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Theorem noseqind 28298
Description: Peano's inductive postulate for surreal sequences. (Contributed by Scott Fenton, 18-Apr-2025.)
Hypotheses
Ref Expression
noseq.1 (𝜑𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω))
noseq.2 (𝜑𝐴 No )
noseqind.3 (𝜑𝐴𝐵)
noseqind.4 ((𝜑𝑦𝐵) → (𝑦 +s 1s ) ∈ 𝐵)
Assertion
Ref Expression
noseqind (𝜑𝑍𝐵)
Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝜑,𝑦   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐵(𝑥)   𝑍(𝑥,𝑦)

Proof of Theorem noseqind
Dummy variables 𝑤 𝑡 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 noseq.1 . . 3 (𝜑𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω))
2 df-ima 5698 . . 3 (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω) = ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)
31, 2eqtrdi 2793 . 2 (𝜑𝑍 = ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω))
4 fveq2 6906 . . . . . . . 8 (𝑧 = ∅ → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑧) = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘∅))
54eleq1d 2826 . . . . . . 7 (𝑧 = ∅ → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑧) ∈ 𝐵 ↔ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘∅) ∈ 𝐵))
6 fveq2 6906 . . . . . . . 8 (𝑧 = 𝑤 → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑧) = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤))
76eleq1d 2826 . . . . . . 7 (𝑧 = 𝑤 → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑧) ∈ 𝐵 ↔ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) ∈ 𝐵))
8 fveq2 6906 . . . . . . . 8 (𝑧 = suc 𝑤 → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑧) = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘suc 𝑤))
98eleq1d 2826 . . . . . . 7 (𝑧 = suc 𝑤 → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑧) ∈ 𝐵 ↔ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘suc 𝑤) ∈ 𝐵))
10 noseq.2 . . . . . . . . 9 (𝜑𝐴 No )
11 fr0g 8476 . . . . . . . . 9 (𝐴 No → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘∅) = 𝐴)
1210, 11syl 17 . . . . . . . 8 (𝜑 → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘∅) = 𝐴)
13 noseqind.3 . . . . . . . 8 (𝜑𝐴𝐵)
1412, 13eqeltrd 2841 . . . . . . 7 (𝜑 → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘∅) ∈ 𝐵)
15 oveq1 7438 . . . . . . . . . . . . 13 (𝑦 = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) → (𝑦 +s 1s ) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) +s 1s ))
1615eleq1d 2826 . . . . . . . . . . . 12 (𝑦 = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) → ((𝑦 +s 1s ) ∈ 𝐵 ↔ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) +s 1s ) ∈ 𝐵))
1716imbi2d 340 . . . . . . . . . . 11 (𝑦 = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) → ((𝜑 → (𝑦 +s 1s ) ∈ 𝐵) ↔ (𝜑 → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) +s 1s ) ∈ 𝐵)))
18 noseqind.4 . . . . . . . . . . . 12 ((𝜑𝑦𝐵) → (𝑦 +s 1s ) ∈ 𝐵)
1918expcom 413 . . . . . . . . . . 11 (𝑦𝐵 → (𝜑 → (𝑦 +s 1s ) ∈ 𝐵))
2017, 19vtoclga 3577 . . . . . . . . . 10 (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) ∈ 𝐵 → (𝜑 → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) +s 1s ) ∈ 𝐵))
2120impcom 407 . . . . . . . . 9 ((𝜑 ∧ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) ∈ 𝐵) → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) +s 1s ) ∈ 𝐵)
22 ovex 7464 . . . . . . . . . . 11 (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) +s 1s ) ∈ V
23 eqid 2737 . . . . . . . . . . . 12 (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)
24 oveq1 7438 . . . . . . . . . . . 12 (𝑡 = 𝑥 → (𝑡 +s 1s ) = (𝑥 +s 1s ))
25 oveq1 7438 . . . . . . . . . . . 12 (𝑡 = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) → (𝑡 +s 1s ) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) +s 1s ))
2623, 24, 25frsucmpt2 8480 . . . . . . . . . . 11 ((𝑤 ∈ ω ∧ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) +s 1s ) ∈ V) → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘suc 𝑤) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) +s 1s ))
2722, 26mpan2 691 . . . . . . . . . 10 (𝑤 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘suc 𝑤) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) +s 1s ))
2827eleq1d 2826 . . . . . . . . 9 (𝑤 ∈ ω → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘suc 𝑤) ∈ 𝐵 ↔ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) +s 1s ) ∈ 𝐵))
2921, 28imbitrrid 246 . . . . . . . 8 (𝑤 ∈ ω → ((𝜑 ∧ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) ∈ 𝐵) → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘suc 𝑤) ∈ 𝐵))
3029expd 415 . . . . . . 7 (𝑤 ∈ ω → (𝜑 → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑤) ∈ 𝐵 → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘suc 𝑤) ∈ 𝐵)))
315, 7, 9, 14, 30finds2 7920 . . . . . 6 (𝑧 ∈ ω → (𝜑 → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑧) ∈ 𝐵))
3231com12 32 . . . . 5 (𝜑 → (𝑧 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑧) ∈ 𝐵))
3332ralrimiv 3145 . . . 4 (𝜑 → ∀𝑧 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑧) ∈ 𝐵)
34 frfnom 8475 . . . . 5 (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω) Fn ω
35 ffnfv 7139 . . . . 5 ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω):ω⟶𝐵 ↔ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω) Fn ω ∧ ∀𝑧 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑧) ∈ 𝐵))
3634, 35mpbiran 709 . . . 4 ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω):ω⟶𝐵 ↔ ∀𝑧 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑧) ∈ 𝐵)
3733, 36sylibr 234 . . 3 (𝜑 → (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω):ω⟶𝐵)
3837frnd 6744 . 2 (𝜑 → ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω) ⊆ 𝐵)
393, 38eqsstrd 4018 1 (𝜑𝑍𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  wss 3951  c0 4333  cmpt 5225  ran crn 5686  cres 5687  cima 5688  suc csuc 6386   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  ωcom 7887  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-sep 5296  ax-nul 5306  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-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-op 4633  df-uni 4908  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-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-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450
This theorem is referenced by:  noseqinds  28299  noseqssno  28300  peano5n0s  28324
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