Theorem seqseq123d 28654

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 7429	. . . . . . 7 ⊢ (𝜑 → (𝑦 + (𝐹‘(𝑥 +s 1s ))) = (𝑦𝑄(𝐹‘(𝑥 +s 1s ))))
3		seqseq123d.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = 𝐺)
4	3	fveq1d 6879	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘(𝑥 +s 1s )) = (𝐺‘(𝑥 +s 1s )))
5	4	oveq2d 7428	. . . . . . 7 ⊢ (𝜑 → (𝑦𝑄(𝐹‘(𝑥 +s 1s ))) = (𝑦𝑄(𝐺‘(𝑥 +s 1s ))))
6	2, 5	eqtrd 2796	. . . . . 6 ⊢ (𝜑 → (𝑦 + (𝐹‘(𝑥 +s 1s ))) = (𝑦𝑄(𝐺‘(𝑥 +s 1s ))))
7	6	opeq2d 4840	. . . . 5 ⊢ (𝜑 → ⟨(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))⟩ = ⟨(𝑥 +s 1s ), (𝑦𝑄(𝐺‘(𝑥 +s 1s )))⟩)
8	7	mpoeq3dv 7491	. . . 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 6884	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑀) = (𝐺‘𝑁))
11	9, 10	opeq12d 4841	. . . 4 ⊢ (𝜑 → ⟨𝑀, (𝐹‘𝑀)⟩ = ⟨𝑁, (𝐺‘𝑁)⟩)
12		rdgeq12 8405	. . . 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 6053	. 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 28652	. 2 ⊢ seqs𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
16		df-seqs 28652	. 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 2821	1 ⊢ (𝜑 → seqs𝑀( + , 𝐹) = seqs𝑁(𝑄, 𝐺))

This proof depends on syntax axioms:   → wi 4   = wceq 1570  Vcvv 3451  ⟨cop 4590   “ cima 5654  ‘cfv 6531  (class class class)co 7412   ∈ cmpo 7414  ωcom 7866  reccrdg 8401   1_s c1s 28174   +_s cadds 28327  seq_scseqs 28651

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

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 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-xp 5657  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-iota 6487  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-seqs 28652

This theorem is used by:  expsval  28793