MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  noseqrdglem Structured version   Visualization version   GIF version

Theorem noseqrdglem 28311
Description: A helper lemma for the value of a recursive defintion generator on surreal sequences. (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))
om2noseq.3 (𝜑𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))
noseqrdg.1 (𝜑𝐴𝑉)
noseqrdg.2 (𝜑𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω))
Assertion
Ref Expression
noseqrdglem ((𝜑𝐵𝑍) → ⟨𝐵, (2nd ‘(𝑅‘(𝐺𝐵)))⟩ ∈ ran 𝑅)
Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑦)   𝑅(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑍(𝑥,𝑦)

Proof of Theorem noseqrdglem
StepHypRef Expression
1 om2noseq.1 . . . . . 6 (𝜑𝐶 No )
2 om2noseq.2 . . . . . 6 (𝜑𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))
3 om2noseq.3 . . . . . 6 (𝜑𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))
41, 2, 3om2noseqf1o 28307 . . . . 5 (𝜑𝐺:ω–1-1-onto𝑍)
5 f1ocnvdm 7305 . . . . 5 ((𝐺:ω–1-1-onto𝑍𝐵𝑍) → (𝐺𝐵) ∈ ω)
64, 5sylan 580 . . . 4 ((𝜑𝐵𝑍) → (𝐺𝐵) ∈ ω)
7 noseqrdg.1 . . . . 5 (𝜑𝐴𝑉)
8 noseqrdg.2 . . . . 5 (𝜑𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω))
91, 2, 3, 7, 8om2noseqrdg 28310 . . . 4 ((𝜑 ∧ (𝐺𝐵) ∈ ω) → (𝑅‘(𝐺𝐵)) = ⟨(𝐺‘(𝐺𝐵)), (2nd ‘(𝑅‘(𝐺𝐵)))⟩)
106, 9syldan 591 . . 3 ((𝜑𝐵𝑍) → (𝑅‘(𝐺𝐵)) = ⟨(𝐺‘(𝐺𝐵)), (2nd ‘(𝑅‘(𝐺𝐵)))⟩)
11 f1ocnvfv2 7297 . . . . 5 ((𝐺:ω–1-1-onto𝑍𝐵𝑍) → (𝐺‘(𝐺𝐵)) = 𝐵)
124, 11sylan 580 . . . 4 ((𝜑𝐵𝑍) → (𝐺‘(𝐺𝐵)) = 𝐵)
1312opeq1d 4879 . . 3 ((𝜑𝐵𝑍) → ⟨(𝐺‘(𝐺𝐵)), (2nd ‘(𝑅‘(𝐺𝐵)))⟩ = ⟨𝐵, (2nd ‘(𝑅‘(𝐺𝐵)))⟩)
1410, 13eqtrd 2777 . 2 ((𝜑𝐵𝑍) → (𝑅‘(𝐺𝐵)) = ⟨𝐵, (2nd ‘(𝑅‘(𝐺𝐵)))⟩)
15 frfnom 8475 . . . . 5 (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω) Fn ω
168fneq1d 6661 . . . . 5 (𝜑 → (𝑅 Fn ω ↔ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω) Fn ω))
1715, 16mpbiri 258 . . . 4 (𝜑𝑅 Fn ω)
1817adantr 480 . . 3 ((𝜑𝐵𝑍) → 𝑅 Fn ω)
1918, 6fnfvelrnd 7102 . 2 ((𝜑𝐵𝑍) → (𝑅‘(𝐺𝐵)) ∈ ran 𝑅)
2014, 19eqeltrrd 2842 1 ((𝜑𝐵𝑍) → ⟨𝐵, (2nd ‘(𝑅‘(𝐺𝐵)))⟩ ∈ ran 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cop 4632  cmpt 5225  ccnv 5684  ran crn 5686  cres 5687  cima 5688   Fn wfn 6556  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  cmpo 7433  ωcom 7887  2nd c2nd 8013  reccrdg 8449   No csur 27684   1s c1s 27868   +s cadds 27992
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-nadd 8704  df-no 27687  df-slt 27688  df-bday 27689  df-sle 27790  df-sslt 27826  df-scut 27828  df-0s 27869  df-1s 27870  df-made 27886  df-old 27887  df-left 27889  df-right 27890  df-norec2 27982  df-adds 27993
This theorem is referenced by:  noseqrdgfn  28312  noseqrdgsuc  28314
  Copyright terms: Public domain W3C validator