Theorem seqseq123d 28705

Description: Equality deduction for the surreal sequence builder. (Contributed by Scott Fenton, 18-Apr-2025.)

Hypotheses

Ref	Expression
seqseq123d.1	⊢ (𝜑 → 𝑀 = 𝑁)
seqseq123d.2	⊢ (𝜑 → + = 𝑄)
seqseq123d.3	⊢ (𝜑 → 𝐹 = 𝐺)

Assertion

Ref	Expression
seqseq123d	⊢ (𝜑 → seqs𝑀( + , 𝐹) = seqs𝑁(𝑄, 𝐺))

Proof of Theorem seqseq123d

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		seqseq123d.2	. . . . . . . 8 ⊢ (𝜑 → + = 𝑄)
2	1	oveqd 7463	. . . . . . 7 ⊢ (𝜑 → (𝑦 + (𝐹‘(𝑥 +s 1s ))) = (𝑦𝑄(𝐹‘(𝑥 +s 1s ))))
3		seqseq123d.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = 𝐺)
4	3	fveq1d 6913	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘(𝑥 +s 1s )) = (𝐺‘(𝑥 +s 1s )))
5	4	oveq2d 7462	. . . . . . 7 ⊢ (𝜑 → (𝑦𝑄(𝐹‘(𝑥 +s 1s ))) = (𝑦𝑄(𝐺‘(𝑥 +s 1s ))))
6	2, 5	eqtrd 2798	. . . . . 6 ⊢ (𝜑 → (𝑦 + (𝐹‘(𝑥 +s 1s ))) = (𝑦𝑄(𝐺‘(𝑥 +s 1s ))))
7	6	opeq2d 4850	. . . . 5 ⊢ (𝜑 → ⟨(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))⟩ = ⟨(𝑥 +s 1s ), (𝑦𝑄(𝐺‘(𝑥 +s 1s )))⟩)
8	7	mpoeq3dv 7525	. . . 4 ⊢ (𝜑 → (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))⟩) = (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑦𝑄(𝐺‘(𝑥 +s 1s )))⟩))
9		seqseq123d.1	. . . . 5 ⊢ (𝜑 → 𝑀 = 𝑁)
10	3, 9	fveq12d 6918	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑀) = (𝐺‘𝑁))
11	9, 10	opeq12d 4851	. . . 4 ⊢ (𝜑 → ⟨𝑀, (𝐹‘𝑀)⟩ = ⟨𝑁, (𝐺‘𝑁)⟩)
12		rdgeq12 8452	. . . 4 ⊢ (((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))⟩) = (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑦𝑄(𝐺‘(𝑥 +s 1s )))⟩) ∧ ⟨𝑀, (𝐹‘𝑀)⟩ = ⟨𝑁, (𝐺‘𝑁)⟩) → rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) = rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑦𝑄(𝐺‘(𝑥 +s 1s )))⟩), ⟨𝑁, (𝐺‘𝑁)⟩))
13	8, 11, 12	syl2anc 596	. . 3 ⊢ (𝜑 → rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) = rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑦𝑄(𝐺‘(𝑥 +s 1s )))⟩), ⟨𝑁, (𝐺‘𝑁)⟩))
14	13	imaeq1d 6064	. 2 ⊢ (𝜑 → (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑦𝑄(𝐺‘(𝑥 +s 1s )))⟩), ⟨𝑁, (𝐺‘𝑁)⟩) “ ω))
15		df-seqs 28703	. 2 ⊢ seqs𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
16		df-seqs 28703	. 2 ⊢ seqs𝑁(𝑄, 𝐺) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑦𝑄(𝐺‘(𝑥 +s 1s )))⟩), ⟨𝑁, (𝐺‘𝑁)⟩) “ ω)
17	14, 15, 16	3eqtr4g 2823	1 ⊢ (𝜑 → seqs𝑀( + , 𝐹) = seqs𝑁(𝑄, 𝐺))

This proof depends on syntax axioms:   → wi 4   = wceq 1570  Vcvv 3454  ⟨cop 4838   “ cima 5659  ‘cfv 6561  (class class class)co 7446   ∈ cmpo 7448  ωcom 7904  reccrdg 8448   1_s c1s 28225   +_s cadds 28378  seq_scseqs 28702

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 4839  df-uni 4871  df-br 5107  df-opab 5172  df-mpt 5191  df-xp 5662  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-iota 6515  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-seqs 28703

This theorem is used by:  expsval  28844