Theorem oasuc 7604

Description: Addition with successor. Definition 8.1 of [TakeutiZaring] p. 56. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
oasuc	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 suc 𝐵) = suc (𝐴 +𝑜 𝐵))

Proof of Theorem oasuc

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rdgsuc 7520	. . 3 ⊢ (𝐵 ∈ On → (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘suc 𝐵) = ((𝑥 ∈ V ↦ suc 𝑥)‘(rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵)))
2	1	adantl 482	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘suc 𝐵) = ((𝑥 ∈ V ↦ suc 𝑥)‘(rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵)))
3		suceloni 7013	. . 3 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On)
4		oav 7591	. . 3 ⊢ ((𝐴 ∈ On ∧ suc 𝐵 ∈ On) → (𝐴 +𝑜 suc 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘suc 𝐵))
5	3, 4	sylan2 491	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 suc 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘suc 𝐵))
6		ovex 6678	. . . 4 ⊢ (𝐴 +𝑜 𝐵) ∈ V
7		suceq 5790	. . . . 5 ⊢ (𝑥 = (𝐴 +𝑜 𝐵) → suc 𝑥 = suc (𝐴 +𝑜 𝐵))
8		eqid 2622	. . . . 5 ⊢ (𝑥 ∈ V ↦ suc 𝑥) = (𝑥 ∈ V ↦ suc 𝑥)
9	6	sucex 7011	. . . . 5 ⊢ suc (𝐴 +𝑜 𝐵) ∈ V
10	7, 8, 9	fvmpt 6282	. . . 4 ⊢ ((𝐴 +𝑜 𝐵) ∈ V → ((𝑥 ∈ V ↦ suc 𝑥)‘(𝐴 +𝑜 𝐵)) = suc (𝐴 +𝑜 𝐵))
11	6, 10	ax-mp 5	. . 3 ⊢ ((𝑥 ∈ V ↦ suc 𝑥)‘(𝐴 +𝑜 𝐵)) = suc (𝐴 +𝑜 𝐵)
12		oav 7591	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵))
13	12	fveq2d 6195	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥 ∈ V ↦ suc 𝑥)‘(𝐴 +𝑜 𝐵)) = ((𝑥 ∈ V ↦ suc 𝑥)‘(rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵)))
14	11, 13	syl5eqr 2670	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc (𝐴 +𝑜 𝐵) = ((𝑥 ∈ V ↦ suc 𝑥)‘(rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵)))
15	2, 5, 14	3eqtr4d 2666	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 suc 𝐵) = suc (𝐴 +𝑜 𝐵))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ↦ cmpt 4729  Oncon0 5723  suc csuc 5725  ‘cfv 5888  (class class class)co 6650  reccrdg 7505   +_𝑜 coa 7557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564

This theorem is referenced by:  oacl  7615  oa0r  7618  o2p2e4  7621  oaordi  7626  oawordri  7630  oawordeulem  7634  oalimcl  7640  oaass  7641  oarec  7642  odi  7659  oeoalem  7676