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Theorem seqseq123d 28307
Description: Equality deduction for the surreal sequence builder. (Contributed by Scott Fenton, 18-Apr-2025.)
Hypotheses
Ref Expression
seqseq123d.1 (𝜑𝑀 = 𝑁)
seqseq123d.2 (𝜑+ = 𝑄)
seqseq123d.3 (𝜑𝐹 = 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.
StepHypRef Expression
1 seqseq123d.2 . . . . . . . 8 (𝜑+ = 𝑄)
21oveqd 7448 . . . . . . 7 (𝜑 → (𝑦 + (𝐹‘(𝑥 +s 1s ))) = (𝑦𝑄(𝐹‘(𝑥 +s 1s ))))
3 seqseq123d.3 . . . . . . . . 9 (𝜑𝐹 = 𝐺)
43fveq1d 6909 . . . . . . . 8 (𝜑 → (𝐹‘(𝑥 +s 1s )) = (𝐺‘(𝑥 +s 1s )))
54oveq2d 7447 . . . . . . 7 (𝜑 → (𝑦𝑄(𝐹‘(𝑥 +s 1s ))) = (𝑦𝑄(𝐺‘(𝑥 +s 1s ))))
62, 5eqtrd 2775 . . . . . 6 (𝜑 → (𝑦 + (𝐹‘(𝑥 +s 1s ))) = (𝑦𝑄(𝐺‘(𝑥 +s 1s ))))
76opeq2d 4885 . . . . 5 (𝜑 → ⟨(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))⟩ = ⟨(𝑥 +s 1s ), (𝑦𝑄(𝐺‘(𝑥 +s 1s )))⟩)
87mpoeq3dv 7512 . . . 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 (𝜑𝑀 = 𝑁)
103, 9fveq12d 6914 . . . . 5 (𝜑 → (𝐹𝑀) = (𝐺𝑁))
119, 10opeq12d 4886 . . . 4 (𝜑 → ⟨𝑀, (𝐹𝑀)⟩ = ⟨𝑁, (𝐺𝑁)⟩)
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, (𝐹𝑀)⟩ = ⟨𝑁, (𝐺𝑁)⟩) → rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))⟩), ⟨𝑀, (𝐹𝑀)⟩) = rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑦𝑄(𝐺‘(𝑥 +s 1s )))⟩), ⟨𝑁, (𝐺𝑁)⟩))
138, 11, 12syl2anc 584 . . 3 (𝜑 → rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))⟩), ⟨𝑀, (𝐹𝑀)⟩) = rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑦𝑄(𝐺‘(𝑥 +s 1s )))⟩), ⟨𝑁, (𝐺𝑁)⟩))
1413imaeq1d 6079 . 2 (𝜑 → (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))⟩), ⟨𝑀, (𝐹𝑀)⟩) “ ω) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑦𝑄(𝐺‘(𝑥 +s 1s )))⟩), ⟨𝑁, (𝐺𝑁)⟩) “ ω))
15 df-seqs 28305 . 2 seqs𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))⟩), ⟨𝑀, (𝐹𝑀)⟩) “ ω)
16 df-seqs 28305 . 2 seqs𝑁(𝑄, 𝐺) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑦𝑄(𝐺‘(𝑥 +s 1s )))⟩), ⟨𝑁, (𝐺𝑁)⟩) “ ω)
1714, 15, 163eqtr4g 2800 1 (𝜑 → seqs𝑀( + , 𝐹) = seqs𝑁(𝑄, 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  Vcvv 3478  cop 4637  cima 5692  cfv 6563  (class class class)co 7431  cmpo 7433  ωcom 7887  reccrdg 8448   1s c1s 27883   +s cadds 28007  seqscseqs 28304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5695  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-iota 6516  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-seqs 28305
This theorem is referenced by:  expsval  28423
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