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Theorem seqeq1 13004
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 fveq2 6330 . . . . 5 (𝑀 = 𝑁 → (𝐹𝑀) = (𝐹𝑁))
2 opeq12 4541 . . . . 5 ((𝑀 = 𝑁 ∧ (𝐹𝑀) = (𝐹𝑁)) → ⟨𝑀, (𝐹𝑀)⟩ = ⟨𝑁, (𝐹𝑁)⟩)
31, 2mpdan 667 . . . 4 (𝑀 = 𝑁 → ⟨𝑀, (𝐹𝑀)⟩ = ⟨𝑁, (𝐹𝑁)⟩)
4 rdgeq2 7659 . . . 4 (⟨𝑀, (𝐹𝑀)⟩ = ⟨𝑁, (𝐹𝑁)⟩ → rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) = rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹𝑁)⟩))
53, 4syl 17 . . 3 (𝑀 = 𝑁 → rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) = rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹𝑁)⟩))
65imaeq1d 5604 . 2 (𝑀 = 𝑁 → (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) “ ω) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹𝑁)⟩) “ ω))
7 df-seq 13002 . 2 seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) “ ω)
8 df-seq 13002 . 2 seq𝑁( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹𝑁)⟩) “ ω)
96, 7, 83eqtr4g 2830 1 (𝑀 = 𝑁 → seq𝑀( + , 𝐹) = seq𝑁( + , 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  Vcvv 3351  cop 4322  cima 5252  cfv 6029  (class class class)co 6791  cmpt2 6793  ωcom 7210  reccrdg 7656  1c1 10137   + caddc 10139  seqcseq 13001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-xp 5255  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5821  df-iota 5992  df-fv 6037  df-wrecs 7557  df-recs 7619  df-rdg 7657  df-seq 13002
This theorem is referenced by:  seqeq1d  13007  seqfn  13013  seq1  13014  seqp1  13016  seqf1olem2  13041  seqid  13046  seqz  13049  iserex  14588  summolem2  14648  summo  14649  zsum  14650  isumsplit  14772  ntrivcvg  14829  ntrivcvgn0  14830  ntrivcvgtail  14832  ntrivcvgmullem  14833  prodmolem2  14865  prodmo  14866  zprod  14867  fprodntriv  14872  ege2le3  15019  gsumval2a  17480  leibpi  24883  dvradcnv2  39065  binomcxplemnotnn0  39074  stirlinglem12  40812
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