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Theorem seqeq3 10406
Description: Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
Assertion
Ref Expression
seqeq3 (𝐹 = 𝐺 → seq𝑀( + , 𝐹) = seq𝑀( + , 𝐺))

Proof of Theorem seqeq3
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 992 . . . . . . . 8 ((𝐹 = 𝐺𝑥 ∈ (ℤ𝑀) ∧ 𝑦 ∈ V) → 𝐹 = 𝐺)
21fveq1d 5498 . . . . . . 7 ((𝐹 = 𝐺𝑥 ∈ (ℤ𝑀) ∧ 𝑦 ∈ V) → (𝐹‘(𝑥 + 1)) = (𝐺‘(𝑥 + 1)))
32oveq2d 5869 . . . . . 6 ((𝐹 = 𝐺𝑥 ∈ (ℤ𝑀) ∧ 𝑦 ∈ V) → (𝑦 + (𝐹‘(𝑥 + 1))) = (𝑦 + (𝐺‘(𝑥 + 1))))
43opeq2d 3772 . . . . 5 ((𝐹 = 𝐺𝑥 ∈ (ℤ𝑀) ∧ 𝑦 ∈ V) → ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩ = ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩)
54mpoeq3dva 5917 . . . 4 (𝐹 = 𝐺 → (𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) = (𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩))
6 fveq1 5495 . . . . 5 (𝐹 = 𝐺 → (𝐹𝑀) = (𝐺𝑀))
76opeq2d 3772 . . . 4 (𝐹 = 𝐺 → ⟨𝑀, (𝐹𝑀)⟩ = ⟨𝑀, (𝐺𝑀)⟩)
8 freceq1 6371 . . . . 5 ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) = (𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩) → frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) = frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩))
9 freceq2 6372 . . . . 5 (⟨𝑀, (𝐹𝑀)⟩ = ⟨𝑀, (𝐺𝑀)⟩ → frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) = frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐺𝑀)⟩))
108, 9sylan9eq 2223 . . . 4 (((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) = (𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩) ∧ ⟨𝑀, (𝐹𝑀)⟩ = ⟨𝑀, (𝐺𝑀)⟩) → frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) = frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐺𝑀)⟩))
115, 7, 10syl2anc 409 . . 3 (𝐹 = 𝐺 → frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) = frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐺𝑀)⟩))
1211rneqd 4840 . 2 (𝐹 = 𝐺 → ran frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) = ran frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐺𝑀)⟩))
13 df-seqfrec 10402 . 2 seq𝑀( + , 𝐹) = ran frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
14 df-seqfrec 10402 . 2 seq𝑀( + , 𝐺) = ran frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐺𝑀)⟩)
1512, 13, 143eqtr4g 2228 1 (𝐹 = 𝐺 → seq𝑀( + , 𝐹) = seq𝑀( + , 𝐺))
Colors of variables: wff set class
Syntax hints:  wi 4  w3a 973   = wceq 1348  wcel 2141  Vcvv 2730  cop 3586  ran crn 4612  cfv 5198  (class class class)co 5853  cmpo 5855  freccfrec 6369  1c1 7775   + caddc 7777  cuz 9487  seqcseq 10401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623  df-iota 5160  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-recs 6284  df-frec 6370  df-seqfrec 10402
This theorem is referenced by:  seqeq3d  10409  cbvsum  11323  fsumadd  11369  cbvprod  11521
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