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Theorem onasuc 8169
 Description: Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Note that this version of oasuc 8165 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 8088 . . . 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 485 . . 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 7607 . . . . 5 (𝐵 ∈ ω → suc 𝐵 ∈ ω)
43adantl 485 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → suc 𝐵 ∈ ω)
54fvresd 6683 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴) ↾ ω)‘suc 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘suc 𝐵))
6 fvres 6682 . . . . 5 (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴) ↾ ω)‘𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵))
76adantl 485 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴) ↾ ω)‘𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵))
87fveq2d 6667 . . 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)))
92, 5, 83eqtr3d 2801 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘suc 𝐵) = ((𝑥 ∈ V ↦ suc 𝑥)‘(rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵)))
10 nnon 7591 . . . 4 (𝐵 ∈ ω → 𝐵 ∈ On)
11 suceloni 7533 . . . 4 (𝐵 ∈ On → suc 𝐵 ∈ On)
1210, 11syl 17 . . 3 (𝐵 ∈ ω → suc 𝐵 ∈ On)
13 oav 8152 . . 3 ((𝐴 ∈ On ∧ suc 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘suc 𝐵))
1412, 13sylan2 595 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 +o suc 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘suc 𝐵))
15 ovex 7189 . . . 4 (𝐴 +o 𝐵) ∈ V
16 suceq 6239 . . . . 5 (𝑥 = (𝐴 +o 𝐵) → suc 𝑥 = suc (𝐴 +o 𝐵))
17 eqid 2758 . . . . 5 (𝑥 ∈ V ↦ suc 𝑥) = (𝑥 ∈ V ↦ suc 𝑥)
1815sucex 7531 . . . . 5 suc (𝐴 +o 𝐵) ∈ V
1916, 17, 18fvmpt 6764 . . . 4 ((𝐴 +o 𝐵) ∈ V → ((𝑥 ∈ V ↦ suc 𝑥)‘(𝐴 +o 𝐵)) = suc (𝐴 +o 𝐵))
2015, 19ax-mp 5 . . 3 ((𝑥 ∈ V ↦ suc 𝑥)‘(𝐴 +o 𝐵)) = suc (𝐴 +o 𝐵)
21 oav 8152 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵))
2210, 21sylan2 595 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 +o 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵))
2322fveq2d 6667 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((𝑥 ∈ V ↦ suc 𝑥)‘(𝐴 +o 𝐵)) = ((𝑥 ∈ V ↦ suc 𝑥)‘(rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵)))
2420, 23syl5eqr 2807 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → suc (𝐴 +o 𝐵) = ((𝑥 ∈ V ↦ suc 𝑥)‘(rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵)))
259, 14, 243eqtr4d 2803 1 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 +o suc 𝐵) = suc (𝐴 +o 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3409   ↦ cmpt 5116   ↾ cres 5530  Oncon0 6174  suc csuc 6176  ‘cfv 6340  (class class class)co 7156  ωcom 7585  reccrdg 8061   +o coa 8115 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7586  df-wrecs 7963  df-recs 8024  df-rdg 8062  df-oadd 8122 This theorem is referenced by:  oa1suc  8172  o2p2e4  8182  nnasuc  8248  rdgeqoa  35101
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