Theorem onmsuc 8450

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 7826	. . . . 5 ⊢ (𝐵 ∈ ω → suc 𝐵 ∈ ω)
2		nnon 7808	. . . . 5 ⊢ (suc 𝐵 ∈ ω → suc 𝐵 ∈ On)
3	1, 2	syl 17	. . . 4 ⊢ (𝐵 ∈ ω → suc 𝐵 ∈ On)
4		omv 8433	. . . 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 6848	. . 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 2771	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·o suc 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅) ↾ ω)‘suc 𝐵))
9		ovex 7385	. . . . 5 ⊢ (𝐴 ·o 𝐵) ∈ V
10		oveq1 7359	. . . . . 6 ⊢ (𝑥 = (𝐴 ·o 𝐵) → (𝑥 +o 𝐴) = ((𝐴 ·o 𝐵) +o 𝐴))
11		eqid 2733	. . . . . 6 ⊢ (𝑥 ∈ V ↦ (𝑥 +o 𝐴)) = (𝑥 ∈ V ↦ (𝑥 +o 𝐴))
12		ovex 7385	. . . . . 6 ⊢ ((𝐴 ·o 𝐵) +o 𝐴) ∈ V
13	10, 11, 12	fvmpt 6935	. . . . 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 7808	. . . . . . 7 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
16		omv 8433	. . . . . . 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 6847	. . . . . . 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 2771	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅) ↾ ω)‘𝐵))
21	20	fveq2d 6832	. . . 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 2782	. . 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 8362	. . . 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 2771	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((𝐴 ·o 𝐵) +o 𝐴) = ((rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅) ↾ ω)‘suc 𝐵))
26	8, 25	eqtr4d 2771	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·o suc 𝐵) = ((𝐴 ·o 𝐵) +o 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3437  ∅c0 4282   ↦ cmpt 5174   ↾ cres 5621  Oncon0 6311  suc csuc 6313  ‘cfv 6486  (class class class)co 7352  ωcom 7802  reccrdg 8334   +_o coa 8388   ·_o comu 8389

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  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 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-omul 8396

This theorem is referenced by:  om1  8463  nnmsuc  8528  onmcl  43449