Theorem seqeq2 10712

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 1023	. . . . . . 7 ⊢ (( + = 𝑄 ∧ 𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ V) → + = 𝑄)
2	1	oveqd 6034	. . . . . 6 ⊢ (( + = 𝑄 ∧ 𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ V) → (𝑦 + (𝐹‘(𝑥 + 1))) = (𝑦𝑄(𝐹‘(𝑥 + 1))))
3	2	opeq2d 3869	. . . . 5 ⊢ (( + = 𝑄 ∧ 𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ V) → ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩ = ⟨(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))⟩)
4	3	mpoeq3dva 6084	. . . 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 6557	. . . 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 4961	. 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 10709	. 2 ⊢ seq𝑀( + , 𝐹) = ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
9		df-seqfrec 10709	. 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 2289	1 ⊢ ( + = 𝑄 → seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹))

Syntax hints:   → wi 4   ∧ w3a 1004   = wceq 1397   ∈ wcel 2202  Vcvv 2802  ⟨cop 3672  ran crn 4726  ‘cfv 5326  (class class class)co 6017   ∈ cmpo 6019  freccfrec 6555  1c1 8032   + caddc 8034  ℤ_≥cuz 9754  seqcseq 10708

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-cnv 4733  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-recs 6470  df-frec 6556  df-seqfrec 10709

This theorem is referenced by:  seqeq2d  10715  resqrex  11586  nninfdc  13073