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Theorem seqeq1 13918
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 6846 . . . . 5 (𝑀 = 𝑁 → (𝐹𝑀) = (𝐹𝑁))
2 opeq12 4836 . . . . 5 ((𝑀 = 𝑁 ∧ (𝐹𝑀) = (𝐹𝑁)) → ⟨𝑀, (𝐹𝑀)⟩ = ⟨𝑁, (𝐹𝑁)⟩)
31, 2mpdan 686 . . . 4 (𝑀 = 𝑁 → ⟨𝑀, (𝐹𝑀)⟩ = ⟨𝑁, (𝐹𝑁)⟩)
4 rdgeq2 8362 . . . 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 6016 . 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 13916 . 2 seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) “ ω)
8 df-seq 13916 . 2 seq𝑁( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹𝑁)⟩) “ ω)
96, 7, 83eqtr4g 2798 1 (𝑀 = 𝑁 → seq𝑀( + , 𝐹) = seq𝑁( + , 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  Vcvv 3447  cop 4596  cima 5640  cfv 6500  (class class class)co 7361  cmpo 7363  ωcom 7806  reccrdg 8359  1c1 11060   + caddc 11062  seqcseq 13915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-xp 5643  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-iota 6452  df-fv 6508  df-ov 7364  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-seq 13916
This theorem is referenced by:  seqeq1d  13921  seqfn  13927  seq1  13928  seqp1  13930  seqf1olem2  13957  seqid  13962  seqz  13965  iserex  15550  summolem2  15609  summo  15610  zsum  15611  isumsplit  15733  ntrivcvg  15790  ntrivcvgn0  15791  ntrivcvgtail  15793  ntrivcvgmullem  15794  prodmolem2  15826  prodmo  15827  zprod  15828  fprodntriv  15833  ege2le3  15980  gsumval2a  18548  leibpi  26315  dvradcnv2  42719  binomcxplemnotnn0  42728  stirlinglem12  44416
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