Theorem onasuc 8528

Description: Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Note that this version of oasuc 8524 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 8437	. . . 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 483	. . 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 7881	. . . . 5 ⊢ (𝐵 ∈ ω → suc 𝐵 ∈ ω)
4	3	adantl 483	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → suc 𝐵 ∈ ω)
5	4	fvresd 6912	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴) ↾ ω)‘suc 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘suc 𝐵))
6		fvres 6911	. . . . 5 ⊢ (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴) ↾ ω)‘𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵))
7	6	adantl 483	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴) ↾ ω)‘𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵))
8	7	fveq2d 6896	. . 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 2781	. 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 7861	. . . 4 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
11		onsuc 7799	. . . 4 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On)
12	10, 11	syl 17	. . 3 ⊢ (𝐵 ∈ ω → suc 𝐵 ∈ On)
13		oav 8511	. . 3 ⊢ ((𝐴 ∈ On ∧ suc 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘suc 𝐵))
14	12, 13	sylan2 594	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 +o suc 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘suc 𝐵))
15		ovex 7442	. . . 4 ⊢ (𝐴 +o 𝐵) ∈ V
16		suceq 6431	. . . . 5 ⊢ (𝑥 = (𝐴 +o 𝐵) → suc 𝑥 = suc (𝐴 +o 𝐵))
17		eqid 2733	. . . . 5 ⊢ (𝑥 ∈ V ↦ suc 𝑥) = (𝑥 ∈ V ↦ suc 𝑥)
18	15	sucex 7794	. . . . 5 ⊢ suc (𝐴 +o 𝐵) ∈ V
19	16, 17, 18	fvmpt 6999	. . . 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 8511	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵))
22	10, 21	sylan2 594	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 +o 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵))
23	22	fveq2d 6896	. . 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 2787	. 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 2783	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 +o suc 𝐵) = suc (𝐴 +o 𝐵))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3475   ↦ cmpt 5232   ↾ cres 5679  Oncon0 6365  suc csuc 6367  ‘cfv 6544  (class class class)co 7409  ωcom 7855  reccrdg 8409   +_o coa 8463

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-oadd 8470

This theorem is referenced by:  oa1suc  8531  o2p2e4  8541  nnasuc  8606  rdgeqoa  36251