Theorem om2noseqsuc 28227

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 7379	. . 3 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)‘𝐴) +s 1s ) ∈ V
3		eqid 2731	. . . 4 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)
4		oveq1 7353	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 +s 1s ) = (𝑥 +s 1s ))
5		oveq1 7353	. . . 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 8359	. . 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 7361	. 2 ⊢ (𝜑 → ((𝐺‘𝐴) +s 1s ) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)‘𝐴) +s 1s ))
12	7, 9, 11	3eqtr4d 2776	1 ⊢ (𝜑 → (𝐺‘suc 𝐴) = ((𝐺‘𝐴) +s 1s ))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ↦ cmpt 5170   ↾ cres 5616  suc csuc 6308  ‘cfv 6481  (class class class)co 7346  ωcom 7796  reccrdg 8328   No csur 27578   1_s c1s 27767   +_s cadds 27902

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329

This theorem is referenced by:  om2noseqlt  28229  om2noseqrdg  28234  noseqrdgsuc  28238