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Theorem seqeq3 14003
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 6896 . . . . . . 7 (𝐹 = 𝐺 → (𝐹‘(𝑥 + 1)) = (𝐺‘(𝑥 + 1)))
21oveq2d 7436 . . . . . 6 (𝐹 = 𝐺 → (𝑦 + (𝐹‘(𝑥 + 1))) = (𝑦 + (𝐺‘(𝑥 + 1))))
32opeq2d 4881 . . . . 5 (𝐹 = 𝐺 → ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩ = ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩)
43mpoeq3dv 7499 . . . 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 6896 . . . . 5 (𝐹 = 𝐺 → (𝐹𝑀) = (𝐺𝑀))
65opeq2d 4881 . . . 4 (𝐹 = 𝐺 → ⟨𝑀, (𝐹𝑀)⟩ = ⟨𝑀, (𝐺𝑀)⟩)
7 rdgeq12 8433 . . . 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 583 . . 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 6062 . 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 13999 . 2 seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) “ ω)
11 df-seq 13999 . 2 seq𝑀( + , 𝐺) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐺‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐺𝑀)⟩) “ ω)
129, 10, 113eqtr4g 2793 1 (𝐹 = 𝐺 → seq𝑀( + , 𝐹) = seq𝑀( + , 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  Vcvv 3471  cop 4635  cima 5681  cfv 6548  (class class class)co 7420  cmpo 7422  ωcom 7870  reccrdg 8429  1c1 11139   + caddc 11141  seqcseq 13998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5684  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-iota 6500  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-seq 13999
This theorem is referenced by:  seqeq3d  14006  cbvprod  15891  iprodmul  15979  geolim3  26273  leibpilem2  26872  basel  27021  faclim  35340  ovoliunnfl  37135  voliunnfl  37137  heiborlem10  37293  binomcxplemnn0  43786  binomcxplemdvsum  43792  binomcxp  43794  fourierdlem112  45606  fouriersw  45619  voliunsge0lem  45860
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