝜑𝑥 ∈ (ℤ𝐾)) → 𝐾 ∈ (ℤ𝑀))
22 uztrn 8928 . . . . . . . . . 10 ((𝑥 ∈ (ℤ𝐾) ∧ 𝐾 ∈ (ℤ𝑀)) → 𝑥 ∈ (ℤ𝑀))
2320, 21, 22syl2anc 403 . . . . . . . . 9 ((𝜑𝑥 ∈ (ℤ𝐾)) → 𝑥 ∈ (ℤ𝑀))
24 iseqhomo.2 . . . . . . . . 9 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
2523, 24syldan 276 . . . . . . . 8 ((𝜑𝑥 ∈ (ℤ𝐾)) → (𝐹𝑥) ∈ 𝑆)
26 iseqhomo.1 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
2719, 25, 26iseq1 9750 . . . . . . 7 (𝜑 → (seq𝐾( + , 𝐹, 𝑆)‘𝐾) = (𝐹𝐾))
28 iseqz.7 . . . . . . 7 (𝜑 → (𝐹𝐾) = 𝑍)
2927, 28eqtrd 2115 . . . . . 6 (𝜑 → (seq𝐾( + , 𝐹, 𝑆)‘𝐾) = 𝑍)
30 iseqeq1 9741 . . . . . . . 8 (𝐾 = 𝑀 → seq𝐾( + , 𝐹, 𝑆) = seq𝑀( + , 𝐹, 𝑆))
3130fveq1d 5253 . . . . . . 7 (𝐾 = 𝑀 → (seq𝐾( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝐾))
3231eqeq1d 2091 . . . . . 6 (𝐾 = 𝑀 → ((seq𝐾( + , 𝐹, 𝑆)‘𝐾) = 𝑍 ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = 𝑍))
3329, 32syl5ibcom 153 . . . . 5 (𝜑 → (𝐾 = 𝑀 → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = 𝑍))
34 eluzel2 8917 . . . . . . . . . 10 (𝐾 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
3517, 34syl 14 . . . . . . . . 9 (𝜑𝑀 ∈ ℤ)
3635adantr 270 . . . . . . . 8 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → 𝑀 ∈ ℤ)
37 simpr 108 . . . . . . . 8 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → 𝐾 ∈ (ℤ‘(𝑀 + 1)))
3824adantlr 461 . . . . . . . 8 (((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
3926adantlr 461 . . . . . . . 8 (((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
4036, 37, 38, 39iseqm1 9760 . . . . . . 7 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = ((seq𝑀( + , 𝐹, 𝑆)‘(𝐾 − 1)) + (𝐹𝐾)))
4128adantr 270 . . . . . . . 8 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → (𝐹𝐾) = 𝑍)
4241oveq2d 5605 . . . . . . 7 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → ((seq𝑀( + , 𝐹, 𝑆)‘(𝐾 − 1)) + (𝐹𝐾)) = ((seq𝑀( + , 𝐹, 𝑆)‘(𝐾 − 1)) + 𝑍))
43 oveq1 5596 . . . . . . . . 9 (𝑥 = (seq𝑀( + , 𝐹, 𝑆)‘(𝐾 − 1)) → (𝑥 + 𝑍) = ((seq𝑀( + , 𝐹, 𝑆)‘(𝐾 − 1)) + 𝑍))
4443eqeq1d 2091 . . . . . . . 8 (𝑥 = (seq𝑀( + , 𝐹, 𝑆)‘(𝐾 − 1)) → ((𝑥 + 𝑍) = 𝑍 ↔ ((seq𝑀( + , 𝐹, 𝑆)‘(𝐾 − 1)) + 𝑍) = 𝑍))
45 iseqz.4 . . . . . . . . . 10 ((𝜑𝑥𝑆) → (𝑥 + 𝑍) = 𝑍)
4645ralrimiva 2440 . . . . . . . . 9 (𝜑 → ∀𝑥𝑆 (𝑥 + 𝑍) = 𝑍)
4746adantr 270 . . . . . . . 8 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → ∀𝑥𝑆 (𝑥 + 𝑍) = 𝑍)
48 eluzp1m1 8935 . . . . . . . . . 10 ((𝑀 ∈ ℤ ∧ 𝐾 ∈ (ℤ‘(𝑀 + 1))) → (𝐾 − 1) ∈ (ℤ𝑀))
4935, 48sylan 277 . . . . . . . . 9 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → (𝐾 − 1) ∈ (ℤ𝑀))
5049, 38, 39iseqcl 9754 . . . . . . . 8 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → (seq𝑀( + , 𝐹, 𝑆)‘(𝐾 − 1)) ∈ 𝑆)
5144, 47, 50rspcdva 2717 . . . . . . 7 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → ((seq𝑀( + , 𝐹, 𝑆)‘(𝐾 − 1)) + 𝑍) = 𝑍)
5240, 42, 513eqtrd 2119 . . . . . 6 ((𝜑𝐾 ∈ (ℤ‘(𝑀 + 1))) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = 𝑍)
5352ex 113 . . . . 5 (𝜑 → (𝐾 ∈ (ℤ‘(𝑀 + 1)) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = 𝑍))
54 uzp1 8945 . . . . . 6 (𝐾 ∈ (ℤ𝑀) → (𝐾 = 𝑀𝐾 ∈ (ℤ‘(𝑀 + 1))))
5517, 54syl 14 . . . . 5 (𝜑 → (𝐾 = 𝑀𝐾 ∈ (ℤ‘(𝑀 + 1))))
5633, 53, 55mpjaod 671 . . . 4 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = 𝑍)
5756a1i 9 . . 3 (𝐾 ∈ ℤ → (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = 𝑍))
58 simpr 108 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ℤ𝐾)) → 𝑘 ∈ (ℤ𝐾))
5917adantr 270 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ℤ𝐾)) → 𝐾 ∈ (ℤ𝑀))
60 uztrn 8928 . . . . . . . . . 10 ((𝑘 ∈ (ℤ𝐾) ∧ 𝐾 ∈ (ℤ𝑀)) → 𝑘 ∈ (ℤ𝑀))
6158, 59, 60syl2anc 403 . . . . . . . . 9 ((𝜑𝑘 ∈ (ℤ𝐾)) → 𝑘 ∈ (ℤ𝑀))
6224adantlr 461 . . . . . . . . 9 (((𝜑𝑘 ∈ (ℤ𝐾)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
6326adantlr 461 . . . . . . . . 9 (((𝜑𝑘 ∈ (ℤ𝐾)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
6461, 62, 63iseqp1 9755 . . . . . . . 8 ((𝜑𝑘 ∈ (ℤ𝐾)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) + (𝐹‘(𝑘 + 1))))
6564adantr 270 . . . . . . 7 (((𝜑𝑘 ∈ (ℤ𝐾)) ∧ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = 𝑍) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) + (𝐹‘(𝑘 + 1))))
66 simpr 108 . . . . . . . 8 (((𝜑𝑘 ∈ (ℤ𝐾)) ∧ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = 𝑍) → (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = 𝑍)
6766oveq1d 5604 . . . . . . 7 (((𝜑𝑘 ∈ (ℤ𝐾)) ∧ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = 𝑍) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) + (𝐹‘(𝑘 + 1))) = (𝑍 + (𝐹‘(𝑘 + 1))))
68 oveq2 5597 . . . . . . . . . 10 (𝑥 = (𝐹‘(𝑘 + 1)) → (𝑍 + 𝑥) = (𝑍 + (𝐹‘(𝑘 + 1))))
6968eqeq1d 2091 . . . . . . . . 9 (𝑥 = (𝐹‘(𝑘 + 1)) → ((𝑍 + 𝑥) = 𝑍 ↔ (𝑍 + (𝐹‘(𝑘 + 1))) = 𝑍))
70 iseqz.3 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → (𝑍 + 𝑥) = 𝑍)
7170ralrimiva 2440 . . . . . . . . . 10 (𝜑 → ∀𝑥𝑆 (𝑍 + 𝑥) = 𝑍)
7271adantr 270 . . . . . . . . 9 ((𝜑𝑘 ∈ (ℤ𝐾)) → ∀𝑥𝑆 (𝑍 + 𝑥) = 𝑍)
73 fveq2 5251 . . . . . . . . . . 11 (𝑥 = (𝑘 + 1) → (𝐹𝑥) = (𝐹‘(𝑘 + 1)))
7473eleq1d 2151 . . . . . . . . . 10 (𝑥 = (𝑘 + 1) → ((𝐹𝑥) ∈ 𝑆 ↔ (𝐹‘(𝑘 + 1)) ∈ 𝑆))
7524ralrimiva 2440 . . . . . . . . . . 11 (𝜑 → ∀𝑥 ∈ (ℤ𝑀)(𝐹𝑥) ∈ 𝑆)
7675adantr 270 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ℤ𝐾)) → ∀𝑥 ∈ (ℤ𝑀)(𝐹𝑥) ∈ 𝑆)
77 peano2uz 8964 . . . . . . . . . . 11 (𝑘 ∈ (ℤ𝑀) → (𝑘 + 1) ∈ (ℤ𝑀))
7861, 77syl 14 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ℤ𝐾)) → (𝑘 + 1) ∈ (ℤ𝑀))
7974, 76, 78rspcdva 2717 . . . . . . . . 9 ((𝜑𝑘 ∈ (ℤ𝐾)) → (𝐹‘(𝑘 + 1)) ∈ 𝑆)
8069, 72, 79rspcdva 2717 . . . . . . . 8 ((𝜑𝑘 ∈ (ℤ𝐾)) → (𝑍 + (𝐹‘(𝑘 + 1))) = 𝑍)
8180adantr 270 . . . . . . 7 (((𝜑𝑘 ∈ (ℤ𝐾)) ∧ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = 𝑍) → (𝑍 + (𝐹‘(𝑘 + 1))) = 𝑍)
8265, 67, 813eqtrd 2119 . . . . . 6 (((𝜑𝑘 ∈ (ℤ𝐾)) ∧ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = 𝑍) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = 𝑍)
8382ex 113 . . . . 5 ((𝜑𝑘 ∈ (ℤ𝐾)) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) = 𝑍 → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = 𝑍))
8483expcom 114 . . . 4 (𝑘 ∈ (ℤ𝐾) → (𝜑 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) = 𝑍 → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = 𝑍)))
8584a2d 26 . . 3 (𝑘 ∈ (ℤ𝐾) → ((𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = 𝑍) → (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = 𝑍)))
866, 9, 12, 15, 57, 85uzind4 8969 . 2 (𝑁 ∈ (ℤ𝐾) → (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = 𝑍))
873, 86mpcom 36 1 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = 𝑍)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wo 662   = wceq 1285  wcel 1434  wral 2353  cfv 4967  (class class class)co 5589  1c1 7252   + caddc 7254  cmin 7554  cz 8644  cuz 8912  ...cfz 9317  seqcseq 9738
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 7337  ax-resscn 7338  ax-1cn 7339  ax-1re 7340  ax-icn 7341  ax-addcl 7342  ax-addrcl 7343  ax-mulcl 7344  ax-addcom 7346  ax-addass 7348  ax-distr 7350  ax-i2m1 7351  ax-0lt1 7352  ax-0id 7354  ax-rnegex 7355  ax-cnre 7357  ax-pre-ltirr 7358  ax-pre-ltwlin 7359  ax-pre-lttrn 7360  ax-pre-ltadd 7362
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 4405  df-rel 4406  df-cnv 4407  df-co 4408  df-dm 4409  df-rn 4410  df-res 4411  df-ima 4412  df-iota 4932  df-fun 4969  df-fn 4970  df-f 4971  df-f1 4972  df-fo 4973  df-f1o 4974  df-fv 4975  df-riota 5545  df-ov 5592  df-oprab 5593  df-mpt2 5594  df-1st 5844  df-2nd 5845  df-recs 6000  df-frec 6086  df-pnf 7425  df-mnf 7426  df-xr 7427  df-ltxr 7428  df-le 7429  df-sub 7556  df-neg 7557  df-inn 8315  df-n0 8564  df-z 8645  df-uz 8913  df-fz 9318  df-iseq 9739
This theorem is referenced by:  ibcval5  10004
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