Theorem onasuc 8438

Description: Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Note that this version of oasuc 8434 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 8351	. . . 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 481	. . 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 7815	. . . . 5 ⊢ (𝐵 ∈ ω → suc 𝐵 ∈ ω)
4	3	adantl 481	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → suc 𝐵 ∈ ω)
5	4	fvresd 6837	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴) ↾ ω)‘suc 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘suc 𝐵))
6		fvres 6836	. . . . 5 ⊢ (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴) ↾ ω)‘𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵))
7	6	adantl 481	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴) ↾ ω)‘𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵))
8	7	fveq2d 6821	. . 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 2774	. 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 7797	. . . 4 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
11		onsuc 7738	. . . 4 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On)
12	10, 11	syl 17	. . 3 ⊢ (𝐵 ∈ ω → suc 𝐵 ∈ On)
13		oav 8421	. . 3 ⊢ ((𝐴 ∈ On ∧ suc 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘suc 𝐵))
14	12, 13	sylan2 593	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 +o suc 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘suc 𝐵))
15		ovex 7374	. . . 4 ⊢ (𝐴 +o 𝐵) ∈ V
16		suceq 6369	. . . . 5 ⊢ (𝑥 = (𝐴 +o 𝐵) → suc 𝑥 = suc (𝐴 +o 𝐵))
17		eqid 2731	. . . . 5 ⊢ (𝑥 ∈ V ↦ suc 𝑥) = (𝑥 ∈ V ↦ suc 𝑥)
18	15	sucex 7734	. . . . 5 ⊢ suc (𝐴 +o 𝐵) ∈ V
19	16, 17, 18	fvmpt 6924	. . . 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 8421	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵))
22	10, 21	sylan2 593	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 +o 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵))
23	22	fveq2d 6821	. . 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 2780	. 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 2776	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 +o suc 𝐵) = suc (𝐴 +o 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ↦ cmpt 5167   ↾ cres 5613  Oncon0 6301  suc csuc 6303  ‘cfv 6476  (class class class)co 7341  ωcom 7791  reccrdg 8323   +_o coa 8377

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-oadd 8384

This theorem is referenced by:  oa1suc  8441  o2p2e4  8451  nnasuc  8516  naddoa  8612  rdgeqoa  37404