Theorem noseqind 28284

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 2787	. 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 2821	. . . . . . 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 2821	. . . . . . 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 2821	. . . . . . 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 8375	. . . . . . . . 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 2836	. . . . . . 7 ⊢ (𝜑 → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘∅) ∈ 𝐵)
15		oveq1 7374	. . . . . . . . . . . . 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 2821	. . . . . . . . . . . 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 3520	. . . . . . . . . 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 7400	. . . . . . . . . . 11 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) +s 1s ) ∈ V
23		eqid 2736	. . . . . . . . . . . 12 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)
24		oveq1 7374	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑥 → (𝑡 +s 1s ) = (𝑥 +s 1s ))
25		oveq1 7374	. . . . . . . . . . . 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 8379	. . . . . . . . . . 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 692	. . . . . . . . . 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 2821	. . . . . . . . 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 7849	. . . . . 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 3128	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑧) ∈ 𝐵)
34		frfnom 8374	. . . . 5 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω) Fn ω
35		ffnfv 7071	. . . . 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 710	. . . 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 6676	. 2 ⊢ (𝜑 → ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω) ⊆ 𝐵)
39	3, 38	eqsstrd 3956	1 ⊢ (𝜑 → 𝑍 ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  Vcvv 3429   ⊆ wss 3889  ∅c0 4273   ↦ cmpt 5166  ran crn 5632   ↾ cres 5633   “ cima 5634  suc csuc 6325   Fn wfn 6493  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  ωcom 7817  reccrdg 8348   No csur 27603   1_s c1s 27798   +_s cadds 27951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  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 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349

This theorem is referenced by:  noseqinds  28285  noseqssno  28286  peano5n0s  28311