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Theorem iseqeq3 9743
Description: Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
Assertion
Ref Expression
iseqeq3 (𝐹 = 𝐺 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐺, 𝑆))

Proof of Theorem iseqeq3
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 939 . . . . . . . 8 ((𝐹 = 𝐺𝑥 ∈ (ℤ𝑀) ∧ 𝑦𝑆) → 𝐹 = 𝐺)
21fveq1d 5253 . . . . . . 7 ((𝐹 = 𝐺𝑥 ∈ (ℤ𝑀) ∧ 𝑦𝑆) → (𝐹‘(𝑥 + 1)) = (𝐺‘(𝑥 + 1)))
32oveq2d 5605 . . . . . 6 ((𝐹 = 𝐺𝑥 ∈ (ℤ𝑀) ∧ 𝑦𝑆) → (𝑦 + (𝐹‘(𝑥 + 1))) = (𝑦 + (𝐺‘(𝑥 + 1))))
43opeq2d 3603 . . . . 5 ((𝐹 = 𝐺𝑥 ∈ (ℤ𝑀) ∧ 𝑦𝑆) → ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩ = ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩)
54mpt2eq3dva 5646 . . . 4 (𝐹 = 𝐺 → (𝑥 ∈ (ℤ𝑀), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) = (𝑥 ∈ (ℤ𝑀), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩))
6 fveq1 5250 . . . . 5 (𝐹 = 𝐺 → (𝐹𝑀) = (𝐺𝑀))
76opeq2d 3603 . . . 4 (𝐹 = 𝐺 → ⟨𝑀, (𝐹𝑀)⟩ = ⟨𝑀, (𝐺𝑀)⟩)
8 freceq1 6087 . . . . 5 ((𝑥 ∈ (ℤ𝑀), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) = (𝑥 ∈ (ℤ𝑀), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩) → frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) = frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩))
9 freceq2 6088 . . . . 5 (⟨𝑀, (𝐹𝑀)⟩ = ⟨𝑀, (𝐺𝑀)⟩ → frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) = frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐺𝑀)⟩))
108, 9sylan9eq 2135 . . . 4 (((𝑥 ∈ (ℤ𝑀), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) = (𝑥 ∈ (ℤ𝑀), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩) ∧ ⟨𝑀, (𝐹𝑀)⟩ = ⟨𝑀, (𝐺𝑀)⟩) → frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) = frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐺𝑀)⟩))
115, 7, 10syl2anc 403 . . 3 (𝐹 = 𝐺 → frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) = frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐺𝑀)⟩))
1211rneqd 4620 . 2 (𝐹 = 𝐺 → ran frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) = ran frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐺𝑀)⟩))
13 df-iseq 9739 . 2 seq𝑀( + , 𝐹, 𝑆) = ran frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
14 df-iseq 9739 . 2 seq𝑀( + , 𝐺, 𝑆) = ran frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐺𝑀)⟩)
1512, 13, 143eqtr4g 2140 1 (𝐹 = 𝐺 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐺, 𝑆))
Colors of variables: wff set class
Syntax hints:  wi 4  w3a 920   = wceq 1285  wcel 1434  cop 3425  ran crn 4400  cfv 4967  (class class class)co 5589  cmpt2 5591  freccfrec 6085  1c1 7252   + caddc 7254  cuz 8912  seqcseq 9738
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-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-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-mpt 3867  df-cnv 4407  df-dm 4409  df-rn 4410  df-res 4411  df-iota 4932  df-fv 4975  df-ov 5592  df-oprab 5593  df-mpt2 5594  df-recs 6000  df-frec 6086  df-iseq 9739
This theorem is referenced by:  expival  9792  sumeq1  10564  sumeq2d  10568  sumeq2  10569
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