Theorem seqsval 28312

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 28308	. . 3 ⊢ seqs𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
2		eqid 2740	. . . . . 6 ⊢ V = V
3		fvoveq1 7471	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝐹‘(𝑧 +s 1s )) = (𝐹‘(𝑥 +s 1s )))
4	3	oveq2d 7464	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑤 + (𝐹‘(𝑧 +s 1s ))) = (𝑤 + (𝐹‘(𝑥 +s 1s ))))
5		oveq1 7455	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑤 + (𝐹‘(𝑥 +s 1s ))) = (𝑦 + (𝐹‘(𝑥 +s 1s ))))
6		eqid 2740	. . . . . . . . 9 ⊢ (𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 +s 1s )))) = (𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 +s 1s ))))
7		ovex 7481	. . . . . . . . 9 ⊢ (𝑦 + (𝐹‘(𝑥 +s 1s ))) ∈ V
8	4, 5, 6, 7	ovmpo 7610	. . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 +s 1s ))))𝑦) = (𝑦 + (𝐹‘(𝑥 +s 1s ))))
9	8	el2v 3495	. . . . . . 7 ⊢ (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 +s 1s ))))𝑦) = (𝑦 + (𝐹‘(𝑥 +s 1s )))
10	9	opeq2i 4901	. . . . . 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 7526	. . . . 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 8467	. . . . 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 6086	. . 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 5713	. . 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 2774	. 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 5963	. 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 2799	1 ⊢ (𝜑 → seqs𝑀( + , 𝐹) = ran 𝑅)

Syntax hints:   → wi 4   = wceq 1537  Vcvv 3488  ⟨cop 4654  ran crn 5701   ↾ cres 5702   “ cima 5703  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  ωcom 7903  reccrdg 8465   1_s c1s 27886   +_s cadds 28010  seq_scseqs 28307

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-seqs 28308

This theorem is referenced by:  seqsfn  28333  seqs1  28334  seqsp1  28335