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Theorem om2noseqsuc 28318
Description: The value of 𝐺 at a successor. (Contributed by Scott Fenton, 18-Apr-2025.)
Hypotheses
Ref Expression
om2noseq.1 (𝜑𝐶 No )
om2noseq.2 (𝜑𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))
om2noseqsuc.3 (𝜑𝐴 ∈ ω)
Assertion
Ref Expression
om2noseqsuc (𝜑 → (𝐺‘suc 𝐴) = ((𝐺𝐴) +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.
StepHypRef Expression
1 om2noseqsuc.3 . . 3 (𝜑𝐴 ∈ ω)
2 ovex 7464 . . 3 (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)‘𝐴) +s 1s ) ∈ V
3 eqid 2735 . . . 4 (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)
4 oveq1 7438 . . . 4 (𝑦 = 𝑥 → (𝑦 +s 1s ) = (𝑥 +s 1s ))
5 oveq1 7438 . . . 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 ))
63, 4, 5frsucmpt2 8479 . . 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 ))
71, 2, 6sylancl 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 (𝜑𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))
98fveq1d 6909 . 2 (𝜑 → (𝐺‘suc 𝐴) = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)‘suc 𝐴))
108fveq1d 6909 . . 3 (𝜑 → (𝐺𝐴) = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)‘𝐴))
1110oveq1d 7446 . 2 (𝜑 → ((𝐺𝐴) +s 1s ) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)‘𝐴) +s 1s ))
127, 9, 113eqtr4d 2785 1 (𝜑 → (𝐺‘suc 𝐴) = ((𝐺𝐴) +s 1s ))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  Vcvv 3478  cmpt 5231  cres 5691  suc csuc 6388  cfv 6563  (class class class)co 7431  ωcom 7887  reccrdg 8448   No csur 27699   1s c1s 27883   +s cadds 28007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449
This theorem is referenced by:  om2noseqlt  28320  om2noseqrdg  28325  noseqrdgsuc  28329
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