Theorem seqseq123d 28293

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 7449	. . . . . . 7 ⊢ (𝜑 → (𝑦 + (𝐹‘(𝑥 +s 1s ))) = (𝑦𝑄(𝐹‘(𝑥 +s 1s ))))
3		seqseq123d.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = 𝐺)
4	3	fveq1d 6907	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘(𝑥 +s 1s )) = (𝐺‘(𝑥 +s 1s )))
5	4	oveq2d 7448	. . . . . . 7 ⊢ (𝜑 → (𝑦𝑄(𝐹‘(𝑥 +s 1s ))) = (𝑦𝑄(𝐺‘(𝑥 +s 1s ))))
6	2, 5	eqtrd 2776	. . . . . 6 ⊢ (𝜑 → (𝑦 + (𝐹‘(𝑥 +s 1s ))) = (𝑦𝑄(𝐺‘(𝑥 +s 1s ))))
7	6	opeq2d 4879	. . . . 5 ⊢ (𝜑 → ⟨(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))⟩ = ⟨(𝑥 +s 1s ), (𝑦𝑄(𝐺‘(𝑥 +s 1s )))⟩)
8	7	mpoeq3dv 7513	. . . 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 6912	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑀) = (𝐺‘𝑁))
11	9, 10	opeq12d 4880	. . . 4 ⊢ (𝜑 → ⟨𝑀, (𝐹‘𝑀)⟩ = ⟨𝑁, (𝐺‘𝑁)⟩)
12		rdgeq12 8454	. . . 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 584	. . 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 6076	. 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 28291	. 2 ⊢ seqs𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
16		df-seqs 28291	. 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 2801	1 ⊢ (𝜑 → seqs𝑀( + , 𝐹) = seqs𝑁(𝑄, 𝐺))

Syntax hints:   → wi 4   = wceq 1539  Vcvv 3479  ⟨cop 4631   “ cima 5687  ‘cfv 6560  (class class class)co 7432   ∈ cmpo 7434  ωcom 7888  reccrdg 8450   1_s c1s 27869   +_s cadds 27993  seq_scseqs 28290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-xp 5690  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-iota 6513  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-seqs 28291

This theorem is referenced by:  expsval  28409