Theorem seqsp1 28706

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 2761	. . . 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 7431	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 +s 1s ) = (𝑦 +s 1s ))
6	5	cbvmptv 5208	. . . . . . 7 ⊢ (𝑥 ∈ V ↦ (𝑥 +s 1s )) = (𝑦 ∈ V ↦ (𝑦 +s 1s ))
7		rdgeq1 8426	. . . . . . 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 6051	. . . . 5 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝑀) “ ω) = (rec((𝑦 ∈ V ↦ (𝑦 +s 1s )), 𝑀) “ ω)
10	4, 9	eqtrdi 2811	. . . 4 ⊢ (𝜑 → 𝑍 = (rec((𝑦 ∈ V ↦ (𝑦 +s 1s )), 𝑀) “ ω))
11		fvexd 6905	. . . 4 ⊢ (𝜑 → (𝐹‘𝑀) ∈ V)
12		eqidd 2761	. . . 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 28683	. . . 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 28703	. . 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 700	. 2 ⊢ (𝜑 → (seqs𝑀( + , 𝐹)‘(𝑁 +s 1s )) = (𝑁(𝑤 ∈ V, 𝑡 ∈ V ↦ (𝑡 + (𝐹‘(𝑤 +s 1s ))))(seqs𝑀( + , 𝐹)‘𝑁)))
16	1	elexd 3473	. . 3 ⊢ (𝜑 → 𝑁 ∈ V)
17		fvex 6903	. . 3 ⊢ (seqs𝑀( + , 𝐹)‘𝑁) ∈ V
18		fvoveq1 7447	. . . . 5 ⊢ (𝑤 = 𝑁 → (𝐹‘(𝑤 +s 1s )) = (𝐹‘(𝑁 +s 1s )))
19	18	oveq2d 7440	. . . 4 ⊢ (𝑤 = 𝑁 → (𝑡 + (𝐹‘(𝑤 +s 1s ))) = (𝑡 + (𝐹‘(𝑁 +s 1s ))))
20		oveq1 7431	. . . 4 ⊢ (𝑡 = (seqs𝑀( + , 𝐹)‘𝑁) → (𝑡 + (𝐹‘(𝑁 +s 1s ))) = ((seqs𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 +s 1s ))))
21		eqid 2760	. . . 4 ⊢ (𝑤 ∈ V, 𝑡 ∈ V ↦ (𝑡 + (𝐹‘(𝑤 +s 1s )))) = (𝑤 ∈ V, 𝑡 ∈ V ↦ (𝑡 + (𝐹‘(𝑤 +s 1s ))))
22		ovex 7457	. . . 4 ⊢ ((seqs𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 +s 1s ))) ∈ V
23	19, 20, 21, 22	ovmpo 7584	. . 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 598	. 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 2795	1 ⊢ (𝜑 → (seqs𝑀( + , 𝐹)‘(𝑁 +s 1s )) = ((seqs𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 +s 1s ))))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  Vcvv 3450  ⟨cop 4834   ↦ cmpt 5185   ↾ cres 5653   “ cima 5654  ‘cfv 6543  (class class class)co 7424   ∈ cmpo 7426  ωcom 7881  reccrdg 8424  Nocsur 28006   1_s c1s 28201   +_s cadds 28354  seq_scseqs 28678

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7882  df-1st 8005  df-2nd 8006  df-frecs 8306  df-wrecs 8337  df-recs 8386  df-rdg 8425  df-1o 8483  df-2o 8484  df-oadd 8487  df-nadd 8682  df-no 28009  df-lts 28010  df-bday 28011  df-les 28111  df-slts 28153  df-cuts 28155  df-0s 28202  df-1s 28203  df-made 28222  df-old 28223  df-left 28225  df-right 28226  df-norec2 28344  df-adds 28355  df-seqs 28679

This theorem is used by:  expsp1  28824