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Theorem seq3homo 9944
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 5310 . . . . 5 (𝑤 = 𝑀 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑤)) = (𝐻‘(seq𝑀( + , 𝐹)‘𝑀)))
3 fveq2 5305 . . . . 5 (𝑤 = 𝑀 → (seq𝑀(𝑄, 𝐺)‘𝑤) = (seq𝑀(𝑄, 𝐺)‘𝑀))
42, 3eqeq12d 2102 . . . 4 (𝑤 = 𝑀 → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑤)) = (seq𝑀(𝑄, 𝐺)‘𝑤) ↔ (𝐻‘(seq𝑀( + , 𝐹)‘𝑀)) = (seq𝑀(𝑄, 𝐺)‘𝑀)))
54imbi2d 228 . . 3 (𝑤 = 𝑀 → ((𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑤)) = (seq𝑀(𝑄, 𝐺)‘𝑤)) ↔ (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑀)) = (seq𝑀(𝑄, 𝐺)‘𝑀))))
6 2fveq3 5310 . . . . 5 (𝑤 = 𝑛 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑤)) = (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)))
7 fveq2 5305 . . . . 5 (𝑤 = 𝑛 → (seq𝑀(𝑄, 𝐺)‘𝑤) = (seq𝑀(𝑄, 𝐺)‘𝑛))
86, 7eqeq12d 2102 . . . 4 (𝑤 = 𝑛 → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑤)) = (seq𝑀(𝑄, 𝐺)‘𝑤) ↔ (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)) = (seq𝑀(𝑄, 𝐺)‘𝑛)))
98imbi2d 228 . . 3 (𝑤 = 𝑛 → ((𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑤)) = (seq𝑀(𝑄, 𝐺)‘𝑤)) ↔ (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)) = (seq𝑀(𝑄, 𝐺)‘𝑛))))
10 2fveq3 5310 . . . . 5 (𝑤 = (𝑛 + 1) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑤)) = (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))))
11 fveq2 5305 . . . . 5 (𝑤 = (𝑛 + 1) → (seq𝑀(𝑄, 𝐺)‘𝑤) = (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1)))
1210, 11eqeq12d 2102 . . . 4 (𝑤 = (𝑛 + 1) → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑤)) = (seq𝑀(𝑄, 𝐺)‘𝑤) ↔ (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1))))
1312imbi2d 228 . . 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 5310 . . . . 5 (𝑤 = 𝑁 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑤)) = (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)))
15 fveq2 5305 . . . . 5 (𝑤 = 𝑁 → (seq𝑀(𝑄, 𝐺)‘𝑤) = (seq𝑀(𝑄, 𝐺)‘𝑁))
1614, 15eqeq12d 2102 . . . 4 (𝑤 = 𝑁 → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑤)) = (seq𝑀(𝑄, 𝐺)‘𝑤) ↔ (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀(𝑄, 𝐺)‘𝑁)))
1716imbi2d 228 . . 3 (𝑤 = 𝑁 → ((𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑤)) = (seq𝑀(𝑄, 𝐺)‘𝑤)) ↔ (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀(𝑄, 𝐺)‘𝑁))))
18 2fveq3 5310 . . . . . . 7 (𝑥 = 𝑀 → (𝐻‘(𝐹𝑥)) = (𝐻‘(𝐹𝑀)))
19 fveq2 5305 . . . . . . 7 (𝑥 = 𝑀 → (𝐺𝑥) = (𝐺𝑀))
2018, 19eqeq12d 2102 . . . . . 6 (𝑥 = 𝑀 → ((𝐻‘(𝐹𝑥)) = (𝐺𝑥) ↔ (𝐻‘(𝐹𝑀)) = (𝐺𝑀)))
21 seq3homo.5 . . . . . . 7 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐻‘(𝐹𝑥)) = (𝐺𝑥))
2221ralrimiva 2446 . . . . . 6 (𝜑 → ∀𝑥 ∈ (ℤ𝑀)(𝐻‘(𝐹𝑥)) = (𝐺𝑥))
23 eluzel2 9024 . . . . . . . 8 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
241, 23syl 14 . . . . . . 7 (𝜑𝑀 ∈ ℤ)
25 uzid 9033 . . . . . . 7 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
2624, 25syl 14 . . . . . 6 (𝜑𝑀 ∈ (ℤ𝑀))
2720, 22, 26rspcdva 2727 . . . . 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 9877 . . . . . 6 (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹𝑀))
3130fveq2d 5309 . . . . 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 9877 . . . . 5 (𝜑 → (seq𝑀(𝑄, 𝐺)‘𝑀) = (𝐺𝑀))
3527, 31, 343eqtr4d 2130 . . . 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 5659 . . . . . 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 108 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → 𝑛 ∈ (ℤ𝑀))
3928adantlr 461 . . . . . . . . . 10 (((𝜑𝑛 ∈ (ℤ𝑀)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
4029adantlr 461 . . . . . . . . . 10 (((𝜑𝑛 ∈ (ℤ𝑀)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
4138, 39, 40seq3p1 9884 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑀)) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
4241fveq2d 5309 . . . . . . . 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 2455 . . . . . . . . . 10 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝐻‘(𝑥 + 𝑦)) = ((𝐻𝑥)𝑄(𝐻𝑦)))
4544adantr 270 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑀)) → ∀𝑥𝑆𝑦𝑆 (𝐻‘(𝑥 + 𝑦)) = ((𝐻𝑥)𝑄(𝐻𝑦)))
46 eqid 2088 . . . . . . . . . . . 12 (ℤ𝑀) = (ℤ𝑀)
4746, 24, 28, 29seqf 9880 . . . . . . . . . . 11 (𝜑 → seq𝑀( + , 𝐹):(ℤ𝑀)⟶𝑆)
4847ffvelrnda 5434 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → (seq𝑀( + , 𝐹)‘𝑛) ∈ 𝑆)
49 fveq2 5305 . . . . . . . . . . . 12 (𝑥 = (𝑛 + 1) → (𝐹𝑥) = (𝐹‘(𝑛 + 1)))
5049eleq1d 2156 . . . . . . . . . . 11 (𝑥 = (𝑛 + 1) → ((𝐹𝑥) ∈ 𝑆 ↔ (𝐹‘(𝑛 + 1)) ∈ 𝑆))
5128ralrimiva 2446 . . . . . . . . . . . 12 (𝜑 → ∀𝑥 ∈ (ℤ𝑀)(𝐹𝑥) ∈ 𝑆)
5251adantr 270 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℤ𝑀)) → ∀𝑥 ∈ (ℤ𝑀)(𝐹𝑥) ∈ 𝑆)
53 peano2uz 9071 . . . . . . . . . . . 12 (𝑛 ∈ (ℤ𝑀) → (𝑛 + 1) ∈ (ℤ𝑀))
5438, 53syl 14 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝑛 + 1) ∈ (ℤ𝑀))
5550, 52, 54rspcdva 2727 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐹‘(𝑛 + 1)) ∈ 𝑆)
56 oveq1 5659 . . . . . . . . . . . . 13 (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → (𝑥 + 𝑦) = ((seq𝑀( + , 𝐹)‘𝑛) + 𝑦))
5756fveq2d 5309 . . . . . . . . . . . 12 (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → (𝐻‘(𝑥 + 𝑦)) = (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)))
58 fveq2 5305 . . . . . . . . . . . . 13 (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → (𝐻𝑥) = (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)))
5958oveq1d 5667 . . . . . . . . . . . 12 (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → ((𝐻𝑥)𝑄(𝐻𝑦)) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻𝑦)))
6057, 59eqeq12d 2102 . . . . . . . . . . 11 (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → ((𝐻‘(𝑥 + 𝑦)) = ((𝐻𝑥)𝑄(𝐻𝑦)) ↔ (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻𝑦))))
61 oveq2 5660 . . . . . . . . . . . . 13 (𝑦 = (𝐹‘(𝑛 + 1)) → ((seq𝑀( + , 𝐹)‘𝑛) + 𝑦) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
6261fveq2d 5309 . . . . . . . . . . . 12 (𝑦 = (𝐹‘(𝑛 + 1)) → (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)) = (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
63 fveq2 5305 . . . . . . . . . . . . 13 (𝑦 = (𝐹‘(𝑛 + 1)) → (𝐻𝑦) = (𝐻‘(𝐹‘(𝑛 + 1))))
6463oveq2d 5668 . . . . . . . . . . . 12 (𝑦 = (𝐹‘(𝑛 + 1)) → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻𝑦)) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1)))))
6562, 64eqeq12d 2102 . . . . . . . . . . 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 2734 . . . . . . . . . 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 403 . . . . . . . . 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 5310 . . . . . . . . . . 11 (𝑥 = (𝑛 + 1) → (𝐻‘(𝐹𝑥)) = (𝐻‘(𝐹‘(𝑛 + 1))))
70 fveq2 5305 . . . . . . . . . . 11 (𝑥 = (𝑛 + 1) → (𝐺𝑥) = (𝐺‘(𝑛 + 1)))
7169, 70eqeq12d 2102 . . . . . . . . . 10 (𝑥 = (𝑛 + 1) → ((𝐻‘(𝐹𝑥)) = (𝐺𝑥) ↔ (𝐻‘(𝐹‘(𝑛 + 1))) = (𝐺‘(𝑛 + 1))))
7222adantr 270 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → ∀𝑥 ∈ (ℤ𝑀)(𝐻‘(𝐹𝑥)) = (𝐺𝑥))
7371, 72, 54rspcdva 2727 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐻‘(𝐹‘(𝑛 + 1))) = (𝐺‘(𝑛 + 1)))
7473oveq2d 5668 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1)))) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐺‘(𝑛 + 1))))
7542, 68, 743eqtrd 2124 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐺‘(𝑛 + 1))))
7632adantlr 461 . . . . . . . 8 (((𝜑𝑛 ∈ (ℤ𝑀)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
7733adantlr 461 . . . . . . . 8 (((𝜑𝑛 ∈ (ℤ𝑀)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
7838, 76, 77seq3p1 9884 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑀)) → (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1)) = ((seq𝑀(𝑄, 𝐺)‘𝑛)𝑄(𝐺‘(𝑛 + 1))))
7975, 78eqeq12d 2102 . . . . . 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 154 . . . . 5 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛)) = (seq𝑀(𝑄, 𝐺)‘𝑛) → (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1))))
8180expcom 114 . . . 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 9076 . 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 102   = wceq 1289  wcel 1438  wral 2359  cfv 5015  (class class class)co 5652  1c1 7351   + caddc 7353  cz 8750  cuz 9019  seqcseq 9852
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-addcom 7445  ax-addass 7447  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-0id 7453  ax-rnegex 7454  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-ltadd 7461
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-frec 6156  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-inn 8423  df-n0 8674  df-z 8751  df-uz 9020  df-iseq 9853  df-seq3 9854
This theorem is referenced by:  seq3distr  9946  efcj  10963
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