Theorem om2noseqsuc 28258

Description: The value of 𝐺 at a successor. (Contributed by Scott Fenton, 18-Apr-2025.)

Hypotheses

Ref	Expression
om2noseq.1	⊢ (𝜑 → 𝐶 ∈ No )
om2noseq.2	⊢ (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))
om2noseqsuc.3	⊢ (𝜑 → 𝐴 ∈ ω)

Assertion

Ref	Expression
om2noseqsuc	⊢ (𝜑 → (𝐺‘suc 𝐴) = ((𝐺‘𝐴) +s 1s ))

Distinct variable group:   𝑥,𝐶

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

Proof of Theorem om2noseqsuc

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		om2noseqsuc.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ ω)
2		ovex 7389	. . 3 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)‘𝐴) +s 1s ) ∈ V
3		eqid 2734	. . . 4 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)
4		oveq1 7363	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 +s 1s ) = (𝑥 +s 1s ))
5		oveq1 7363	. . . 4 ⊢ (𝑦 = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)‘𝐴) → (𝑦 +s 1s ) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)‘𝐴) +s 1s ))
6	3, 4, 5	frsucmpt2 8369	. . 3 ⊢ ((𝐴 ∈ ω ∧ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)‘𝐴) +s 1s ) ∈ V) → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)‘suc 𝐴) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)‘𝐴) +s 1s ))
7	1, 2, 6	sylancl 586	. 2 ⊢ (𝜑 → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)‘suc 𝐴) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)‘𝐴) +s 1s ))
8		om2noseq.2	. . 3 ⊢ (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))
9	8	fveq1d 6834	. 2 ⊢ (𝜑 → (𝐺‘suc 𝐴) = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)‘suc 𝐴))
10	8	fveq1d 6834	. . 3 ⊢ (𝜑 → (𝐺‘𝐴) = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)‘𝐴))
11	10	oveq1d 7371	. 2 ⊢ (𝜑 → ((𝐺‘𝐴) +s 1s ) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)‘𝐴) +s 1s ))
12	7, 9, 11	3eqtr4d 2779	1 ⊢ (𝜑 → (𝐺‘suc 𝐴) = ((𝐺‘𝐴) +s 1s ))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  Vcvv 3438   ↦ cmpt 5177   ↾ cres 5624  suc csuc 6317  ‘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:  om2noseqlt  28260  om2noseqrdg  28265  noseqrdgsuc  28269