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Theorem seqeq1 12752
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 6153 . . . . 5 (𝑀 = 𝑁 → (𝐹𝑀) = (𝐹𝑁))
2 opeq12 4377 . . . . 5 ((𝑀 = 𝑁 ∧ (𝐹𝑀) = (𝐹𝑁)) → ⟨𝑀, (𝐹𝑀)⟩ = ⟨𝑁, (𝐹𝑁)⟩)
31, 2mpdan 701 . . . 4 (𝑀 = 𝑁 → ⟨𝑀, (𝐹𝑀)⟩ = ⟨𝑁, (𝐹𝑁)⟩)
4 rdgeq2 7460 . . . 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 5429 . 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 12750 . 2 seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) “ ω)
8 df-seq 12750 . 2 seq𝑁( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹𝑁)⟩) “ ω)
96, 7, 83eqtr4g 2680 1 (𝑀 = 𝑁 → seq𝑀( + , 𝐹) = seq𝑁( + , 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  Vcvv 3189  cop 4159  cima 5082  cfv 5852  (class class class)co 6610  cmpt2 6612  ωcom 7019  reccrdg 7457  1c1 9889   + caddc 9891  seqcseq 12749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-xp 5085  df-cnv 5087  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-iota 5815  df-fv 5860  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-seq 12750
This theorem is referenced by:  seqeq1d  12755  seqfn  12761  seq1  12762  seqp1  12764  seqf1olem2  12789  seqid  12794  seqz  12797  iserex  14329  summolem2  14388  summo  14389  zsum  14390  isumsplit  14508  ntrivcvg  14565  ntrivcvgn0  14566  ntrivcvgtail  14568  ntrivcvgmullem  14569  prodmolem2  14601  prodmo  14602  zprod  14603  fprodntriv  14608  ege2le3  14756  gsumval2a  17211  leibpi  24586  dvradcnv2  38063  binomcxplemnotnn0  38072  stirlinglem12  39635
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