Theorem noseqp1 28242

Description: One plus an element of 𝑍 is an element of 𝑍. (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 )
noseqp1.3	⊢ (𝜑 → 𝐵 ∈ 𝑍)

Assertion

Ref	Expression
noseqp1	⊢ (𝜑 → (𝐵 +s 1s ) ∈ 𝑍)

Distinct variable group:   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝑍(𝑥)

Proof of Theorem noseqp1

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		noseqp1.3	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑍)
2		noseq.1	. . . . 5 ⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω))
3	1, 2	eleqtrd 2837	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω))
4		df-ima 5672	. . . 4 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω) = ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)
5	3, 4	eleqtrdi 2845	. . 3 ⊢ (𝜑 → 𝐵 ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω))
6		frfnom 8454	. . . 4 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω) Fn ω
7		fvelrnb 6944	. . . 4 ⊢ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω) Fn ω → (𝐵 ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω) ↔ ∃𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑦) = 𝐵))
8	6, 7	ax-mp 5	. . 3 ⊢ (𝐵 ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω) ↔ ∃𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑦) = 𝐵)
9	5, 8	sylib 218	. 2 ⊢ (𝜑 → ∃𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑦) = 𝐵)
10		ovex 7443	. . . . . . 7 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑦) +s 1s ) ∈ V
11		eqid 2736	. . . . . . . 8 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)
12		oveq1 7417	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑧 +s 1s ) = (𝑥 +s 1s ))
13		oveq1 7417	. . . . . . . 8 ⊢ (𝑧 = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑦) → (𝑧 +s 1s ) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑦) +s 1s ))
14	11, 12, 13	frsucmpt2 8459	. . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑦) +s 1s ) ∈ V) → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘suc 𝑦) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑦) +s 1s ))
15	10, 14	mpan2 691	. . . . . 6 ⊢ (𝑦 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘suc 𝑦) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑦) +s 1s ))
16	15	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘suc 𝑦) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑦) +s 1s ))
17		peano2 7891	. . . . . . . 8 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
18		fnfvelrn 7075	. . . . . . . 8 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω) Fn ω ∧ suc 𝑦 ∈ ω) → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘suc 𝑦) ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω))
19	6, 17, 18	sylancr 587	. . . . . . 7 ⊢ (𝑦 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘suc 𝑦) ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω))
20	19	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘suc 𝑦) ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω))
21	2, 4	eqtrdi 2787	. . . . . . 7 ⊢ (𝜑 → 𝑍 = ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω))
22	21	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → 𝑍 = ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω))
23	20, 22	eleqtrrd 2838	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘suc 𝑦) ∈ 𝑍)
24	16, 23	eqeltrrd 2836	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑦) +s 1s ) ∈ 𝑍)
25		oveq1 7417	. . . . 5 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑦) = 𝐵 → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑦) +s 1s ) = (𝐵 +s 1s ))
26	25	eleq1d 2820	. . . 4 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑦) = 𝐵 → ((((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑦) +s 1s ) ∈ 𝑍 ↔ (𝐵 +s 1s ) ∈ 𝑍))
27	24, 26	syl5ibcom 245	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑦) = 𝐵 → (𝐵 +s 1s ) ∈ 𝑍))
28	27	impr 454	. 2 ⊢ ((𝜑 ∧ (𝑦 ∈ ω ∧ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑦) = 𝐵)) → (𝐵 +s 1s ) ∈ 𝑍)
29	9, 28	rexlimddv 3148	1 ⊢ (𝜑 → (𝐵 +s 1s ) ∈ 𝑍)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3061  Vcvv 3464   ↦ cmpt 5206  ran crn 5660   ↾ cres 5661   “ cima 5662  suc csuc 6359   Fn wfn 6531  ‘cfv 6536  (class class class)co 7410  ωcom 7866  reccrdg 8428   No csur 27608   1_s c1s 27792   +_s cadds 27923

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 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-om 7867  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429

This theorem is referenced by:  noseqinds  28244  noseqrdgsuc  28259  peano2n0s  28280