Theorem om2noseqsuc 28196

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 7382	. . 3 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)‘𝐴) +s 1s ) ∈ V
3		eqid 2729	. . . 4 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)
4		oveq1 7356	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 +s 1s ) = (𝑥 +s 1s ))
5		oveq1 7356	. . . 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 8362	. . 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 6824	. 2 ⊢ (𝜑 → (𝐺‘suc 𝐴) = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)‘suc 𝐴))
10	8	fveq1d 6824	. . 3 ⊢ (𝜑 → (𝐺‘𝐴) = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)‘𝐴))
11	10	oveq1d 7364	. 2 ⊢ (𝜑 → ((𝐺‘𝐴) +s 1s ) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)‘𝐴) +s 1s ))
12	7, 9, 11	3eqtr4d 2774	1 ⊢ (𝜑 → (𝐺‘suc 𝐴) = ((𝐺‘𝐴) +s 1s ))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3436   ↦ cmpt 5173   ↾ cres 5621  suc csuc 6309  ‘cfv 6482  (class class class)co 7349  ωcom 7799  reccrdg 8331   No csur 27549   1_s c1s 27737   +_s cadds 27871

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 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-om 7800  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332

This theorem is referenced by:  om2noseqlt  28198  om2noseqrdg  28203  noseqrdgsuc  28207