Theorem seqeq1 14187

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.

Step	Hyp	Ref	Expression
1		fveq2 6916	. . . . 5 ⊢ (𝑀 = 𝑁 → (𝐹‘𝑀) = (𝐹‘𝑁))
2		opeq12 4847	. . . . 5 ⊢ ((𝑀 = 𝑁 ∧ (𝐹‘𝑀) = (𝐹‘𝑁)) → ⟨𝑀, (𝐹‘𝑀)⟩ = ⟨𝑁, (𝐹‘𝑁)⟩)
3	1, 2	mpdan 700	. . . 4 ⊢ (𝑀 = 𝑁 → ⟨𝑀, (𝐹‘𝑀)⟩ = ⟨𝑁, (𝐹‘𝑁)⟩)
4		rdgeq2 8458	. . . 4 ⊢ (⟨𝑀, (𝐹‘𝑀)⟩ = ⟨𝑁, (𝐹‘𝑁)⟩ → rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) = rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹‘𝑁)⟩))
5	3, 4	syl 18	. . 3 ⊢ (𝑀 = 𝑁 → rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) = rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹‘𝑁)⟩))
6	5	imaeq1d 6069	. 2 ⊢ (𝑀 = 𝑁 → (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹‘𝑁)⟩) “ ω))
7		df-seq 14185	. 2 ⊢ seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
8		df-seq 14185	. 2 ⊢ seq𝑁( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹‘𝑁)⟩) “ ω)
9	6, 7, 8	3eqtr4g 2823	1 ⊢ (𝑀 = 𝑁 → seq𝑀( + , 𝐹) = seq𝑁( + , 𝐹))

This proof depends on syntax axioms:   → wi 4   = wceq 1570  Vcvv 3454  ⟨cop 4840   “ cima 5661  ‘cfv 6566  (class class class)co 7451   ∈ cmpo 7453  ωcom 7911  reccrdg 8455  1c1 11240   + caddc 11242  seqcseq 14184

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-xp 5664  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-iota 6520  df-fv 6574  df-ov 7454  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-seq 14185

This theorem is used by:  seqeq1d  14190  seqfn  14196  seq1  14197  seqp1  14199  seqf1olem2  14225  seqid  14230  seqz  14233  iserex  15868  summolem2  15926  summo  15927  zsum  15928  isumsplit  16053  ntrivcvg  16110  ntrivcvgn0  16111  ntrivcvgtail  16113  ntrivcvgmullem  16114  prodmolem2  16146  prodmo  16147  zprod  16148  fprodntriv  16153  ege2le3  16300  gsumval2a  18924  leibpi  27327  dvradcnv2  45592  binomcxplemnotnn0  45601  stirlinglem12  47368