Theorem seqseq123d 28260

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 7441	. . . . . . 7 ⊢ (𝜑 → (𝑦 + (𝐹‘(𝑥 +s 1s ))) = (𝑦𝑄(𝐹‘(𝑥 +s 1s ))))
3		seqseq123d.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = 𝐺)
4	3	fveq1d 6903	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘(𝑥 +s 1s )) = (𝐺‘(𝑥 +s 1s )))
5	4	oveq2d 7440	. . . . . . 7 ⊢ (𝜑 → (𝑦𝑄(𝐹‘(𝑥 +s 1s ))) = (𝑦𝑄(𝐺‘(𝑥 +s 1s ))))
6	2, 5	eqtrd 2766	. . . . . 6 ⊢ (𝜑 → (𝑦 + (𝐹‘(𝑥 +s 1s ))) = (𝑦𝑄(𝐺‘(𝑥 +s 1s ))))
7	6	opeq2d 4886	. . . . 5 ⊢ (𝜑 → ⟨(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))⟩ = ⟨(𝑥 +s 1s ), (𝑦𝑄(𝐺‘(𝑥 +s 1s )))⟩)
8	7	mpoeq3dv 7504	. . . 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 6908	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑀) = (𝐺‘𝑁))
11	9, 10	opeq12d 4887	. . . 4 ⊢ (𝜑 → ⟨𝑀, (𝐹‘𝑀)⟩ = ⟨𝑁, (𝐺‘𝑁)⟩)
12		rdgeq12 8443	. . . 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 582	. . 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 6068	. 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 28258	. 2 ⊢ seqs𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
16		df-seqs 28258	. 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 2791	1 ⊢ (𝜑 → seqs𝑀( + , 𝐹) = seqs𝑁(𝑄, 𝐺))

Syntax hints:   → wi 4   = wceq 1534  Vcvv 3462  ⟨cop 4639   “ cima 5685  ‘cfv 6554  (class class class)co 7424   ∈ cmpo 7426  ωcom 7876  reccrdg 8439   1_s c1s 27853   +_s cadds 27973  seq_scseqs 28257

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 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-mpt 5237  df-xp 5688  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-iota 6506  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-seqs 28258

This theorem is referenced by: (None)