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Theorem seq3fveq2 9953
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 9507 . . 3 (𝑁 ∈ (ℤ𝐾) → 𝑁 ∈ (𝐾...𝑁))
31, 2syl 14 . 2 (𝜑𝑁 ∈ (𝐾...𝑁))
4 eleq1 2151 . . . . . 6 (𝑧 = 𝐾 → (𝑧 ∈ (𝐾...𝑁) ↔ 𝐾 ∈ (𝐾...𝑁)))
5 fveq2 5318 . . . . . . 7 (𝑧 = 𝐾 → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘𝐾))
6 fveq2 5318 . . . . . . 7 (𝑧 = 𝐾 → (seq𝐾( + , 𝐺)‘𝑧) = (seq𝐾( + , 𝐺)‘𝐾))
75, 6eqeq12d 2103 . . . . . 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 2151 . . . . . 6 (𝑧 = 𝑤 → (𝑧 ∈ (𝐾...𝑁) ↔ 𝑤 ∈ (𝐾...𝑁)))
11 fveq2 5318 . . . . . . 7 (𝑧 = 𝑤 → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘𝑤))
12 fveq2 5318 . . . . . . 7 (𝑧 = 𝑤 → (seq𝐾( + , 𝐺)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑤))
1311, 12eqeq12d 2103 . . . . . 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 2151 . . . . . 6 (𝑧 = (𝑤 + 1) → (𝑧 ∈ (𝐾...𝑁) ↔ (𝑤 + 1) ∈ (𝐾...𝑁)))
17 fveq2 5318 . . . . . . 7 (𝑧 = (𝑤 + 1) → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘(𝑤 + 1)))
18 fveq2 5318 . . . . . . 7 (𝑧 = (𝑤 + 1) → (seq𝐾( + , 𝐺)‘𝑧) = (seq𝐾( + , 𝐺)‘(𝑤 + 1)))
1917, 18eqeq12d 2103 . . . . . 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 2151 . . . . . 6 (𝑧 = 𝑁 → (𝑧 ∈ (𝐾...𝑁) ↔ 𝑁 ∈ (𝐾...𝑁)))
23 fveq2 5318 . . . . . . 7 (𝑧 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘𝑁))
24 fveq2 5318 . . . . . . 7 (𝑧 = 𝑁 → (seq𝐾( + , 𝐺)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑁))
2523, 24eqeq12d 2103 . . . . . 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 9089 . . . . . . . 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 9938 . . . . . 6 (𝜑 → (seq𝐾( + , 𝐺)‘𝐾) = (𝐺𝐾))
3528, 34eqtr4d 2124 . . . . 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 9512 . . . . . . . 8 ((𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁)) → 𝑤 ∈ (𝐾...𝑁))
3837adantl 272 . . . . . . 7 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝑤 ∈ (𝐾...𝑁))
3938expr 368 . . . . . 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 5673 . . . . . 6 ((seq𝑀( + , 𝐹)‘𝑤) = (seq𝐾( + , 𝐺)‘𝑤) → ((seq𝑀( + , 𝐹)‘𝑤) + (𝐹‘(𝑤 + 1))) = ((seq𝐾( + , 𝐺)‘𝑤) + (𝐹‘(𝑤 + 1))))
42 simprl 499 . . . . . . . . 9 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝑤 ∈ (ℤ𝐾))
4329adantr 271 . . . . . . . . 9 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝐾 ∈ (ℤ𝑀))
44 uztrn 9096 . . . . . . . . 9 ((𝑤 ∈ (ℤ𝐾) ∧ 𝐾 ∈ (ℤ𝑀)) → 𝑤 ∈ (ℤ𝑀))
4542, 43, 44syl2anc 404 . . . . . . . 8 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝑤 ∈ (ℤ𝑀))
46 seq3fveq2.f . . . . . . . . 9 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
4746adantlr 462 . . . . . . . 8 (((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
4833adantlr 462 . . . . . . . 8 (((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
4945, 47, 48seq3p1 9945 . . . . . . 7 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (seq𝑀( + , 𝐹)‘(𝑤 + 1)) = ((seq𝑀( + , 𝐹)‘𝑤) + (𝐹‘(𝑤 + 1))))
5032adantlr 462 . . . . . . . . 9 (((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) ∧ 𝑥 ∈ (ℤ𝐾)) → (𝐺𝑥) ∈ 𝑆)
5142, 50, 48seq3p1 9945 . . . . . . . 8 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (seq𝐾( + , 𝐺)‘(𝑤 + 1)) = ((seq𝐾( + , 𝐺)‘𝑤) + (𝐺‘(𝑤 + 1))))
52 fveq2 5318 . . . . . . . . . . 11 (𝑘 = (𝑤 + 1) → (𝐹𝑘) = (𝐹‘(𝑤 + 1)))
53 fveq2 5318 . . . . . . . . . . 11 (𝑘 = (𝑤 + 1) → (𝐺𝑘) = (𝐺‘(𝑤 + 1)))
5452, 53eqeq12d 2103 . . . . . . . . . 10 (𝑘 = (𝑤 + 1) → ((𝐹𝑘) = (𝐺𝑘) ↔ (𝐹‘(𝑤 + 1)) = (𝐺‘(𝑤 + 1))))
55 seq3fveq2.4 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝐹𝑘) = (𝐺𝑘))
5655ralrimiva 2447 . . . . . . . . . . 11 (𝜑 → ∀𝑘 ∈ ((𝐾 + 1)...𝑁)(𝐹𝑘) = (𝐺𝑘))
5756adantr 271 . . . . . . . . . 10 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → ∀𝑘 ∈ ((𝐾 + 1)...𝑁)(𝐹𝑘) = (𝐺𝑘))
58 eluzp1p1 9105 . . . . . . . . . . . 12 (𝑤 ∈ (ℤ𝐾) → (𝑤 + 1) ∈ (ℤ‘(𝐾 + 1)))
5958ad2antrl 475 . . . . . . . . . . 11 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (𝑤 + 1) ∈ (ℤ‘(𝐾 + 1)))
60 elfzuz3 9498 . . . . . . . . . . . 12 ((𝑤 + 1) ∈ (𝐾...𝑁) → 𝑁 ∈ (ℤ‘(𝑤 + 1)))
6160ad2antll 476 . . . . . . . . . . 11 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝑁 ∈ (ℤ‘(𝑤 + 1)))
62 elfzuzb 9495 . . . . . . . . . . 11 ((𝑤 + 1) ∈ ((𝐾 + 1)...𝑁) ↔ ((𝑤 + 1) ∈ (ℤ‘(𝐾 + 1)) ∧ 𝑁 ∈ (ℤ‘(𝑤 + 1))))
6359, 61, 62sylanbrc 409 . . . . . . . . . 10 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (𝑤 + 1) ∈ ((𝐾 + 1)...𝑁))
6454, 57, 63rspcdva 2728 . . . . . . . . 9 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (𝐹‘(𝑤 + 1)) = (𝐺‘(𝑤 + 1)))
6564oveq2d 5682 . . . . . . . 8 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → ((seq𝐾( + , 𝐺)‘𝑤) + (𝐹‘(𝑤 + 1))) = ((seq𝐾( + , 𝐺)‘𝑤) + (𝐺‘(𝑤 + 1))))
6651, 65eqtr4d 2124 . . . . . . 7 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (seq𝐾( + , 𝐺)‘(𝑤 + 1)) = ((seq𝐾( + , 𝐺)‘𝑤) + (𝐹‘(𝑤 + 1))))
6749, 66eqeq12d 2103 . . . . . 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 527 . . . 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 9137 . . 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 1290  wcel 1439  wral 2360  cfv 5028  (class class class)co 5666  1c1 7412   + caddc 7414  cz 8811  cuz 9080  ...cfz 9485  seqcseq 9913
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 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416  ax-cnex 7497  ax-resscn 7498  ax-1cn 7499  ax-1re 7500  ax-icn 7501  ax-addcl 7502  ax-addrcl 7503  ax-mulcl 7504  ax-addcom 7506  ax-addass 7508  ax-distr 7510  ax-i2m1 7511  ax-0lt1 7512  ax-0id 7514  ax-rnegex 7515  ax-cnre 7517  ax-pre-ltirr 7518  ax-pre-ltwlin 7519  ax-pre-lttrn 7520  ax-pre-ltadd 7522
This theorem depends on definitions:  df-bi 116  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-iord 4202  df-on 4204  df-ilim 4205  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-frec 6170  df-pnf 7585  df-mnf 7586  df-xr 7587  df-ltxr 7588  df-le 7589  df-sub 7716  df-neg 7717  df-inn 8484  df-n0 8735  df-z 8812  df-uz 9081  df-fz 9486  df-iseq 9914  df-seq3 9915
This theorem is referenced by:  seq3feq2  9954  seq3fveq  9956
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