Theorem seqeq1 13995

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 6891	. . . . 5 ⊢ (𝑀 = 𝑁 → (𝐹‘𝑀) = (𝐹‘𝑁))
2		opeq12 4871	. . . . 5 ⊢ ((𝑀 = 𝑁 ∧ (𝐹‘𝑀) = (𝐹‘𝑁)) → ⟨𝑀, (𝐹‘𝑀)⟩ = ⟨𝑁, (𝐹‘𝑁)⟩)
3	1, 2	mpdan 686	. . . 4 ⊢ (𝑀 = 𝑁 → ⟨𝑀, (𝐹‘𝑀)⟩ = ⟨𝑁, (𝐹‘𝑁)⟩)
4		rdgeq2 8426	. . . 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 17	. . 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 6056	. 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 13993	. 2 ⊢ seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
8		df-seq 13993	. 2 ⊢ seq𝑁( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑁, (𝐹‘𝑁)⟩) “ ω)
9	6, 7, 8	3eqtr4g 2792	1 ⊢ (𝑀 = 𝑁 → seq𝑀( + , 𝐹) = seq𝑁( + , 𝐹))

Syntax hints:   → wi 4   = wceq 1534  Vcvv 3469  ⟨cop 4630   “ cima 5675  ‘cfv 6542  (class class class)co 7414   ∈ cmpo 7416  ωcom 7864  reccrdg 8423  1c1 11133   + caddc 11135  seqcseq 13992

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 2698

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 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-xp 5678  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-iota 6494  df-fv 6550  df-ov 7417  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-seq 13993

This theorem is referenced by:  seqeq1d  13998  seqfn  14004  seq1  14005  seqp1  14007  seqf1olem2  14033  seqid  14038  seqz  14041  iserex  15629  summolem2  15688  summo  15689  zsum  15690  isumsplit  15812  ntrivcvg  15869  ntrivcvgn0  15870  ntrivcvgtail  15872  ntrivcvgmullem  15873  prodmolem2  15905  prodmo  15906  zprod  15907  fprodntriv  15912  ege2le3  16060  gsumval2a  18638  leibpi  26867  dvradcnv2  43756  binomcxplemnotnn0  43765  stirlinglem12  45445