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Theorem onasuc 7763
Description: Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Note that this version of oasuc 7759 does not need Replacement.) (Contributed by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
onasuc ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 +𝑜 suc 𝐵) = suc (𝐴 +𝑜 𝐵))

Proof of Theorem onasuc
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 frsuc 7686 . . . 4 (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴) ↾ ω)‘suc 𝐵) = ((𝑥 ∈ V ↦ suc 𝑥)‘((rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴) ↾ ω)‘𝐵)))
21adantl 467 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴) ↾ ω)‘suc 𝐵) = ((𝑥 ∈ V ↦ suc 𝑥)‘((rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴) ↾ ω)‘𝐵)))
3 peano2 7234 . . . . 5 (𝐵 ∈ ω → suc 𝐵 ∈ ω)
43adantl 467 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → suc 𝐵 ∈ ω)
5 fvres 6349 . . . 4 (suc 𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴) ↾ ω)‘suc 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘suc 𝐵))
64, 5syl 17 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴) ↾ ω)‘suc 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘suc 𝐵))
7 fvres 6349 . . . . 5 (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴) ↾ ω)‘𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵))
87adantl 467 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴) ↾ ω)‘𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵))
98fveq2d 6337 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((𝑥 ∈ V ↦ suc 𝑥)‘((rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴) ↾ ω)‘𝐵)) = ((𝑥 ∈ V ↦ suc 𝑥)‘(rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵)))
102, 6, 93eqtr3d 2813 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘suc 𝐵) = ((𝑥 ∈ V ↦ suc 𝑥)‘(rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵)))
11 nnon 7219 . . . 4 (𝐵 ∈ ω → 𝐵 ∈ On)
12 suceloni 7161 . . . 4 (𝐵 ∈ On → suc 𝐵 ∈ On)
1311, 12syl 17 . . 3 (𝐵 ∈ ω → suc 𝐵 ∈ On)
14 oav 7746 . . 3 ((𝐴 ∈ On ∧ suc 𝐵 ∈ On) → (𝐴 +𝑜 suc 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘suc 𝐵))
1513, 14sylan2 574 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 +𝑜 suc 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘suc 𝐵))
16 ovex 6824 . . . 4 (𝐴 +𝑜 𝐵) ∈ V
17 suceq 5934 . . . . 5 (𝑥 = (𝐴 +𝑜 𝐵) → suc 𝑥 = suc (𝐴 +𝑜 𝐵))
18 eqid 2771 . . . . 5 (𝑥 ∈ V ↦ suc 𝑥) = (𝑥 ∈ V ↦ suc 𝑥)
1916sucex 7159 . . . . 5 suc (𝐴 +𝑜 𝐵) ∈ V
2017, 18, 19fvmpt 6425 . . . 4 ((𝐴 +𝑜 𝐵) ∈ V → ((𝑥 ∈ V ↦ suc 𝑥)‘(𝐴 +𝑜 𝐵)) = suc (𝐴 +𝑜 𝐵))
2116, 20ax-mp 5 . . 3 ((𝑥 ∈ V ↦ suc 𝑥)‘(𝐴 +𝑜 𝐵)) = suc (𝐴 +𝑜 𝐵)
22 oav 7746 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵))
2311, 22sylan2 574 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 +𝑜 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵))
2423fveq2d 6337 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((𝑥 ∈ V ↦ suc 𝑥)‘(𝐴 +𝑜 𝐵)) = ((𝑥 ∈ V ↦ suc 𝑥)‘(rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵)))
2521, 24syl5eqr 2819 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → suc (𝐴 +𝑜 𝐵) = ((𝑥 ∈ V ↦ suc 𝑥)‘(rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵)))
2610, 15, 253eqtr4d 2815 1 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 +𝑜 suc 𝐵) = suc (𝐴 +𝑜 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  cmpt 4864  cres 5252  Oncon0 5867  suc csuc 5869  cfv 6032  (class class class)co 6794  ωcom 7213  reccrdg 7659   +𝑜 coa 7711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-pss 3740  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5824  df-ord 5870  df-on 5871  df-lim 5872  df-suc 5873  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-ov 6797  df-oprab 6798  df-mpt2 6799  df-om 7214  df-wrecs 7560  df-recs 7622  df-rdg 7660  df-oadd 7718
This theorem is referenced by:  oa1suc  7766  nnasuc  7841  rdgeqoa  33556
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