Theorem seqseq123d 28437

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 7417	. . . . . . 7 ⊢ (𝜑 → (𝑦 + (𝐹‘(𝑥 +s 1s ))) = (𝑦𝑄(𝐹‘(𝑥 +s 1s ))))
3		seqseq123d.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = 𝐺)
4	3	fveq1d 6873	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘(𝑥 +s 1s )) = (𝐺‘(𝑥 +s 1s )))
5	4	oveq2d 7416	. . . . . . 7 ⊢ (𝜑 → (𝑦𝑄(𝐹‘(𝑥 +s 1s ))) = (𝑦𝑄(𝐺‘(𝑥 +s 1s ))))
6	2, 5	eqtrd 2800	. . . . . 6 ⊢ (𝜑 → (𝑦 + (𝐹‘(𝑥 +s 1s ))) = (𝑦𝑄(𝐺‘(𝑥 +s 1s ))))
7	6	opeq2d 4841	. . . . 5 ⊢ (𝜑 → ⟨(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))⟩ = ⟨(𝑥 +s 1s ), (𝑦𝑄(𝐺‘(𝑥 +s 1s )))⟩)
8	7	mpoeq3dv 7479	. . . 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 6878	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑀) = (𝐺‘𝑁))
11	9, 10	opeq12d 4842	. . . 4 ⊢ (𝜑 → ⟨𝑀, (𝐹‘𝑀)⟩ = ⟨𝑁, (𝐺‘𝑁)⟩)
12		rdgeq12 8388	. . . 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 595	. . 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 6052	. 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 28435	. 2 ⊢ seqs𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
16		df-seqs 28435	. 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 2825	1 ⊢ (𝜑 → seqs𝑀( + , 𝐹) = seqs𝑁(𝑄, 𝐺))

Syntax hints:   → wi 4   = wceq 1563  Vcvv 3457  ⟨cop 4591   “ cima 5655  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402  ωcom 7850  reccrdg 8384   1_s c1s 27957   +_s cadds 28110  seq_scseqs 28434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-xp 5658  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-iota 6481  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-seqs 28435

This theorem is referenced by:  expsval  28576