Theorem seqsval 28294

Description: The value of the surreal sequence builder. (Contributed by Scott Fenton, 18-Apr-2025.)

Hypothesis

Ref	Expression
seqsval.1	⊢ (𝜑 → 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 +s 1s ))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩) ↾ ω))

Assertion

Ref	Expression
seqsval	⊢ (𝜑 → seqs𝑀( + , 𝐹) = ran 𝑅)

Distinct variable groups:   𝑤,𝐹,𝑥,𝑦,𝑧   𝑤, + ,𝑥,𝑦,𝑧   𝑥,𝑀,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝑅(𝑥,𝑦,𝑧,𝑤)   𝑀(𝑧,𝑤)

Proof of Theorem seqsval

Step	Hyp	Ref	Expression
1		df-seqs 28290	. . 3 ⊢ seqs𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
2		eqid 2737	. . . . . 6 ⊢ V = V
3		fvoveq1 7454	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝐹‘(𝑧 +s 1s )) = (𝐹‘(𝑥 +s 1s )))
4	3	oveq2d 7447	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑤 + (𝐹‘(𝑧 +s 1s ))) = (𝑤 + (𝐹‘(𝑥 +s 1s ))))
5		oveq1 7438	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑤 + (𝐹‘(𝑥 +s 1s ))) = (𝑦 + (𝐹‘(𝑥 +s 1s ))))
6		eqid 2737	. . . . . . . . 9 ⊢ (𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 +s 1s )))) = (𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 +s 1s ))))
7		ovex 7464	. . . . . . . . 9 ⊢ (𝑦 + (𝐹‘(𝑥 +s 1s ))) ∈ V
8	4, 5, 6, 7	ovmpo 7593	. . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 +s 1s ))))𝑦) = (𝑦 + (𝐹‘(𝑥 +s 1s ))))
9	8	el2v 3487	. . . . . . 7 ⊢ (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 +s 1s ))))𝑦) = (𝑦 + (𝐹‘(𝑥 +s 1s )))
10	9	opeq2i 4877	. . . . . 6 ⊢ ⟨(𝑥 +s 1s ), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 +s 1s ))))𝑦)⟩ = ⟨(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))⟩
11	2, 2, 10	mpoeq123i 7509	. . . . 5 ⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 +s 1s ))))𝑦)⟩) = (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))⟩)
12		rdgeq1 8451	. . . . 5 ⊢ ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 +s 1s ))))𝑦)⟩) = (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))⟩) → rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 +s 1s ))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩) = rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))⟩), ⟨𝑀, (𝐹‘𝑀)⟩))
13	11, 12	ax-mp 5	. . . 4 ⊢ rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 +s 1s ))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩) = rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
14	13	imaeq1i 6075	. . 3 ⊢ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 +s 1s ))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
15		df-ima 5698	. . 3 ⊢ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 +s 1s ))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω) = ran (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 +s 1s ))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩) ↾ ω)
16	1, 14, 15	3eqtr2i 2771	. 2 ⊢ seqs𝑀( + , 𝐹) = ran (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 +s 1s ))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩) ↾ ω)
17		seqsval.1	. . 3 ⊢ (𝜑 → 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 +s 1s ))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩) ↾ ω))
18	17	rneqd 5949	. 2 ⊢ (𝜑 → ran 𝑅 = ran (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 +s 1s ))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩) ↾ ω))
19	16, 18	eqtr4id 2796	1 ⊢ (𝜑 → seqs𝑀( + , 𝐹) = ran 𝑅)

Syntax hints:   → wi 4   = wceq 1540  Vcvv 3480  ⟨cop 4632  ran crn 5686   ↾ cres 5687   “ cima 5688  ‘cfv 6561  (class class class)co 7431   ∈ cmpo 7433  ωcom 7887  reccrdg 8449   1_s c1s 27868   +_s cadds 27992  seq_scseqs 28289

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-seqs 28290

This theorem is referenced by:  seqsfn  28315  seqs1  28316  seqsp1  28317