Theorem onasuc 8519

Description: Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Note that this version of oasuc 8515 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.

Step	Hyp	Ref	Expression
1		frsuc 8430	. . . 4 ⊢ (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴) ↾ ω)‘suc 𝐵) = ((𝑥 ∈ V ↦ suc 𝑥)‘((rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴) ↾ ω)‘𝐵)))
2	1	adantl 487	. . 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 7892	. . . . 5 ⊢ (𝐵 ∈ ω → suc 𝐵 ∈ ω)
4	3	adantl 487	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → suc 𝐵 ∈ ω)
5	4	fvresd 6905	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴) ↾ ω)‘suc 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘suc 𝐵))
6		fvres 6904	. . . . 5 ⊢ (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴) ↾ ω)‘𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵))
7	6	adantl 487	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴) ↾ ω)‘𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵))
8	7	fveq2d 6889	. . 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)))
9	2, 5, 8	3eqtr3d 2808	. 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 7874	. . . 4 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
11		onsuc 7815	. . . 4 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On)
12	10, 11	syl 18	. . 3 ⊢ (𝐵 ∈ ω → suc 𝐵 ∈ On)
13		oav 8502	. . 3 ⊢ ((𝐴 ∈ On ∧ suc 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘suc 𝐵))
14	12, 13	sylan2 605	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 +o suc 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘suc 𝐵))
15		ovex 7452	. . . 4 ⊢ (𝐴 +o 𝐵) ∈ V
16		suceq 6433	. . . . 5 ⊢ (𝑥 = (𝐴 +o 𝐵) → suc 𝑥 = suc (𝐴 +o 𝐵))
17		eqid 2765	. . . . 5 ⊢ (𝑥 ∈ V ↦ suc 𝑥) = (𝑥 ∈ V ↦ suc 𝑥)
18	15	sucex 7811	. . . . 5 ⊢ suc (𝐴 +o 𝐵) ∈ V
19	16, 17, 18	fvmpt 6993	. . . 4 ⊢ ((𝐴 +o 𝐵) ∈ V → ((𝑥 ∈ V ↦ suc 𝑥)‘(𝐴 +o 𝐵)) = suc (𝐴 +o 𝐵))
20	15, 19	ax-mp 5	. . 3 ⊢ ((𝑥 ∈ V ↦ suc 𝑥)‘(𝐴 +o 𝐵)) = suc (𝐴 +o 𝐵)
21		oav 8502	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵))
22	10, 21	sylan2 605	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 +o 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵))
23	22	fveq2d 6889	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((𝑥 ∈ V ↦ suc 𝑥)‘(𝐴 +o 𝐵)) = ((𝑥 ∈ V ↦ suc 𝑥)‘(rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵)))
24	20, 23	eqtr3id 2814	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → suc (𝐴 +o 𝐵) = ((𝑥 ∈ V ↦ suc 𝑥)‘(rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵)))
25	9, 14, 24	3eqtr4d 2810	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 +o suc 𝐵) = suc (𝐴 +o 𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  Vcvv 3457   ↦ cmpt 5194   ↾ cres 5665  Oncon0 6364  suc csuc 6366  ‘cfv 6540  (class class class)co 7419  ωcom 7868  reccrdg 8402   +_o coa 8456

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-oadd 8463

This theorem is used by:  oa1suc  8522  o2p2e4  8532  nnasuc  8598  naddoa  8695  rdgeqoa  38073