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Theorem iseqoveq 9592
 Description: Equality theorem for the sequence builder operation. This is similar to iseqeq2 9577 but + and 𝑄 only have to agree on elements of 𝑆. (Contributed by Jim Kingdon, 21-Apr-2022.)
Hypotheses
Ref Expression
iseqoveq.m (𝜑𝑀 ∈ ℤ)
iseqoveq.f ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
iseqoveq.eq ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑥𝑄𝑦))
iseqoveq.plcl ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
iseqoveq.qcl ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
Assertion
Ref Expression
iseqoveq (𝜑 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀(𝑄, 𝐹, 𝑆))
Distinct variable groups:   𝑥, + ,𝑦   𝑥,𝐹,𝑦   𝑥,𝑀,𝑦   𝑥,𝑄,𝑦   𝑥,𝑆,𝑦   𝜑,𝑥,𝑦

Proof of Theorem iseqoveq
Dummy variables 𝑘 𝑤 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2083 . . . 4 (ℤ𝑀) = (ℤ𝑀)
2 iseqoveq.m . . . 4 (𝜑𝑀 ∈ ℤ)
3 iseqoveq.f . . . 4 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
4 iseqoveq.plcl . . . 4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
51, 2, 3, 4iseqfcl 9587 . . 3 (𝜑 → seq𝑀( + , 𝐹, 𝑆):(ℤ𝑀)⟶𝑆)
6 ffn 5097 . . 3 (seq𝑀( + , 𝐹, 𝑆):(ℤ𝑀)⟶𝑆 → seq𝑀( + , 𝐹, 𝑆) Fn (ℤ𝑀))
75, 6syl 14 . 2 (𝜑 → seq𝑀( + , 𝐹, 𝑆) Fn (ℤ𝑀))
8 iseqoveq.qcl . . . 4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
91, 2, 3, 8iseqfcl 9587 . . 3 (𝜑 → seq𝑀(𝑄, 𝐹, 𝑆):(ℤ𝑀)⟶𝑆)
10 ffn 5097 . . 3 (seq𝑀(𝑄, 𝐹, 𝑆):(ℤ𝑀)⟶𝑆 → seq𝑀(𝑄, 𝐹, 𝑆) Fn (ℤ𝑀))
119, 10syl 14 . 2 (𝜑 → seq𝑀(𝑄, 𝐹, 𝑆) Fn (ℤ𝑀))
12 fveq2 5229 . . . . . 6 (𝑤 = 𝑀 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑀))
13 fveq2 5229 . . . . . 6 (𝑤 = 𝑀 → (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑤) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑀))
1412, 13eqeq12d 2097 . . . . 5 (𝑤 = 𝑀 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑤) ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑀)))
1514imbi2d 228 . . . 4 (𝑤 = 𝑀 → ((𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑤)) ↔ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑀))))
16 fveq2 5229 . . . . . 6 (𝑤 = 𝑘 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑘))
17 fveq2 5229 . . . . . 6 (𝑤 = 𝑘 → (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑤) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑘))
1816, 17eqeq12d 2097 . . . . 5 (𝑤 = 𝑘 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑤) ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑘)))
1918imbi2d 228 . . . 4 (𝑤 = 𝑘 → ((𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑤)) ↔ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑘))))
20 fveq2 5229 . . . . . 6 (𝑤 = (𝑘 + 1) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)))
21 fveq2 5229 . . . . . 6 (𝑤 = (𝑘 + 1) → (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑤) = (seq𝑀(𝑄, 𝐹, 𝑆)‘(𝑘 + 1)))
2220, 21eqeq12d 2097 . . . . 5 (𝑤 = (𝑘 + 1) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑤) ↔ (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = (seq𝑀(𝑄, 𝐹, 𝑆)‘(𝑘 + 1))))
2322imbi2d 228 . . . 4 (𝑤 = (𝑘 + 1) → ((𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑤)) ↔ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = (seq𝑀(𝑄, 𝐹, 𝑆)‘(𝑘 + 1)))))
24 fveq2 5229 . . . . . 6 (𝑤 = 𝑛 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑛))
25 fveq2 5229 . . . . . 6 (𝑤 = 𝑛 → (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑤) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑛))
2624, 25eqeq12d 2097 . . . . 5 (𝑤 = 𝑛 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑤) ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑛)))
2726imbi2d 228 . . . 4 (𝑤 = 𝑛 → ((𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑤)) ↔ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑛))))
28 simpr 108 . . . . . . 7 ((𝜑𝑀 ∈ ℤ) → 𝑀 ∈ ℤ)
293adantlr 461 . . . . . . 7 (((𝜑𝑀 ∈ ℤ) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
304adantlr 461 . . . . . . 7 (((𝜑𝑀 ∈ ℤ) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
3128, 29, 30iseq1 9585 . . . . . 6 ((𝜑𝑀 ∈ ℤ) → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (𝐹𝑀))
328adantlr 461 . . . . . . 7 (((𝜑𝑀 ∈ ℤ) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
3328, 29, 32iseq1 9585 . . . . . 6 ((𝜑𝑀 ∈ ℤ) → (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑀) = (𝐹𝑀))
3431, 33eqtr4d 2118 . . . . 5 ((𝜑𝑀 ∈ ℤ) → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑀))
3534expcom 114 . . . 4 (𝑀 ∈ ℤ → (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑀)))
36 simpr 108 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (ℤ𝑀)) → 𝑘 ∈ (ℤ𝑀))
3736adantr 270 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (ℤ𝑀)) ∧ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑘)) → 𝑘 ∈ (ℤ𝑀))
383adantlr 461 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (ℤ𝑀)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
3938adantlr 461 . . . . . . . . . . 11 ((((𝜑𝑘 ∈ (ℤ𝑀)) ∧ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑘)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
404adantlr 461 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (ℤ𝑀)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
4140adantlr 461 . . . . . . . . . . 11 ((((𝜑𝑘 ∈ (ℤ𝑀)) ∧ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑘)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
4237, 39, 41iseqcl 9589 . . . . . . . . . 10 (((𝜑𝑘 ∈ (ℤ𝑀)) ∧ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑘)) → (seq𝑀( + , 𝐹, 𝑆)‘𝑘) ∈ 𝑆)
43 fveq2 5229 . . . . . . . . . . . 12 (𝑥 = (𝑘 + 1) → (𝐹𝑥) = (𝐹‘(𝑘 + 1)))
4443eleq1d 2151 . . . . . . . . . . 11 (𝑥 = (𝑘 + 1) → ((𝐹𝑥) ∈ 𝑆 ↔ (𝐹‘(𝑘 + 1)) ∈ 𝑆))
453ralrimiva 2439 . . . . . . . . . . . 12 (𝜑 → ∀𝑥 ∈ (ℤ𝑀)(𝐹𝑥) ∈ 𝑆)
4645ad2antrr 472 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (ℤ𝑀)) ∧ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑘)) → ∀𝑥 ∈ (ℤ𝑀)(𝐹𝑥) ∈ 𝑆)
47 peano2uz 8804 . . . . . . . . . . . 12 (𝑘 ∈ (ℤ𝑀) → (𝑘 + 1) ∈ (ℤ𝑀))
4847ad2antlr 473 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (ℤ𝑀)) ∧ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑘)) → (𝑘 + 1) ∈ (ℤ𝑀))
4944, 46, 48rspcdva 2715 . . . . . . . . . 10 (((𝜑𝑘 ∈ (ℤ𝑀)) ∧ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑘)) → (𝐹‘(𝑘 + 1)) ∈ 𝑆)
50 iseqoveq.eq . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑥𝑄𝑦))
5150ralrimivva 2448 . . . . . . . . . . 11 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) = (𝑥𝑄𝑦))
5251ad2antrr 472 . . . . . . . . . 10 (((𝜑𝑘 ∈ (ℤ𝑀)) ∧ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑘)) → ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) = (𝑥𝑄𝑦))
53 oveq1 5570 . . . . . . . . . . . 12 (𝑥 = (seq𝑀( + , 𝐹, 𝑆)‘𝑘) → (𝑥 + 𝑦) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) + 𝑦))
54 oveq1 5570 . . . . . . . . . . . 12 (𝑥 = (seq𝑀( + , 𝐹, 𝑆)‘𝑘) → (𝑥𝑄𝑦) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑘)𝑄𝑦))
5553, 54eqeq12d 2097 . . . . . . . . . . 11 (𝑥 = (seq𝑀( + , 𝐹, 𝑆)‘𝑘) → ((𝑥 + 𝑦) = (𝑥𝑄𝑦) ↔ ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) + 𝑦) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑘)𝑄𝑦)))
56 oveq2 5571 . . . . . . . . . . . 12 (𝑦 = (𝐹‘(𝑘 + 1)) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) + 𝑦) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) + (𝐹‘(𝑘 + 1))))
57 oveq2 5571 . . . . . . . . . . . 12 (𝑦 = (𝐹‘(𝑘 + 1)) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑘)𝑄𝑦) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑘)𝑄(𝐹‘(𝑘 + 1))))
5856, 57eqeq12d 2097 . . . . . . . . . . 11 (𝑦 = (𝐹‘(𝑘 + 1)) → (((seq𝑀( + , 𝐹, 𝑆)‘𝑘) + 𝑦) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑘)𝑄𝑦) ↔ ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) + (𝐹‘(𝑘 + 1))) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑘)𝑄(𝐹‘(𝑘 + 1)))))
5955, 58rspc2va 2722 . . . . . . . . . 10 ((((seq𝑀( + , 𝐹, 𝑆)‘𝑘) ∈ 𝑆 ∧ (𝐹‘(𝑘 + 1)) ∈ 𝑆) ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) = (𝑥𝑄𝑦)) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) + (𝐹‘(𝑘 + 1))) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑘)𝑄(𝐹‘(𝑘 + 1))))
6042, 49, 52, 59syl21anc 1169 . . . . . . . . 9 (((𝜑𝑘 ∈ (ℤ𝑀)) ∧ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑘)) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) + (𝐹‘(𝑘 + 1))) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑘)𝑄(𝐹‘(𝑘 + 1))))
61 simpr 108 . . . . . . . . . 10 (((𝜑𝑘 ∈ (ℤ𝑀)) ∧ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑘)) → (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑘))
6261oveq1d 5578 . . . . . . . . 9 (((𝜑𝑘 ∈ (ℤ𝑀)) ∧ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑘)) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑘)𝑄(𝐹‘(𝑘 + 1))) = ((seq𝑀(𝑄, 𝐹, 𝑆)‘𝑘)𝑄(𝐹‘(𝑘 + 1))))
6360, 62eqtrd 2115 . . . . . . . 8 (((𝜑𝑘 ∈ (ℤ𝑀)) ∧ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑘)) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) + (𝐹‘(𝑘 + 1))) = ((seq𝑀(𝑄, 𝐹, 𝑆)‘𝑘)𝑄(𝐹‘(𝑘 + 1))))
6436, 38, 40iseqp1 9590 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ℤ𝑀)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) + (𝐹‘(𝑘 + 1))))
658adantlr 461 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (ℤ𝑀)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
6636, 38, 65iseqp1 9590 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ℤ𝑀)) → (seq𝑀(𝑄, 𝐹, 𝑆)‘(𝑘 + 1)) = ((seq𝑀(𝑄, 𝐹, 𝑆)‘𝑘)𝑄(𝐹‘(𝑘 + 1))))
6764, 66eqeq12d 2097 . . . . . . . . 9 ((𝜑𝑘 ∈ (ℤ𝑀)) → ((seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = (seq𝑀(𝑄, 𝐹, 𝑆)‘(𝑘 + 1)) ↔ ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) + (𝐹‘(𝑘 + 1))) = ((seq𝑀(𝑄, 𝐹, 𝑆)‘𝑘)𝑄(𝐹‘(𝑘 + 1)))))
6867adantr 270 . . . . . . . 8 (((𝜑𝑘 ∈ (ℤ𝑀)) ∧ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑘)) → ((seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = (seq𝑀(𝑄, 𝐹, 𝑆)‘(𝑘 + 1)) ↔ ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) + (𝐹‘(𝑘 + 1))) = ((seq𝑀(𝑄, 𝐹, 𝑆)‘𝑘)𝑄(𝐹‘(𝑘 + 1)))))
6963, 68mpbird 165 . . . . . . 7 (((𝜑𝑘 ∈ (ℤ𝑀)) ∧ (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑘)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = (seq𝑀(𝑄, 𝐹, 𝑆)‘(𝑘 + 1)))
7069ex 113 . . . . . 6 ((𝜑𝑘 ∈ (ℤ𝑀)) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑘) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = (seq𝑀(𝑄, 𝐹, 𝑆)‘(𝑘 + 1))))
7170expcom 114 . . . . 5 (𝑘 ∈ (ℤ𝑀) → (𝜑 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑘) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑘) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = (seq𝑀(𝑄, 𝐹, 𝑆)‘(𝑘 + 1)))))
7271a2d 26 . . . 4 (𝑘 ∈ (ℤ𝑀) → ((𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑘) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑘)) → (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘(𝑘 + 1)) = (seq𝑀(𝑄, 𝐹, 𝑆)‘(𝑘 + 1)))))
7315, 19, 23, 27, 35, 72uzind4 8809 . . 3 (𝑛 ∈ (ℤ𝑀) → (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑛)))
7473impcom 123 . 2 ((𝜑𝑛 ∈ (ℤ𝑀)) → (seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq𝑀(𝑄, 𝐹, 𝑆)‘𝑛))
757, 11, 74eqfnfvd 5320 1 (𝜑 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀(𝑄, 𝐹, 𝑆))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1285   ∈ wcel 1434  ∀wral 2353   Fn wfn 4947  ⟶wf 4948  ‘cfv 4952  (class class class)co 5563  1c1 7096   + caddc 7098  ℤcz 8484  ℤ≥cuz 8752  seqcseq 9573 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 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-addcom 7190  ax-addass 7192  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-0id 7198  ax-rnegex 7199  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-ltadd 7206 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 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-frec 6060  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-inn 8159  df-n0 8408  df-z 8485  df-uz 8753  df-iseq 9574 This theorem is referenced by: (None)
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