Theorem om2noseqsuc 28448

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 7433	. . 3 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)‘𝐴) +s 1s ) ∈ V
3		eqid 2765	. . . 4 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)
4		oveq1 7407	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 +s 1s ) = (𝑥 +s 1s ))
5		oveq1 7407	. . . 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 8415	. . 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 597	. 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 6873	. 2 ⊢ (𝜑 → (𝐺‘suc 𝐴) = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)‘suc 𝐴))
10	8	fveq1d 6873	. . 3 ⊢ (𝜑 → (𝐺‘𝐴) = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)‘𝐴))
11	10	oveq1d 7415	. 2 ⊢ (𝜑 → ((𝐺‘𝐴) +s 1s ) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)‘𝐴) +s 1s ))
12	7, 9, 11	3eqtr4d 2810	1 ⊢ (𝜑 → (𝐺‘suc 𝐴) = ((𝐺‘𝐴) +s 1s ))

Syntax hints:   → wi 4   = wceq 1563   ∈ wcel 2145  Vcvv 3457   ↦ cmpt 5186   ↾ cres 5654  suc csuc 6352  ‘cfv 6525  (class class class)co 7400  ωcom 7850  reccrdg 8384   No csur 27762   1_s c1s 27957   +_s cadds 28110

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-om 7851  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385

This theorem is referenced by:  om2noseqlt  28450  om2noseqrdg  28455  noseqrdgsuc  28459