Theorem noseqind 28193

Description: Peano's inductive postulate for surreal sequences. (Contributed by Scott Fenton, 18-Apr-2025.)

Hypotheses

Ref	Expression
noseq.1	⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω))
noseq.2	⊢ (𝜑 → 𝐴 ∈ No )
noseqind.3	⊢ (𝜑 → 𝐴 ∈ 𝐵)
noseqind.4	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦 +s 1s ) ∈ 𝐵)

Assertion

Ref	Expression
noseqind	⊢ (𝜑 → 𝑍 ⊆ 𝐵)

Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝜑,𝑦   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐵(𝑥)   𝑍(𝑥,𝑦)

Proof of Theorem noseqind

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

Step	Hyp	Ref	Expression
1		noseq.1	. . 3 ⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω))
2		df-ima 5654	. . 3 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω) = ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)
3	1, 2	eqtrdi 2781	. 2 ⊢ (𝜑 → 𝑍 = ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω))
4		fveq2 6861	. . . . . . . 8 ⊢ (𝑧 = ∅ → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑧) = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘∅))
5	4	eleq1d 2814	. . . . . . 7 ⊢ (𝑧 = ∅ → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑧) ∈ 𝐵 ↔ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘∅) ∈ 𝐵))
6		fveq2 6861	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑧) = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤))
7	6	eleq1d 2814	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑧) ∈ 𝐵 ↔ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) ∈ 𝐵))
8		fveq2 6861	. . . . . . . 8 ⊢ (𝑧 = suc 𝑤 → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑧) = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘suc 𝑤))
9	8	eleq1d 2814	. . . . . . 7 ⊢ (𝑧 = suc 𝑤 → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑧) ∈ 𝐵 ↔ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘suc 𝑤) ∈ 𝐵))
10		noseq.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ No )
11		fr0g 8407	. . . . . . . . 9 ⊢ (𝐴 ∈ No → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘∅) = 𝐴)
12	10, 11	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘∅) = 𝐴)
13		noseqind.3	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
14	12, 13	eqeltrd 2829	. . . . . . 7 ⊢ (𝜑 → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘∅) ∈ 𝐵)
15		oveq1 7397	. . . . . . . . . . . . 13 ⊢ (𝑦 = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) → (𝑦 +s 1s ) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) +s 1s ))
16	15	eleq1d 2814	. . . . . . . . . . . 12 ⊢ (𝑦 = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) → ((𝑦 +s 1s ) ∈ 𝐵 ↔ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) +s 1s ) ∈ 𝐵))
17	16	imbi2d 340	. . . . . . . . . . 11 ⊢ (𝑦 = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) → ((𝜑 → (𝑦 +s 1s ) ∈ 𝐵) ↔ (𝜑 → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) +s 1s ) ∈ 𝐵)))
18		noseqind.4	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦 +s 1s ) ∈ 𝐵)
19	18	expcom 413	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐵 → (𝜑 → (𝑦 +s 1s ) ∈ 𝐵))
20	17, 19	vtoclga 3546	. . . . . . . . . 10 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) ∈ 𝐵 → (𝜑 → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) +s 1s ) ∈ 𝐵))
21	20	impcom 407	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) ∈ 𝐵) → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) +s 1s ) ∈ 𝐵)
22		ovex 7423	. . . . . . . . . . 11 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) +s 1s ) ∈ V
23		eqid 2730	. . . . . . . . . . . 12 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)
24		oveq1 7397	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑥 → (𝑡 +s 1s ) = (𝑥 +s 1s ))
25		oveq1 7397	. . . . . . . . . . . 12 ⊢ (𝑡 = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) → (𝑡 +s 1s ) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) +s 1s ))
26	23, 24, 25	frsucmpt2 8411	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ω ∧ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) +s 1s ) ∈ V) → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘suc 𝑤) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) +s 1s ))
27	22, 26	mpan2 691	. . . . . . . . . 10 ⊢ (𝑤 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘suc 𝑤) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) +s 1s ))
28	27	eleq1d 2814	. . . . . . . . 9 ⊢ (𝑤 ∈ ω → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘suc 𝑤) ∈ 𝐵 ↔ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) +s 1s ) ∈ 𝐵))
29	21, 28	imbitrrid 246	. . . . . . . 8 ⊢ (𝑤 ∈ ω → ((𝜑 ∧ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) ∈ 𝐵) → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘suc 𝑤) ∈ 𝐵))
30	29	expd 415	. . . . . . 7 ⊢ (𝑤 ∈ ω → (𝜑 → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) ∈ 𝐵 → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘suc 𝑤) ∈ 𝐵)))
31	5, 7, 9, 14, 30	finds2 7877	. . . . . 6 ⊢ (𝑧 ∈ ω → (𝜑 → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑧) ∈ 𝐵))
32	31	com12 32	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑧) ∈ 𝐵))
33	32	ralrimiv 3125	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑧) ∈ 𝐵)
34		frfnom 8406	. . . . 5 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω) Fn ω
35		ffnfv 7094	. . . . 5 ⊢ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω):ω⟶𝐵 ↔ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω) Fn ω ∧ ∀𝑧 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑧) ∈ 𝐵))
36	34, 35	mpbiran 709	. . . 4 ⊢ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω):ω⟶𝐵 ↔ ∀𝑧 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑧) ∈ 𝐵)
37	33, 36	sylibr 234	. . 3 ⊢ (𝜑 → (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω):ω⟶𝐵)
38	37	frnd 6699	. 2 ⊢ (𝜑 → ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω) ⊆ 𝐵)
39	3, 38	eqsstrd 3984	1 ⊢ (𝜑 → 𝑍 ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  Vcvv 3450   ⊆ wss 3917  ∅c0 4299   ↦ cmpt 5191  ran crn 5642   ↾ cres 5643   “ cima 5644  suc csuc 6337   Fn wfn 6509  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  ωcom 7845  reccrdg 8380   No csur 27558   1_s c1s 27742   +_s cadds 27873

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381

This theorem is referenced by:  noseqinds  28194  noseqssno  28195  peano5n0s  28219