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Theorem seqeq1 10421
Description: Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
Assertion
Ref Expression
seqeq1 (𝑀 = 𝑁 → seq𝑀( + , 𝐹) = seq𝑁( + , 𝐹))

Proof of Theorem seqeq1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 19 . . . . . 6 (𝑀 = 𝑁𝑀 = 𝑁)
2 fveq2 5510 . . . . . 6 (𝑀 = 𝑁 → (𝐹𝑀) = (𝐹𝑁))
31, 2opeq12d 3784 . . . . 5 (𝑀 = 𝑁 → ⟨𝑀, (𝐹𝑀)⟩ = ⟨𝑁, (𝐹𝑁)⟩)
4 freceq2 6387 . . . . 5 (⟨𝑀, (𝐹𝑀)⟩ = ⟨𝑁, (𝐹𝑁)⟩ → frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) = frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹𝑁)⟩))
53, 4syl 14 . . . 4 (𝑀 = 𝑁 → frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) = frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹𝑁)⟩))
6 fveq2 5510 . . . . . 6 (𝑀 = 𝑁 → (ℤ𝑀) = (ℤ𝑁))
7 eqid 2177 . . . . . 6 V = V
8 mpoeq12 5928 . . . . . 6 (((ℤ𝑀) = (ℤ𝑁) ∧ V = V) → (𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) = (𝑥 ∈ (ℤ𝑁), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩))
96, 7, 8sylancl 413 . . . . 5 (𝑀 = 𝑁 → (𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) = (𝑥 ∈ (ℤ𝑁), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩))
10 freceq1 6386 . . . . 5 ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) = (𝑥 ∈ (ℤ𝑁), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) → frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹𝑁)⟩) = frec((𝑥 ∈ (ℤ𝑁), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹𝑁)⟩))
119, 10syl 14 . . . 4 (𝑀 = 𝑁 → frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹𝑁)⟩) = frec((𝑥 ∈ (ℤ𝑁), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹𝑁)⟩))
125, 11eqtrd 2210 . . 3 (𝑀 = 𝑁 → frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) = frec((𝑥 ∈ (ℤ𝑁), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹𝑁)⟩))
1312rneqd 4851 . 2 (𝑀 = 𝑁 → ran frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) = ran frec((𝑥 ∈ (ℤ𝑁), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹𝑁)⟩))
14 df-seqfrec 10419 . 2 seq𝑀( + , 𝐹) = ran frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
15 df-seqfrec 10419 . 2 seq𝑁( + , 𝐹) = ran frec((𝑥 ∈ (ℤ𝑁), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹𝑁)⟩)
1613, 14, 153eqtr4g 2235 1 (𝑀 = 𝑁 → seq𝑀( + , 𝐹) = seq𝑁( + , 𝐹))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  Vcvv 2737  cop 3594  ran crn 4623  cfv 5211  (class class class)co 5868  cmpo 5870  freccfrec 6384  1c1 7790   + caddc 7792  cuz 9504  seqcseq 10418
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-cnv 4630  df-dm 4632  df-rn 4633  df-res 4634  df-iota 5173  df-fv 5219  df-oprab 5872  df-mpo 5873  df-recs 6299  df-frec 6385  df-seqfrec 10419
This theorem is referenced by:  seqeq1d  10424  seq3f1olemqsum  10473  seq3id  10481  seq3z  10484  iserex  11318  summodclem2  11361  summodc  11362  zsumdc  11363  isumsplit  11470  ntrivcvgap  11527  ntrivcvgap0  11528  prodmodclem2  11556  prodmodc  11557  zproddc  11558  fprodntrivap  11563  ege2le3  11650
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