Theorem seqeq2 10703

Description: Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)

Assertion

Ref	Expression
seqeq2	⊢ ( + = 𝑄 → seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹))

Proof of Theorem seqeq2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1021	. . . . . . 7 ⊢ (( + = 𝑄 ∧ 𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ V) → + = 𝑄)
2	1	oveqd 6030	. . . . . 6 ⊢ (( + = 𝑄 ∧ 𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ V) → (𝑦 + (𝐹‘(𝑥 + 1))) = (𝑦𝑄(𝐹‘(𝑥 + 1))))
3	2	opeq2d 3867	. . . . 5 ⊢ (( + = 𝑄 ∧ 𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ V) → ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩ = ⟨(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))⟩)
4	3	mpoeq3dva 6080	. . . 4 ⊢ ( + = 𝑄 → (𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) = (𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))⟩))
5		freceq1 6553	. . . 4 ⊢ ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) = (𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))⟩) → frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) = frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩))
6	4, 5	syl 14	. . 3 ⊢ ( + = 𝑄 → frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) = frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩))
7	6	rneqd 4959	. 2 ⊢ ( + = 𝑄 → ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) = ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩))
8		df-seqfrec 10700	. 2 ⊢ seq𝑀( + , 𝐹) = ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
9		df-seqfrec 10700	. 2 ⊢ seq𝑀(𝑄, 𝐹) = ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
10	7, 8, 9	3eqtr4g 2287	1 ⊢ ( + = 𝑄 → seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹))

Syntax hints:   → wi 4   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  Vcvv 2800  ⟨cop 3670  ran crn 4724  ‘cfv 5324  (class class class)co 6013   ∈ cmpo 6015  freccfrec 6551  1c1 8023   + caddc 8025  ℤ_≥cuz 9745  seqcseq 10699

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-cnv 4731  df-dm 4733  df-rn 4734  df-res 4735  df-iota 5284  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-recs 6466  df-frec 6552  df-seqfrec 10700

This theorem is referenced by:  seqeq2d  10706  resqrex  11577  nninfdc  13064