Theorem onmsuc 8321

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 7711	. . . . 5 ⊢ (𝐵 ∈ ω → suc 𝐵 ∈ ω)
2		nnon 7693	. . . . 5 ⊢ (suc 𝐵 ∈ ω → suc 𝐵 ∈ On)
3	1, 2	syl 17	. . . 4 ⊢ (𝐵 ∈ ω → suc 𝐵 ∈ On)
4		omv 8304	. . . 4 ⊢ ((𝐴 ∈ On ∧ suc 𝐵 ∈ On) → (𝐴 ·o suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘suc 𝐵))
5	3, 4	sylan2 592	. . 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 6776	. . 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 2781	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·o suc 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅) ↾ ω)‘suc 𝐵))
9		ovex 7288	. . . . 5 ⊢ (𝐴 ·o 𝐵) ∈ V
10		oveq1 7262	. . . . . 6 ⊢ (𝑥 = (𝐴 ·o 𝐵) → (𝑥 +o 𝐴) = ((𝐴 ·o 𝐵) +o 𝐴))
11		eqid 2738	. . . . . 6 ⊢ (𝑥 ∈ V ↦ (𝑥 +o 𝐴)) = (𝑥 ∈ V ↦ (𝑥 +o 𝐴))
12		ovex 7288	. . . . . 6 ⊢ ((𝐴 ·o 𝐵) +o 𝐴) ∈ V
13	10, 11, 12	fvmpt 6857	. . . . 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 7693	. . . . . . 7 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
16		omv 8304	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵))
17	15, 16	sylan2 592	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵))
18		fvres 6775	. . . . . . 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 2781	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅) ↾ ω)‘𝐵))
21	20	fveq2d 6760	. . . 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 2793	. . 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 8238	. . . 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 2781	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((𝐴 ·o 𝐵) +o 𝐴) = ((rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅) ↾ ω)‘suc 𝐵))
26	8, 25	eqtr4d 2781	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·o suc 𝐵) = ((𝐴 ·o 𝐵) +o 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Vcvv 3422  ∅c0 4253   ↦ cmpt 5153   ↾ cres 5582  Oncon0 6251  suc csuc 6253  ‘cfv 6418  (class class class)co 7255  ωcom 7687  reccrdg 8211   +_o coa 8264   ·_o comu 8265

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-omul 8272

This theorem is referenced by:  om1  8335  nnmsuc  8400