Theorem noseqind 28382

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 5660	. . 3 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω) = ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)
3	1, 2	eqtrdi 2813	. 2 ⊢ (𝜑 → 𝑍 = ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω))
4		fveq2 6867	. . . . . . . 8 ⊢ (𝑧 = ∅ → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑧) = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘∅))
5	4	eleq1d 2847	. . . . . . 7 ⊢ (𝑧 = ∅ → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑧) ∈ 𝐵 ↔ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘∅) ∈ 𝐵))
6		fveq2 6867	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑧) = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤))
7	6	eleq1d 2847	. . . . . . 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 6867	. . . . . . . 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 2847	. . . . . . 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 2862	. . . . . . 7 ⊢ (𝜑 → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘∅) ∈ 𝐵)
15		oveq1 7403	. . . . . . . . . . . . 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 2847	. . . . . . . . . . . 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 342	. . . . . . . . . . 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 417	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐵 → (𝜑 → (𝑦 +s 1s ) ∈ 𝐵))
20	17, 19	vtoclga 3541	. . . . . . . . . 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 411	. . . . . . . . 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 7429	. . . . . . . . . . 11 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) +s 1s ) ∈ V
23		eqid 2762	. . . . . . . . . . . 12 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)
24		oveq1 7403	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑥 → (𝑡 +s 1s ) = (𝑥 +s 1s ))
25		oveq1 7403	. . . . . . . . . . . 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 701	. . . . . . . . . 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 2847	. . . . . . . . 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 248	. . . . . . . 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 419	. . . . . . 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 7879	. . . . . 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 3153	. . . 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 7100	. . . . 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 719	. . . 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 236	. . 3 ⊢ (𝜑 → (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω):ω⟶𝐵)
38	37	frnd 6700	. 2 ⊢ (𝜑 → ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω) ⊆ 𝐵)
39	3, 38	eqsstrd 3970	1 ⊢ (𝜑 → 𝑍 ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∀wral 3076  Vcvv 3454   ⊆ wss 3904  ∅c0 4285   ↦ cmpt 5181  ran crn 5648   ↾ cres 5649   “ cima 5650  suc csuc 6348   Fn wfn 6516  ⟶wf 6517  ‘cfv 6521  (class class class)co 7396  ωcom 7846  reccrdg 8380   No csur 27701   1_s c1s 27896   +_s cadds 28049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-om 7847  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381

This theorem is referenced by:  noseqinds  28383  noseqssno  28384  peano5n0s  28409