Theorem noseqp1 28252

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 2836	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω))
4		df-ima 5635	. . . 4 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω) = ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)
5	3, 4	eleqtrdi 2844	. . 3 ⊢ (𝜑 → 𝐵 ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω))
6		frfnom 8364	. . . 4 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω) Fn ω
7		fvelrnb 6892	. . . 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 7389	. . . . . . 7 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑦) +s 1s ) ∈ V
11		eqid 2734	. . . . . . . 8 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)
12		oveq1 7363	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑧 +s 1s ) = (𝑥 +s 1s ))
13		oveq1 7363	. . . . . . . 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 8369	. . . . . . 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 7830	. . . . . . . 8 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
18		fnfvelrn 7023	. . . . . . . 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 2785	. . . . . . 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 2837	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘suc 𝑦) ∈ 𝑍)
24	16, 23	eqeltrrd 2835	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑦) +s 1s ) ∈ 𝑍)
25		oveq1 7363	. . . . 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 2819	. . . 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 3141	1 ⊢ (𝜑 → (𝐵 +s 1s ) ∈ 𝑍)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3058  Vcvv 3438   ↦ cmpt 5177  ran crn 5623   ↾ cres 5624   “ cima 5625  suc csuc 6317   Fn wfn 6485  ‘cfv 6490  (class class class)co 7356  ωcom 7806  reccrdg 8338   No csur 27605   1_s c1s 27794   +_s cadds 27929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339

This theorem is referenced by:  noseqinds  28254  noseqrdgsuc  28269  peano2n0s  28291