Theorem onmsuc 8567

Description: Multiplication with successor. Theorem 4J(A2) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)

Assertion

Ref	Expression
onmsuc	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·o suc 𝐵) = ((𝐴 ·o 𝐵) +o 𝐴))

Proof of Theorem onmsuc

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		peano2 7912	. . . . 5 ⊢ (𝐵 ∈ ω → suc 𝐵 ∈ ω)
2		nnon 7893	. . . . 5 ⊢ (suc 𝐵 ∈ ω → suc 𝐵 ∈ On)
3	1, 2	syl 17	. . . 4 ⊢ (𝐵 ∈ ω → suc 𝐵 ∈ On)
4		omv 8550	. . . 4 ⊢ ((𝐴 ∈ On ∧ suc 𝐵 ∈ On) → (𝐴 ·o suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘suc 𝐵))
5	3, 4	sylan2 593	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·o suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘suc 𝐵))
6	1	adantl 481	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → suc 𝐵 ∈ ω)
7	6	fvresd 6926	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅) ↾ ω)‘suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘suc 𝐵))
8	5, 7	eqtr4d 2780	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·o suc 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅) ↾ ω)‘suc 𝐵))
9		ovex 7464	. . . . 5 ⊢ (𝐴 ·o 𝐵) ∈ V
10		oveq1 7438	. . . . . 6 ⊢ (𝑥 = (𝐴 ·o 𝐵) → (𝑥 +o 𝐴) = ((𝐴 ·o 𝐵) +o 𝐴))
11		eqid 2737	. . . . . 6 ⊢ (𝑥 ∈ V ↦ (𝑥 +o 𝐴)) = (𝑥 ∈ V ↦ (𝑥 +o 𝐴))
12		ovex 7464	. . . . . 6 ⊢ ((𝐴 ·o 𝐵) +o 𝐴) ∈ V
13	10, 11, 12	fvmpt 7016	. . . . 5 ⊢ ((𝐴 ·o 𝐵) ∈ V → ((𝑥 ∈ V ↦ (𝑥 +o 𝐴))‘(𝐴 ·o 𝐵)) = ((𝐴 ·o 𝐵) +o 𝐴))
14	9, 13	ax-mp 5	. . . 4 ⊢ ((𝑥 ∈ V ↦ (𝑥 +o 𝐴))‘(𝐴 ·o 𝐵)) = ((𝐴 ·o 𝐵) +o 𝐴)
15		nnon 7893	. . . . . . 7 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
16		omv 8550	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵))
17	15, 16	sylan2 593	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵))
18		fvres 6925	. . . . . . 7 ⊢ (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅) ↾ ω)‘𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵))
19	18	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅) ↾ ω)‘𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵))
20	17, 19	eqtr4d 2780	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅) ↾ ω)‘𝐵))
21	20	fveq2d 6910	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((𝑥 ∈ V ↦ (𝑥 +o 𝐴))‘(𝐴 ·o 𝐵)) = ((𝑥 ∈ V ↦ (𝑥 +o 𝐴))‘((rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅) ↾ ω)‘𝐵)))
22	14, 21	eqtr3id 2791	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((𝐴 ·o 𝐵) +o 𝐴) = ((𝑥 ∈ V ↦ (𝑥 +o 𝐴))‘((rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅) ↾ ω)‘𝐵)))
23		frsuc 8477	. . . 4 ⊢ (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅) ↾ ω)‘suc 𝐵) = ((𝑥 ∈ V ↦ (𝑥 +o 𝐴))‘((rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅) ↾ ω)‘𝐵)))
24	23	adantl 481	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅) ↾ ω)‘suc 𝐵) = ((𝑥 ∈ V ↦ (𝑥 +o 𝐴))‘((rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅) ↾ ω)‘𝐵)))
25	22, 24	eqtr4d 2780	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((𝐴 ·o 𝐵) +o 𝐴) = ((rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅) ↾ ω)‘suc 𝐵))
26	8, 25	eqtr4d 2780	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·o suc 𝐵) = ((𝐴 ·o 𝐵) +o 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3480  ∅c0 4333   ↦ cmpt 5225   ↾ cres 5687  Oncon0 6384  suc csuc 6386  ‘cfv 6561  (class class class)co 7431  ωcom 7887  reccrdg 8449   +_o coa 8503   ·_o comu 8504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-omul 8511

This theorem is referenced by:  om1  8580  nnmsuc  8645  onmcl  43344