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Theorem seq3fveq2 10300
 Description: Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)
Hypotheses
Ref Expression
seq3fveq2.1 (𝜑𝐾 ∈ (ℤ𝑀))
seq3fveq2.2 (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (𝐺𝐾))
seq3fveq2.f ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
seq3fveq2.g ((𝜑𝑥 ∈ (ℤ𝐾)) → (𝐺𝑥) ∈ 𝑆)
seq3fveq2.pl ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
seq3fveq2.3 (𝜑𝑁 ∈ (ℤ𝐾))
seq3fveq2.4 ((𝜑𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝐹𝑘) = (𝐺𝑘))
Assertion
Ref Expression
seq3fveq2 (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝐾( + , 𝐺)‘𝑁))
Distinct variable groups:   𝑥,𝑘,𝑦,𝐹   𝑘,𝐺,𝑥,𝑦   𝑘,𝐾,𝑥,𝑦   𝑘,𝑁,𝑥,𝑦   𝜑,𝑘,𝑥,𝑦   𝑘,𝑀,𝑥,𝑦   + ,𝑘,𝑥,𝑦   𝑆,𝑘,𝑥,𝑦

Proof of Theorem seq3fveq2
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 seq3fveq2.3 . . 3 (𝜑𝑁 ∈ (ℤ𝐾))
2 eluzfz2 9870 . . 3 (𝑁 ∈ (ℤ𝐾) → 𝑁 ∈ (𝐾...𝑁))
31, 2syl 14 . 2 (𝜑𝑁 ∈ (𝐾...𝑁))
4 eleq1 2203 . . . . . 6 (𝑧 = 𝐾 → (𝑧 ∈ (𝐾...𝑁) ↔ 𝐾 ∈ (𝐾...𝑁)))
5 fveq2 5432 . . . . . . 7 (𝑧 = 𝐾 → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘𝐾))
6 fveq2 5432 . . . . . . 7 (𝑧 = 𝐾 → (seq𝐾( + , 𝐺)‘𝑧) = (seq𝐾( + , 𝐺)‘𝐾))
75, 6eqeq12d 2155 . . . . . 6 (𝑧 = 𝐾 → ((seq𝑀( + , 𝐹)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑧) ↔ (seq𝑀( + , 𝐹)‘𝐾) = (seq𝐾( + , 𝐺)‘𝐾)))
84, 7imbi12d 233 . . . . 5 (𝑧 = 𝐾 → ((𝑧 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑧)) ↔ (𝐾 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝐾( + , 𝐺)‘𝐾))))
98imbi2d 229 . . . 4 (𝑧 = 𝐾 → ((𝜑 → (𝑧 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑧))) ↔ (𝜑 → (𝐾 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝐾( + , 𝐺)‘𝐾)))))
10 eleq1 2203 . . . . . 6 (𝑧 = 𝑤 → (𝑧 ∈ (𝐾...𝑁) ↔ 𝑤 ∈ (𝐾...𝑁)))
11 fveq2 5432 . . . . . . 7 (𝑧 = 𝑤 → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘𝑤))
12 fveq2 5432 . . . . . . 7 (𝑧 = 𝑤 → (seq𝐾( + , 𝐺)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑤))
1311, 12eqeq12d 2155 . . . . . 6 (𝑧 = 𝑤 → ((seq𝑀( + , 𝐹)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑧) ↔ (seq𝑀( + , 𝐹)‘𝑤) = (seq𝐾( + , 𝐺)‘𝑤)))
1410, 13imbi12d 233 . . . . 5 (𝑧 = 𝑤 → ((𝑧 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑧)) ↔ (𝑤 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝐾( + , 𝐺)‘𝑤))))
1514imbi2d 229 . . . 4 (𝑧 = 𝑤 → ((𝜑 → (𝑧 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑧))) ↔ (𝜑 → (𝑤 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝐾( + , 𝐺)‘𝑤)))))
16 eleq1 2203 . . . . . 6 (𝑧 = (𝑤 + 1) → (𝑧 ∈ (𝐾...𝑁) ↔ (𝑤 + 1) ∈ (𝐾...𝑁)))
17 fveq2 5432 . . . . . . 7 (𝑧 = (𝑤 + 1) → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘(𝑤 + 1)))
18 fveq2 5432 . . . . . . 7 (𝑧 = (𝑤 + 1) → (seq𝐾( + , 𝐺)‘𝑧) = (seq𝐾( + , 𝐺)‘(𝑤 + 1)))
1917, 18eqeq12d 2155 . . . . . 6 (𝑧 = (𝑤 + 1) → ((seq𝑀( + , 𝐹)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑧) ↔ (seq𝑀( + , 𝐹)‘(𝑤 + 1)) = (seq𝐾( + , 𝐺)‘(𝑤 + 1))))
2016, 19imbi12d 233 . . . . 5 (𝑧 = (𝑤 + 1) → ((𝑧 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑧)) ↔ ((𝑤 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘(𝑤 + 1)) = (seq𝐾( + , 𝐺)‘(𝑤 + 1)))))
2120imbi2d 229 . . . 4 (𝑧 = (𝑤 + 1) → ((𝜑 → (𝑧 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑧))) ↔ (𝜑 → ((𝑤 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘(𝑤 + 1)) = (seq𝐾( + , 𝐺)‘(𝑤 + 1))))))
22 eleq1 2203 . . . . . 6 (𝑧 = 𝑁 → (𝑧 ∈ (𝐾...𝑁) ↔ 𝑁 ∈ (𝐾...𝑁)))
23 fveq2 5432 . . . . . . 7 (𝑧 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘𝑁))
24 fveq2 5432 . . . . . . 7 (𝑧 = 𝑁 → (seq𝐾( + , 𝐺)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑁))
2523, 24eqeq12d 2155 . . . . . 6 (𝑧 = 𝑁 → ((seq𝑀( + , 𝐹)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑧) ↔ (seq𝑀( + , 𝐹)‘𝑁) = (seq𝐾( + , 𝐺)‘𝑁)))
2622, 25imbi12d 233 . . . . 5 (𝑧 = 𝑁 → ((𝑧 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑧)) ↔ (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝐾( + , 𝐺)‘𝑁))))
2726imbi2d 229 . . . 4 (𝑧 = 𝑁 → ((𝜑 → (𝑧 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑧))) ↔ (𝜑 → (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝐾( + , 𝐺)‘𝑁)))))
28 seq3fveq2.2 . . . . . 6 (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (𝐺𝐾))
29 seq3fveq2.1 . . . . . . . 8 (𝜑𝐾 ∈ (ℤ𝑀))
30 eluzelz 9386 . . . . . . . 8 (𝐾 ∈ (ℤ𝑀) → 𝐾 ∈ ℤ)
3129, 30syl 14 . . . . . . 7 (𝜑𝐾 ∈ ℤ)
32 seq3fveq2.g . . . . . . 7 ((𝜑𝑥 ∈ (ℤ𝐾)) → (𝐺𝑥) ∈ 𝑆)
33 seq3fveq2.pl . . . . . . 7 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
3431, 32, 33seq3-1 10291 . . . . . 6 (𝜑 → (seq𝐾( + , 𝐺)‘𝐾) = (𝐺𝐾))
3528, 34eqtr4d 2176 . . . . 5 (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝐾( + , 𝐺)‘𝐾))
3635a1i13 24 . . . 4 (𝐾 ∈ ℤ → (𝜑 → (𝐾 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝐾( + , 𝐺)‘𝐾))))
37 peano2fzr 9875 . . . . . . . 8 ((𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁)) → 𝑤 ∈ (𝐾...𝑁))
3837adantl 275 . . . . . . 7 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝑤 ∈ (𝐾...𝑁))
3938expr 373 . . . . . 6 ((𝜑𝑤 ∈ (ℤ𝐾)) → ((𝑤 + 1) ∈ (𝐾...𝑁) → 𝑤 ∈ (𝐾...𝑁)))
4039imim1d 75 . . . . 5 ((𝜑𝑤 ∈ (ℤ𝐾)) → ((𝑤 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝐾( + , 𝐺)‘𝑤)) → ((𝑤 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝐾( + , 𝐺)‘𝑤))))
41 oveq1 5792 . . . . . 6 ((seq𝑀( + , 𝐹)‘𝑤) = (seq𝐾( + , 𝐺)‘𝑤) → ((seq𝑀( + , 𝐹)‘𝑤) + (𝐹‘(𝑤 + 1))) = ((seq𝐾( + , 𝐺)‘𝑤) + (𝐹‘(𝑤 + 1))))
42 simprl 521 . . . . . . . . 9 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝑤 ∈ (ℤ𝐾))
4329adantr 274 . . . . . . . . 9 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝐾 ∈ (ℤ𝑀))
44 uztrn 9393 . . . . . . . . 9 ((𝑤 ∈ (ℤ𝐾) ∧ 𝐾 ∈ (ℤ𝑀)) → 𝑤 ∈ (ℤ𝑀))
4542, 43, 44syl2anc 409 . . . . . . . 8 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝑤 ∈ (ℤ𝑀))
46 seq3fveq2.f . . . . . . . . 9 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
4746adantlr 469 . . . . . . . 8 (((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
4833adantlr 469 . . . . . . . 8 (((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
4945, 47, 48seq3p1 10293 . . . . . . 7 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (seq𝑀( + , 𝐹)‘(𝑤 + 1)) = ((seq𝑀( + , 𝐹)‘𝑤) + (𝐹‘(𝑤 + 1))))
5032adantlr 469 . . . . . . . . 9 (((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) ∧ 𝑥 ∈ (ℤ𝐾)) → (𝐺𝑥) ∈ 𝑆)
5142, 50, 48seq3p1 10293 . . . . . . . 8 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (seq𝐾( + , 𝐺)‘(𝑤 + 1)) = ((seq𝐾( + , 𝐺)‘𝑤) + (𝐺‘(𝑤 + 1))))
52 fveq2 5432 . . . . . . . . . . 11 (𝑘 = (𝑤 + 1) → (𝐹𝑘) = (𝐹‘(𝑤 + 1)))
53 fveq2 5432 . . . . . . . . . . 11 (𝑘 = (𝑤 + 1) → (𝐺𝑘) = (𝐺‘(𝑤 + 1)))
5452, 53eqeq12d 2155 . . . . . . . . . 10 (𝑘 = (𝑤 + 1) → ((𝐹𝑘) = (𝐺𝑘) ↔ (𝐹‘(𝑤 + 1)) = (𝐺‘(𝑤 + 1))))
55 seq3fveq2.4 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝐹𝑘) = (𝐺𝑘))
5655ralrimiva 2509 . . . . . . . . . . 11 (𝜑 → ∀𝑘 ∈ ((𝐾 + 1)...𝑁)(𝐹𝑘) = (𝐺𝑘))
5756adantr 274 . . . . . . . . . 10 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → ∀𝑘 ∈ ((𝐾 + 1)...𝑁)(𝐹𝑘) = (𝐺𝑘))
58 eluzp1p1 9402 . . . . . . . . . . . 12 (𝑤 ∈ (ℤ𝐾) → (𝑤 + 1) ∈ (ℤ‘(𝐾 + 1)))
5958ad2antrl 482 . . . . . . . . . . 11 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (𝑤 + 1) ∈ (ℤ‘(𝐾 + 1)))
60 elfzuz3 9861 . . . . . . . . . . . 12 ((𝑤 + 1) ∈ (𝐾...𝑁) → 𝑁 ∈ (ℤ‘(𝑤 + 1)))
6160ad2antll 483 . . . . . . . . . . 11 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝑁 ∈ (ℤ‘(𝑤 + 1)))
62 elfzuzb 9858 . . . . . . . . . . 11 ((𝑤 + 1) ∈ ((𝐾 + 1)...𝑁) ↔ ((𝑤 + 1) ∈ (ℤ‘(𝐾 + 1)) ∧ 𝑁 ∈ (ℤ‘(𝑤 + 1))))
6359, 61, 62sylanbrc 414 . . . . . . . . . 10 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (𝑤 + 1) ∈ ((𝐾 + 1)...𝑁))
6454, 57, 63rspcdva 2799 . . . . . . . . 9 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (𝐹‘(𝑤 + 1)) = (𝐺‘(𝑤 + 1)))
6564oveq2d 5801 . . . . . . . 8 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → ((seq𝐾( + , 𝐺)‘𝑤) + (𝐹‘(𝑤 + 1))) = ((seq𝐾( + , 𝐺)‘𝑤) + (𝐺‘(𝑤 + 1))))
6651, 65eqtr4d 2176 . . . . . . 7 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (seq𝐾( + , 𝐺)‘(𝑤 + 1)) = ((seq𝐾( + , 𝐺)‘𝑤) + (𝐹‘(𝑤 + 1))))
6749, 66eqeq12d 2155 . . . . . 6 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → ((seq𝑀( + , 𝐹)‘(𝑤 + 1)) = (seq𝐾( + , 𝐺)‘(𝑤 + 1)) ↔ ((seq𝑀( + , 𝐹)‘𝑤) + (𝐹‘(𝑤 + 1))) = ((seq𝐾( + , 𝐺)‘𝑤) + (𝐹‘(𝑤 + 1)))))
6841, 67syl5ibr 155 . . . . 5 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → ((seq𝑀( + , 𝐹)‘𝑤) = (seq𝐾( + , 𝐺)‘𝑤) → (seq𝑀( + , 𝐹)‘(𝑤 + 1)) = (seq𝐾( + , 𝐺)‘(𝑤 + 1))))
6940, 68animpimp2impd 549 . . . 4 (𝑤 ∈ (ℤ𝐾) → ((𝜑 → (𝑤 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝐾( + , 𝐺)‘𝑤))) → (𝜑 → ((𝑤 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘(𝑤 + 1)) = (seq𝐾( + , 𝐺)‘(𝑤 + 1))))))
709, 15, 21, 27, 36, 69uzind4 9437 . . 3 (𝑁 ∈ (ℤ𝐾) → (𝜑 → (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝐾( + , 𝐺)‘𝑁))))
711, 70mpcom 36 . 2 (𝜑 → (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝐾( + , 𝐺)‘𝑁)))
723, 71mpd 13 1 (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝐾( + , 𝐺)‘𝑁))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332   ∈ wcel 1481  ∀wral 2417  ‘cfv 5134  (class class class)co 5785  1c1 7672   + caddc 7674  ℤcz 9105  ℤ≥cuz 9377  ...cfz 9848  seqcseq 10276 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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4052  ax-sep 4055  ax-nul 4063  ax-pow 4107  ax-pr 4141  ax-un 4365  ax-setind 4462  ax-iinf 4512  ax-cnex 7762  ax-resscn 7763  ax-1cn 7764  ax-1re 7765  ax-icn 7766  ax-addcl 7767  ax-addrcl 7768  ax-mulcl 7769  ax-addcom 7771  ax-addass 7773  ax-distr 7775  ax-i2m1 7776  ax-0lt1 7777  ax-0id 7779  ax-rnegex 7780  ax-cnre 7782  ax-pre-ltirr 7783  ax-pre-ltwlin 7784  ax-pre-lttrn 7785  ax-pre-ltadd 7787 This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2692  df-sbc 2915  df-csb 3009  df-dif 3079  df-un 3081  df-in 3083  df-ss 3090  df-nul 3370  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-int 3781  df-iun 3824  df-br 3939  df-opab 3999  df-mpt 4000  df-tr 4036  df-id 4225  df-iord 4298  df-on 4300  df-ilim 4301  df-suc 4303  df-iom 4515  df-xp 4556  df-rel 4557  df-cnv 4558  df-co 4559  df-dm 4560  df-rn 4561  df-res 4562  df-ima 4563  df-iota 5099  df-fun 5136  df-fn 5137  df-f 5138  df-f1 5139  df-fo 5140  df-f1o 5141  df-fv 5142  df-riota 5741  df-ov 5788  df-oprab 5789  df-mpo 5790  df-1st 6049  df-2nd 6050  df-recs 6213  df-frec 6299  df-pnf 7853  df-mnf 7854  df-xr 7855  df-ltxr 7856  df-le 7857  df-sub 7986  df-neg 7987  df-inn 8772  df-n0 9029  df-z 9106  df-uz 9378  df-fz 9849  df-seqfrec 10277 This theorem is referenced by:  seq3feq2  10301  seq3fveq  10302
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