Theorem noseqind 28162

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 5644	. . 3 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω) = ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)
3	1, 2	eqtrdi 2780	. 2 ⊢ (𝜑 → 𝑍 = ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω))
4		fveq2 6840	. . . . . . . 8 ⊢ (𝑧 = ∅ → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑧) = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘∅))
5	4	eleq1d 2813	. . . . . . 7 ⊢ (𝑧 = ∅ → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑧) ∈ 𝐵 ↔ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘∅) ∈ 𝐵))
6		fveq2 6840	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑧) = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤))
7	6	eleq1d 2813	. . . . . . 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 6840	. . . . . . . 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 2813	. . . . . . 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 8381	. . . . . . . . 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 2828	. . . . . . 7 ⊢ (𝜑 → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘∅) ∈ 𝐵)
15		oveq1 7376	. . . . . . . . . . . . 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 2813	. . . . . . . . . . . 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 3540	. . . . . . . . . 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 7402	. . . . . . . . . . 11 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) +s 1s ) ∈ V
23		eqid 2729	. . . . . . . . . . . 12 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)
24		oveq1 7376	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑥 → (𝑡 +s 1s ) = (𝑥 +s 1s ))
25		oveq1 7376	. . . . . . . . . . . 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 8385	. . . . . . . . . . 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 2813	. . . . . . . . 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 7854	. . . . . 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 3124	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑧) ∈ 𝐵)
34		frfnom 8380	. . . . 5 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω) Fn ω
35		ffnfv 7073	. . . . 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 6678	. 2 ⊢ (𝜑 → ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω) ⊆ 𝐵)
39	3, 38	eqsstrd 3978	1 ⊢ (𝜑 → 𝑍 ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3444   ⊆ wss 3911  ∅c0 4292   ↦ cmpt 5183  ran crn 5632   ↾ cres 5633   “ cima 5634  suc csuc 6322   Fn wfn 6494  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369  ωcom 7822  reccrdg 8354   No csur 27527   1_s c1s 27711   +_s cadds 27842

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 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-om 7823  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355

This theorem is referenced by:  noseqinds  28163  noseqssno  28164  peano5n0s  28188