Theorem seqsp1 28466

Description: The value of the surreal sequence builder at a successor. (Contributed by Scott Fenton, 19-Apr-2025.)

Hypotheses

Ref	Expression
seqsp1.1	⊢ (𝜑 → 𝑀 ∈ No )
seqsp1.2	⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝑀) “ ω))
seqsp1.3	⊢ (𝜑 → 𝑁 ∈ 𝑍)

Assertion

Ref	Expression
seqsp1	⊢ (𝜑 → (seqs𝑀( + , 𝐹)‘(𝑁 +s 1s )) = ((seqs𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 +s 1s ))))

Proof of Theorem seqsp1

Dummy variables 𝑦 𝑧 𝑤 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		seqsp1.3	. . 3 ⊢ (𝜑 → 𝑁 ∈ 𝑍)
2		seqsp1.1	. . . 4 ⊢ (𝜑 → 𝑀 ∈ No )
3		eqidd 2770	. . . 4 ⊢ (𝜑 → (rec((𝑦 ∈ V ↦ (𝑦 +s 1s )), 𝑀) ↾ ω) = (rec((𝑦 ∈ V ↦ (𝑦 +s 1s )), 𝑀) ↾ ω))
4		seqsp1.2	. . . . 5 ⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝑀) “ ω))
5		oveq1 7415	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 +s 1s ) = (𝑦 +s 1s ))
6	5	cbvmptv 5216	. . . . . . 7 ⊢ (𝑥 ∈ V ↦ (𝑥 +s 1s )) = (𝑦 ∈ V ↦ (𝑦 +s 1s ))
7		rdgeq1 8394	. . . . . . 7 ⊢ ((𝑥 ∈ V ↦ (𝑥 +s 1s )) = (𝑦 ∈ V ↦ (𝑦 +s 1s )) → rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝑀) = rec((𝑦 ∈ V ↦ (𝑦 +s 1s )), 𝑀))
8	6, 7	ax-mp 5	. . . . . 6 ⊢ rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝑀) = rec((𝑦 ∈ V ↦ (𝑦 +s 1s )), 𝑀)
9	8	imaeq1i 6057	. . . . 5 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝑀) “ ω) = (rec((𝑦 ∈ V ↦ (𝑦 +s 1s )), 𝑀) “ ω)
10	4, 9	eqtrdi 2820	. . . 4 ⊢ (𝜑 → 𝑍 = (rec((𝑦 ∈ V ↦ (𝑦 +s 1s )), 𝑀) “ ω))
11		fvexd 6894	. . . 4 ⊢ (𝜑 → (𝐹‘𝑀) ∈ V)
12		eqidd 2770	. . . 4 ⊢ (𝜑 → (rec((𝑦 ∈ V, 𝑧 ∈ V ↦ ⟨(𝑦 +s 1s ), (𝑦(𝑤 ∈ V, 𝑡 ∈ V ↦ (𝑡 + (𝐹‘(𝑤 +s 1s ))))𝑧)⟩), ⟨𝑀, (𝐹‘𝑀)⟩) ↾ ω) = (rec((𝑦 ∈ V, 𝑧 ∈ V ↦ ⟨(𝑦 +s 1s ), (𝑦(𝑤 ∈ V, 𝑡 ∈ V ↦ (𝑡 + (𝐹‘(𝑤 +s 1s ))))𝑧)⟩), ⟨𝑀, (𝐹‘𝑀)⟩) ↾ ω))
13	12	seqsval 28443	. . . 4 ⊢ (𝜑 → seqs𝑀( + , 𝐹) = ran (rec((𝑦 ∈ V, 𝑧 ∈ V ↦ ⟨(𝑦 +s 1s ), (𝑦(𝑤 ∈ V, 𝑡 ∈ V ↦ (𝑡 + (𝐹‘(𝑤 +s 1s ))))𝑧)⟩), ⟨𝑀, (𝐹‘𝑀)⟩) ↾ ω))
14	2, 3, 10, 11, 12, 13	noseqrdgsuc 28463	. . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍) → (seqs𝑀( + , 𝐹)‘(𝑁 +s 1s )) = (𝑁(𝑤 ∈ V, 𝑡 ∈ V ↦ (𝑡 + (𝐹‘(𝑤 +s 1s ))))(seqs𝑀( + , 𝐹)‘𝑁)))
15	1, 14	mpdan 699	. 2 ⊢ (𝜑 → (seqs𝑀( + , 𝐹)‘(𝑁 +s 1s )) = (𝑁(𝑤 ∈ V, 𝑡 ∈ V ↦ (𝑡 + (𝐹‘(𝑤 +s 1s ))))(seqs𝑀( + , 𝐹)‘𝑁)))
16	1	elexd 3486	. . 3 ⊢ (𝜑 → 𝑁 ∈ V)
17		fvex 6892	. . 3 ⊢ (seqs𝑀( + , 𝐹)‘𝑁) ∈ V
18		fvoveq1 7431	. . . . 5 ⊢ (𝑤 = 𝑁 → (𝐹‘(𝑤 +s 1s )) = (𝐹‘(𝑁 +s 1s )))
19	18	oveq2d 7424	. . . 4 ⊢ (𝑤 = 𝑁 → (𝑡 + (𝐹‘(𝑤 +s 1s ))) = (𝑡 + (𝐹‘(𝑁 +s 1s ))))
20		oveq1 7415	. . . 4 ⊢ (𝑡 = (seqs𝑀( + , 𝐹)‘𝑁) → (𝑡 + (𝐹‘(𝑁 +s 1s ))) = ((seqs𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 +s 1s ))))
21		eqid 2769	. . . 4 ⊢ (𝑤 ∈ V, 𝑡 ∈ V ↦ (𝑡 + (𝐹‘(𝑤 +s 1s )))) = (𝑤 ∈ V, 𝑡 ∈ V ↦ (𝑡 + (𝐹‘(𝑤 +s 1s ))))
22		ovex 7441	. . . 4 ⊢ ((seqs𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 +s 1s ))) ∈ V
23	19, 20, 21, 22	ovmpo 7568	. . 3 ⊢ ((𝑁 ∈ V ∧ (seqs𝑀( + , 𝐹)‘𝑁) ∈ V) → (𝑁(𝑤 ∈ V, 𝑡 ∈ V ↦ (𝑡 + (𝐹‘(𝑤 +s 1s ))))(seqs𝑀( + , 𝐹)‘𝑁)) = ((seqs𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 +s 1s ))))
24	16, 17, 23	sylancl 597	. 2 ⊢ (𝜑 → (𝑁(𝑤 ∈ V, 𝑡 ∈ V ↦ (𝑡 + (𝐹‘(𝑤 +s 1s ))))(seqs𝑀( + , 𝐹)‘𝑁)) = ((seqs𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 +s 1s ))))
25	15, 24	eqtrd 2804	1 ⊢ (𝜑 → (seqs𝑀( + , 𝐹)‘(𝑁 +s 1s )) = ((seqs𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 +s 1s ))))

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  Vcvv 3463  ⟨cop 4597   ↦ cmpt 5193   ↾ cres 5661   “ cima 5662  ‘cfv 6534  (class class class)co 7408   ∈ cmpo 7410  ωcom 7858  reccrdg 8392   No csur 27766   1_s c1s 27961   +_s cadds 28114  seq_scseqs 28438

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-ot 4600  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6300  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-2o 8450  df-oadd 8453  df-nadd 8648  df-no 27769  df-lts 27770  df-bday 27771  df-les 27871  df-slts 27913  df-cuts 27915  df-0s 27962  df-1s 27963  df-made 27982  df-old 27983  df-left 27985  df-right 27986  df-norec2 28104  df-adds 28115  df-seqs 28439

This theorem is referenced by:  expsp1  28584