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Theorem seq3homo 10496
Description: Apply a homomorphism to a sequence. (Contributed by Jim Kingdon, 10-Oct-2022.)
Hypotheses
Ref Expression
seq3homo.1 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
seq3homo.2 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
seq3homo.3 (𝜑𝑁 ∈ (ℤ𝑀))
seq3homo.4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝐻‘(𝑥 + 𝑦)) = ((𝐻𝑥)𝑄(𝐻𝑦)))
seq3homo.5 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐻‘(𝐹𝑥)) = (𝐺𝑥))
seq3homo.g ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
seq3homo.qcl ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
Assertion
Ref Expression
seq3homo (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀(𝑄, 𝐺)‘𝑁))
Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝐻,𝑦   𝑥,𝑀,𝑦   𝑥,𝑁,𝑦   𝜑,𝑥,𝑦   𝑥,𝐺   𝑥, + ,𝑦   𝑥,𝑄,𝑦   𝑥,𝑆,𝑦   𝑦,𝐺

Proof of Theorem seq3homo
Dummy variables 𝑛 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 seq3homo.3 . 2 (𝜑𝑁 ∈ (ℤ𝑀))
2 2fveq3 5516 . . . . 5 (𝑤 = 𝑀 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑤)) = (𝐻‘(seq𝑀( + , 𝐹)‘𝑀)))
3 fveq2 5511 . . . . 5 (𝑤 = 𝑀 → (seq𝑀(𝑄, 𝐺)‘𝑤) = (seq𝑀(𝑄, 𝐺)‘𝑀))
42, 3eqeq12d 2192 . . . 4 (𝑤 = 𝑀 → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑤)) = (seq𝑀(𝑄, 𝐺)‘𝑤) ↔ (𝐻‘(seq𝑀( + , 𝐹)‘𝑀)) = (seq𝑀(𝑄, 𝐺)‘𝑀)))
54imbi2d 230 . . 3 (𝑤 = 𝑀 → ((𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑤)) = (seq𝑀(𝑄, 𝐺)‘𝑤)) ↔ (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑀)) = (seq𝑀(𝑄, 𝐺)‘𝑀))))
6 2fveq3 5516 . . . . 5 (𝑤 = 𝑛 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑤)) = (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)))
7 fveq2 5511 . . . . 5 (𝑤 = 𝑛 → (seq𝑀(𝑄, 𝐺)‘𝑤) = (seq𝑀(𝑄, 𝐺)‘𝑛))
86, 7eqeq12d 2192 . . . 4 (𝑤 = 𝑛 → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑤)) = (seq𝑀(𝑄, 𝐺)‘𝑤) ↔ (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)) = (seq𝑀(𝑄, 𝐺)‘𝑛)))
98imbi2d 230 . . 3 (𝑤 = 𝑛 → ((𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑤)) = (seq𝑀(𝑄, 𝐺)‘𝑤)) ↔ (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)) = (seq𝑀(𝑄, 𝐺)‘𝑛))))
10 2fveq3 5516 . . . . 5 (𝑤 = (𝑛 + 1) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑤)) = (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))))
11 fveq2 5511 . . . . 5 (𝑤 = (𝑛 + 1) → (seq𝑀(𝑄, 𝐺)‘𝑤) = (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1)))
1210, 11eqeq12d 2192 . . . 4 (𝑤 = (𝑛 + 1) → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑤)) = (seq𝑀(𝑄, 𝐺)‘𝑤) ↔ (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1))))
1312imbi2d 230 . . 3 (𝑤 = (𝑛 + 1) → ((𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑤)) = (seq𝑀(𝑄, 𝐺)‘𝑤)) ↔ (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1)))))
14 2fveq3 5516 . . . . 5 (𝑤 = 𝑁 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑤)) = (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)))
15 fveq2 5511 . . . . 5 (𝑤 = 𝑁 → (seq𝑀(𝑄, 𝐺)‘𝑤) = (seq𝑀(𝑄, 𝐺)‘𝑁))
1614, 15eqeq12d 2192 . . . 4 (𝑤 = 𝑁 → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑤)) = (seq𝑀(𝑄, 𝐺)‘𝑤) ↔ (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀(𝑄, 𝐺)‘𝑁)))
1716imbi2d 230 . . 3 (𝑤 = 𝑁 → ((𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑤)) = (seq𝑀(𝑄, 𝐺)‘𝑤)) ↔ (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀(𝑄, 𝐺)‘𝑁))))
18 2fveq3 5516 . . . . . . 7 (𝑥 = 𝑀 → (𝐻‘(𝐹𝑥)) = (𝐻‘(𝐹𝑀)))
19 fveq2 5511 . . . . . . 7 (𝑥 = 𝑀 → (𝐺𝑥) = (𝐺𝑀))
2018, 19eqeq12d 2192 . . . . . 6 (𝑥 = 𝑀 → ((𝐻‘(𝐹𝑥)) = (𝐺𝑥) ↔ (𝐻‘(𝐹𝑀)) = (𝐺𝑀)))
21 seq3homo.5 . . . . . . 7 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐻‘(𝐹𝑥)) = (𝐺𝑥))
2221ralrimiva 2550 . . . . . 6 (𝜑 → ∀𝑥 ∈ (ℤ𝑀)(𝐻‘(𝐹𝑥)) = (𝐺𝑥))
23 eluzel2 9522 . . . . . . . 8 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
241, 23syl 14 . . . . . . 7 (𝜑𝑀 ∈ ℤ)
25 uzid 9531 . . . . . . 7 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
2624, 25syl 14 . . . . . 6 (𝜑𝑀 ∈ (ℤ𝑀))
2720, 22, 26rspcdva 2846 . . . . 5 (𝜑 → (𝐻‘(𝐹𝑀)) = (𝐺𝑀))
28 seq3homo.2 . . . . . . 7 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
29 seq3homo.1 . . . . . . 7 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
3024, 28, 29seq3-1 10446 . . . . . 6 (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹𝑀))
3130fveq2d 5515 . . . . 5 (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑀)) = (𝐻‘(𝐹𝑀)))
32 seq3homo.g . . . . . 6 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
33 seq3homo.qcl . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
3424, 32, 33seq3-1 10446 . . . . 5 (𝜑 → (seq𝑀(𝑄, 𝐺)‘𝑀) = (𝐺𝑀))
3527, 31, 343eqtr4d 2220 . . . 4 (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑀)) = (seq𝑀(𝑄, 𝐺)‘𝑀))
3635a1i 9 . . 3 (𝑀 ∈ ℤ → (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑀)) = (seq𝑀(𝑄, 𝐺)‘𝑀)))
37 oveq1 5876 . . . . . 6 ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛)) = (seq𝑀(𝑄, 𝐺)‘𝑛) → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐺‘(𝑛 + 1))) = ((seq𝑀(𝑄, 𝐺)‘𝑛)𝑄(𝐺‘(𝑛 + 1))))
38 simpr 110 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → 𝑛 ∈ (ℤ𝑀))
3928adantlr 477 . . . . . . . . . 10 (((𝜑𝑛 ∈ (ℤ𝑀)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
4029adantlr 477 . . . . . . . . . 10 (((𝜑𝑛 ∈ (ℤ𝑀)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
4138, 39, 40seq3p1 10448 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑀)) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
4241fveq2d 5515 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
43 seq3homo.4 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝐻‘(𝑥 + 𝑦)) = ((𝐻𝑥)𝑄(𝐻𝑦)))
4443ralrimivva 2559 . . . . . . . . . 10 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝐻‘(𝑥 + 𝑦)) = ((𝐻𝑥)𝑄(𝐻𝑦)))
4544adantr 276 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑀)) → ∀𝑥𝑆𝑦𝑆 (𝐻‘(𝑥 + 𝑦)) = ((𝐻𝑥)𝑄(𝐻𝑦)))
46 eqid 2177 . . . . . . . . . . . 12 (ℤ𝑀) = (ℤ𝑀)
4746, 24, 28, 29seqf 10447 . . . . . . . . . . 11 (𝜑 → seq𝑀( + , 𝐹):(ℤ𝑀)⟶𝑆)
4847ffvelcdmda 5647 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → (seq𝑀( + , 𝐹)‘𝑛) ∈ 𝑆)
49 fveq2 5511 . . . . . . . . . . . 12 (𝑥 = (𝑛 + 1) → (𝐹𝑥) = (𝐹‘(𝑛 + 1)))
5049eleq1d 2246 . . . . . . . . . . 11 (𝑥 = (𝑛 + 1) → ((𝐹𝑥) ∈ 𝑆 ↔ (𝐹‘(𝑛 + 1)) ∈ 𝑆))
5128ralrimiva 2550 . . . . . . . . . . . 12 (𝜑 → ∀𝑥 ∈ (ℤ𝑀)(𝐹𝑥) ∈ 𝑆)
5251adantr 276 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℤ𝑀)) → ∀𝑥 ∈ (ℤ𝑀)(𝐹𝑥) ∈ 𝑆)
53 peano2uz 9572 . . . . . . . . . . . 12 (𝑛 ∈ (ℤ𝑀) → (𝑛 + 1) ∈ (ℤ𝑀))
5438, 53syl 14 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝑛 + 1) ∈ (ℤ𝑀))
5550, 52, 54rspcdva 2846 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐹‘(𝑛 + 1)) ∈ 𝑆)
56 oveq1 5876 . . . . . . . . . . . . 13 (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → (𝑥 + 𝑦) = ((seq𝑀( + , 𝐹)‘𝑛) + 𝑦))
5756fveq2d 5515 . . . . . . . . . . . 12 (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → (𝐻‘(𝑥 + 𝑦)) = (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)))
58 fveq2 5511 . . . . . . . . . . . . 13 (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → (𝐻𝑥) = (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)))
5958oveq1d 5884 . . . . . . . . . . . 12 (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → ((𝐻𝑥)𝑄(𝐻𝑦)) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻𝑦)))
6057, 59eqeq12d 2192 . . . . . . . . . . 11 (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → ((𝐻‘(𝑥 + 𝑦)) = ((𝐻𝑥)𝑄(𝐻𝑦)) ↔ (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻𝑦))))
61 oveq2 5877 . . . . . . . . . . . . 13 (𝑦 = (𝐹‘(𝑛 + 1)) → ((seq𝑀( + , 𝐹)‘𝑛) + 𝑦) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
6261fveq2d 5515 . . . . . . . . . . . 12 (𝑦 = (𝐹‘(𝑛 + 1)) → (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)) = (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
63 fveq2 5511 . . . . . . . . . . . . 13 (𝑦 = (𝐹‘(𝑛 + 1)) → (𝐻𝑦) = (𝐻‘(𝐹‘(𝑛 + 1))))
6463oveq2d 5885 . . . . . . . . . . . 12 (𝑦 = (𝐹‘(𝑛 + 1)) → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻𝑦)) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1)))))
6562, 64eqeq12d 2192 . . . . . . . . . . 11 (𝑦 = (𝐹‘(𝑛 + 1)) → ((𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻𝑦)) ↔ (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1))))))
6660, 65rspc2v 2854 . . . . . . . . . 10 (((seq𝑀( + , 𝐹)‘𝑛) ∈ 𝑆 ∧ (𝐹‘(𝑛 + 1)) ∈ 𝑆) → (∀𝑥𝑆𝑦𝑆 (𝐻‘(𝑥 + 𝑦)) = ((𝐻𝑥)𝑄(𝐻𝑦)) → (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1))))))
6748, 55, 66syl2anc 411 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑀)) → (∀𝑥𝑆𝑦𝑆 (𝐻‘(𝑥 + 𝑦)) = ((𝐻𝑥)𝑄(𝐻𝑦)) → (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1))))))
6845, 67mpd 13 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1)))))
69 2fveq3 5516 . . . . . . . . . . 11 (𝑥 = (𝑛 + 1) → (𝐻‘(𝐹𝑥)) = (𝐻‘(𝐹‘(𝑛 + 1))))
70 fveq2 5511 . . . . . . . . . . 11 (𝑥 = (𝑛 + 1) → (𝐺𝑥) = (𝐺‘(𝑛 + 1)))
7169, 70eqeq12d 2192 . . . . . . . . . 10 (𝑥 = (𝑛 + 1) → ((𝐻‘(𝐹𝑥)) = (𝐺𝑥) ↔ (𝐻‘(𝐹‘(𝑛 + 1))) = (𝐺‘(𝑛 + 1))))
7222adantr 276 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → ∀𝑥 ∈ (ℤ𝑀)(𝐻‘(𝐹𝑥)) = (𝐺𝑥))
7371, 72, 54rspcdva 2846 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐻‘(𝐹‘(𝑛 + 1))) = (𝐺‘(𝑛 + 1)))
7473oveq2d 5885 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1)))) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐺‘(𝑛 + 1))))
7542, 68, 743eqtrd 2214 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐺‘(𝑛 + 1))))
7632adantlr 477 . . . . . . . 8 (((𝜑𝑛 ∈ (ℤ𝑀)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
7733adantlr 477 . . . . . . . 8 (((𝜑𝑛 ∈ (ℤ𝑀)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
7838, 76, 77seq3p1 10448 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑀)) → (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1)) = ((seq𝑀(𝑄, 𝐺)‘𝑛)𝑄(𝐺‘(𝑛 + 1))))
7975, 78eqeq12d 2192 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1)) ↔ ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐺‘(𝑛 + 1))) = ((seq𝑀(𝑄, 𝐺)‘𝑛)𝑄(𝐺‘(𝑛 + 1)))))
8037, 79syl5ibr 156 . . . . 5 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛)) = (seq𝑀(𝑄, 𝐺)‘𝑛) → (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1))))
8180expcom 116 . . . 4 (𝑛 ∈ (ℤ𝑀) → (𝜑 → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛)) = (seq𝑀(𝑄, 𝐺)‘𝑛) → (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1)))))
8281a2d 26 . . 3 (𝑛 ∈ (ℤ𝑀) → ((𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)) = (seq𝑀(𝑄, 𝐺)‘𝑛)) → (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1)))))
835, 9, 13, 17, 36, 82uzind4 9577 . 2 (𝑁 ∈ (ℤ𝑀) → (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀(𝑄, 𝐺)‘𝑁)))
841, 83mpcom 36 1 (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀(𝑄, 𝐺)‘𝑁))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  wral 2455  cfv 5212  (class class class)co 5869  1c1 7803   + caddc 7805  cz 9242  cuz 9517  seqcseq 10431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-ltadd 7918
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-inn 8909  df-n0 9166  df-z 9243  df-uz 9518  df-seqfrec 10432
This theorem is referenced by:  seqfeq3  10498  seq3distr  10499  efcj  11665
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