Theorem seqsp1 28236

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 2732	. . . 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 7348	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 +s 1s ) = (𝑦 +s 1s ))
6	5	cbvmptv 5190	. . . . . . 7 ⊢ (𝑥 ∈ V ↦ (𝑥 +s 1s )) = (𝑦 ∈ V ↦ (𝑦 +s 1s ))
7		rdgeq1 8325	. . . . . . 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 6001	. . . . 5 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝑀) “ ω) = (rec((𝑦 ∈ V ↦ (𝑦 +s 1s )), 𝑀) “ ω)
10	4, 9	eqtrdi 2782	. . . 4 ⊢ (𝜑 → 𝑍 = (rec((𝑦 ∈ V ↦ (𝑦 +s 1s )), 𝑀) “ ω))
11		fvexd 6832	. . . 4 ⊢ (𝜑 → (𝐹‘𝑀) ∈ V)
12		eqidd 2732	. . . 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 28213	. . . 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 28233	. . 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 687	. 2 ⊢ (𝜑 → (seqs𝑀( + , 𝐹)‘(𝑁 +s 1s )) = (𝑁(𝑤 ∈ V, 𝑡 ∈ V ↦ (𝑡 + (𝐹‘(𝑤 +s 1s ))))(seqs𝑀( + , 𝐹)‘𝑁)))
16	1	elexd 3460	. . 3 ⊢ (𝜑 → 𝑁 ∈ V)
17		fvex 6830	. . 3 ⊢ (seqs𝑀( + , 𝐹)‘𝑁) ∈ V
18		fvoveq1 7364	. . . . 5 ⊢ (𝑤 = 𝑁 → (𝐹‘(𝑤 +s 1s )) = (𝐹‘(𝑁 +s 1s )))
19	18	oveq2d 7357	. . . 4 ⊢ (𝑤 = 𝑁 → (𝑡 + (𝐹‘(𝑤 +s 1s ))) = (𝑡 + (𝐹‘(𝑁 +s 1s ))))
20		oveq1 7348	. . . 4 ⊢ (𝑡 = (seqs𝑀( + , 𝐹)‘𝑁) → (𝑡 + (𝐹‘(𝑁 +s 1s ))) = ((seqs𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 +s 1s ))))
21		eqid 2731	. . . 4 ⊢ (𝑤 ∈ V, 𝑡 ∈ V ↦ (𝑡 + (𝐹‘(𝑤 +s 1s )))) = (𝑤 ∈ V, 𝑡 ∈ V ↦ (𝑡 + (𝐹‘(𝑤 +s 1s ))))
22		ovex 7374	. . . 4 ⊢ ((seqs𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 +s 1s ))) ∈ V
23	19, 20, 21, 22	ovmpo 7501	. . 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 586	. 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 2766	1 ⊢ (𝜑 → (seqs𝑀( + , 𝐹)‘(𝑁 +s 1s )) = ((seqs𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 +s 1s ))))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  Vcvv 3436  ⟨cop 4577   ↦ cmpt 5167   ↾ cres 5613   “ cima 5614  ‘cfv 6476  (class class class)co 7341   ∈ cmpo 7343  ωcom 7791  reccrdg 8323   No csur 27573   1_s c1s 27762   +_s cadds 27897  seq_scseqs 28208

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-tp 4576  df-op 4578  df-ot 4580  df-uni 4855  df-int 4893  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-se 5565  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-2o 8381  df-oadd 8384  df-nadd 8576  df-no 27576  df-slt 27577  df-bday 27578  df-sle 27679  df-sslt 27716  df-scut 27718  df-0s 27763  df-1s 27764  df-made 27783  df-old 27784  df-left 27786  df-right 27787  df-norec2 27887  df-adds 27898  df-seqs 28209

This theorem is referenced by:  expsp1  28347