Theorem onmsuc 8454

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 7830	. . . . 5 ⊢ (𝐵 ∈ ω → suc 𝐵 ∈ ω)
2		nnon 7812	. . . . 5 ⊢ (suc 𝐵 ∈ ω → suc 𝐵 ∈ On)
3	1, 2	syl 17	. . . 4 ⊢ (𝐵 ∈ ω → suc 𝐵 ∈ On)
4		omv 8437	. . . 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 6846	. . 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 2767	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·o suc 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅) ↾ ω)‘suc 𝐵))
9		ovex 7386	. . . . 5 ⊢ (𝐴 ·o 𝐵) ∈ V
10		oveq1 7360	. . . . . 6 ⊢ (𝑥 = (𝐴 ·o 𝐵) → (𝑥 +o 𝐴) = ((𝐴 ·o 𝐵) +o 𝐴))
11		eqid 2729	. . . . . 6 ⊢ (𝑥 ∈ V ↦ (𝑥 +o 𝐴)) = (𝑥 ∈ V ↦ (𝑥 +o 𝐴))
12		ovex 7386	. . . . . 6 ⊢ ((𝐴 ·o 𝐵) +o 𝐴) ∈ V
13	10, 11, 12	fvmpt 6934	. . . . 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 7812	. . . . . . 7 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
16		omv 8437	. . . . . . 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 6845	. . . . . . 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 2767	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅) ↾ ω)‘𝐵))
21	20	fveq2d 6830	. . . 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 2778	. . 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 8366	. . . 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 2767	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((𝐴 ·o 𝐵) +o 𝐴) = ((rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅) ↾ ω)‘suc 𝐵))
26	8, 25	eqtr4d 2767	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·o suc 𝐵) = ((𝐴 ·o 𝐵) +o 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3438  ∅c0 4286   ↦ cmpt 5176   ↾ cres 5625  Oncon0 6311  suc csuc 6313  ‘cfv 6486  (class class class)co 7353  ωcom 7806  reccrdg 8338   +_o coa 8392   ·_o comu 8393

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-omul 8400

This theorem is referenced by:  om1  8467  nnmsuc  8532  onmcl  43307