MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  seqeq3 Structured version   Visualization version   GIF version

Theorem seqeq3 12746
Description: Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
Assertion
Ref Expression
seqeq3 (𝐹 = 𝐺 → seq𝑀( + , 𝐹) = seq𝑀( + , 𝐺))

Proof of Theorem seqeq3
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq1 6147 . . . . . . 7 (𝐹 = 𝐺 → (𝐹‘(𝑥 + 1)) = (𝐺‘(𝑥 + 1)))
21oveq2d 6620 . . . . . 6 (𝐹 = 𝐺 → (𝑦 + (𝐹‘(𝑥 + 1))) = (𝑦 + (𝐺‘(𝑥 + 1))))
32opeq2d 4377 . . . . 5 (𝐹 = 𝐺 → ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩ = ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩)
43mpt2eq3dv 6674 . . . 4 (𝐹 = 𝐺 → (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) = (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩))
5 fveq1 6147 . . . . 5 (𝐹 = 𝐺 → (𝐹𝑀) = (𝐺𝑀))
65opeq2d 4377 . . . 4 (𝐹 = 𝐺 → ⟨𝑀, (𝐹𝑀)⟩ = ⟨𝑀, (𝐺𝑀)⟩)
7 rdgeq12 7454 . . . 4 (((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) = (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩) ∧ ⟨𝑀, (𝐹𝑀)⟩ = ⟨𝑀, (𝐺𝑀)⟩) → rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) = rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐺𝑀)⟩))
84, 6, 7syl2anc 692 . . 3 (𝐹 = 𝐺 → rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) = rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐺𝑀)⟩))
98imaeq1d 5424 . 2 (𝐹 = 𝐺 → (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) “ ω) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐺𝑀)⟩) “ ω))
10 df-seq 12742 . 2 seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) “ ω)
11 df-seq 12742 . 2 seq𝑀( + , 𝐺) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐺𝑀)⟩) “ ω)
129, 10, 113eqtr4g 2680 1 (𝐹 = 𝐺 → seq𝑀( + , 𝐹) = seq𝑀( + , 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  Vcvv 3186  cop 4154  cima 5077  cfv 5847  (class class class)co 6604  cmpt2 6606  ωcom 7012  reccrdg 7450  1c1 9881   + caddc 9883  seqcseq 12741
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 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-xp 5080  df-cnv 5082  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-iota 5810  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-seq 12742
This theorem is referenced by:  seqeq3d  12749  cbvprod  14570  iprodmul  14659  geolim3  23998  leibpilem2  24568  basel  24716  faclim  31337  ovoliunnfl  33080  voliunnfl  33082  heiborlem10  33248  binomcxplemnn0  38027  binomcxplemdvsum  38033  binomcxp  38035  fourierdlem112  39739  fouriersw  39752  voliunsge0lem  39993
  Copyright terms: Public domain W3C validator