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Theorem seqeq1 13356
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 6646 . . . . 5 (𝑀 = 𝑁 → (𝐹𝑀) = (𝐹𝑁))
2 opeq12 4781 . . . . 5 ((𝑀 = 𝑁 ∧ (𝐹𝑀) = (𝐹𝑁)) → ⟨𝑀, (𝐹𝑀)⟩ = ⟨𝑁, (𝐹𝑁)⟩)
31, 2mpdan 685 . . . 4 (𝑀 = 𝑁 → ⟨𝑀, (𝐹𝑀)⟩ = ⟨𝑁, (𝐹𝑁)⟩)
4 rdgeq2 8026 . . . 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 5904 . 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 13354 . 2 seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) “ ω)
8 df-seq 13354 . 2 seq𝑁( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹𝑁)⟩) “ ω)
96, 7, 83eqtr4g 2880 1 (𝑀 = 𝑁 → seq𝑀( + , 𝐹) = seq𝑁( + , 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  Vcvv 3473  cop 4549  cima 5534  cfv 6331  (class class class)co 7133  cmpo 7135  ωcom 7558  reccrdg 8023  1c1 10516   + caddc 10518  seqcseq 13353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rab 3134  df-v 3475  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-mpt 5123  df-xp 5537  df-cnv 5539  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-pred 6124  df-iota 6290  df-fv 6339  df-wrecs 7925  df-recs 7986  df-rdg 8024  df-seq 13354
This theorem is referenced by:  seqeq1d  13359  seqfn  13365  seq1  13366  seqp1  13368  seqf1olem2  13395  seqid  13400  seqz  13403  iserex  14993  summolem2  15053  summo  15054  zsum  15055  isumsplit  15175  ntrivcvg  15233  ntrivcvgn0  15234  ntrivcvgtail  15236  ntrivcvgmullem  15237  prodmolem2  15269  prodmo  15270  zprod  15271  fprodntriv  15276  ege2le3  15423  gsumval2a  17874  leibpi  25507  dvradcnv2  40834  binomcxplemnotnn0  40843  stirlinglem12  42518
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