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Theorem noseqp1 28312
Description: One plus an element of 𝑍 is an element of 𝑍. (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 )
noseqp1.3 (𝜑𝐵𝑍)
Assertion
Ref Expression
noseqp1 (𝜑 → (𝐵 +s 1s ) ∈ 𝑍)
Distinct variable group:   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝑍(𝑥)

Proof of Theorem noseqp1
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 noseqp1.3 . . . . 5 (𝜑𝐵𝑍)
2 noseq.1 . . . . 5 (𝜑𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω))
31, 2eleqtrd 2841 . . . 4 (𝜑𝐵 ∈ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω))
4 df-ima 5702 . . . 4 (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω) = ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)
53, 4eleqtrdi 2849 . . 3 (𝜑𝐵 ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω))
6 frfnom 8474 . . . 4 (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω) Fn ω
7 fvelrnb 6969 . . . 4 ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω) Fn ω → (𝐵 ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω) ↔ ∃𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑦) = 𝐵))
86, 7ax-mp 5 . . 3 (𝐵 ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω) ↔ ∃𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑦) = 𝐵)
95, 8sylib 218 . 2 (𝜑 → ∃𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑦) = 𝐵)
10 ovex 7464 . . . . . . 7 (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑦) +s 1s ) ∈ V
11 eqid 2735 . . . . . . . 8 (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)
12 oveq1 7438 . . . . . . . 8 (𝑧 = 𝑥 → (𝑧 +s 1s ) = (𝑥 +s 1s ))
13 oveq1 7438 . . . . . . . 8 (𝑧 = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑦) → (𝑧 +s 1s ) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑦) +s 1s ))
1411, 12, 13frsucmpt2 8479 . . . . . . 7 ((𝑦 ∈ ω ∧ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑦) +s 1s ) ∈ V) → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘suc 𝑦) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑦) +s 1s ))
1510, 14mpan2 691 . . . . . 6 (𝑦 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘suc 𝑦) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑦) +s 1s ))
1615adantl 481 . . . . 5 ((𝜑𝑦 ∈ ω) → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘suc 𝑦) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑦) +s 1s ))
17 peano2 7913 . . . . . . . 8 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
18 fnfvelrn 7100 . . . . . . . 8 (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω) Fn ω ∧ suc 𝑦 ∈ ω) → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘suc 𝑦) ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω))
196, 17, 18sylancr 587 . . . . . . 7 (𝑦 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘suc 𝑦) ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω))
2019adantl 481 . . . . . 6 ((𝜑𝑦 ∈ ω) → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘suc 𝑦) ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω))
212, 4eqtrdi 2791 . . . . . . 7 (𝜑𝑍 = ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω))
2221adantr 480 . . . . . 6 ((𝜑𝑦 ∈ ω) → 𝑍 = ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω))
2320, 22eleqtrrd 2842 . . . . 5 ((𝜑𝑦 ∈ ω) → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘suc 𝑦) ∈ 𝑍)
2416, 23eqeltrrd 2840 . . . 4 ((𝜑𝑦 ∈ ω) → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑦) +s 1s ) ∈ 𝑍)
25 oveq1 7438 . . . . 5 (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑦) = 𝐵 → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑦) +s 1s ) = (𝐵 +s 1s ))
2625eleq1d 2824 . . . 4 (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑦) = 𝐵 → ((((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑦) +s 1s ) ∈ 𝑍 ↔ (𝐵 +s 1s ) ∈ 𝑍))
2724, 26syl5ibcom 245 . . 3 ((𝜑𝑦 ∈ ω) → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑦) = 𝐵 → (𝐵 +s 1s ) ∈ 𝑍))
2827impr 454 . 2 ((𝜑 ∧ (𝑦 ∈ ω ∧ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) ↾ ω)‘𝑦) = 𝐵)) → (𝐵 +s 1s ) ∈ 𝑍)
299, 28rexlimddv 3159 1 (𝜑 → (𝐵 +s 1s ) ∈ 𝑍)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wrex 3068  Vcvv 3478  cmpt 5231  ran crn 5690  cres 5691  cima 5692  suc csuc 6388   Fn wfn 6558  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:  noseqinds  28314  noseqrdgsuc  28329  peano2n0s  28350
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