Theorem onmsuc 8513

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 7865	. . . . 5 ⊢ (𝐵 ∈ ω → suc 𝐵 ∈ ω)
2		nnon 7845	. . . . 5 ⊢ (suc 𝐵 ∈ ω → suc 𝐵 ∈ On)
3	1, 2	syl 17	. . . 4 ⊢ (𝐵 ∈ ω → suc 𝐵 ∈ On)
4		omv 8496	. . . 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 482	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → suc 𝐵 ∈ ω)
7	6	fvresd 6899	. . 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 2775	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·o suc 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅) ↾ ω)‘suc 𝐵))
9		ovex 7427	. . . . 5 ⊢ (𝐴 ·o 𝐵) ∈ V
10		oveq1 7401	. . . . . 6 ⊢ (𝑥 = (𝐴 ·o 𝐵) → (𝑥 +o 𝐴) = ((𝐴 ·o 𝐵) +o 𝐴))
11		eqid 2732	. . . . . 6 ⊢ (𝑥 ∈ V ↦ (𝑥 +o 𝐴)) = (𝑥 ∈ V ↦ (𝑥 +o 𝐴))
12		ovex 7427	. . . . . 6 ⊢ ((𝐴 ·o 𝐵) +o 𝐴) ∈ V
13	10, 11, 12	fvmpt 6985	. . . . 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 7845	. . . . . . 7 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
16		omv 8496	. . . . . . 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 6898	. . . . . . 7 ⊢ (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅) ↾ ω)‘𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵))
19	18	adantl 482	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅) ↾ ω)‘𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵))
20	17, 19	eqtr4d 2775	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅) ↾ ω)‘𝐵))
21	20	fveq2d 6883	. . . 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 2786	. . 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 8421	. . . 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 482	. . 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 2775	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((𝐴 ·o 𝐵) +o 𝐴) = ((rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅) ↾ ω)‘suc 𝐵))
26	8, 25	eqtr4d 2775	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·o suc 𝐵) = ((𝐴 ·o 𝐵) +o 𝐴))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474  ∅c0 4319   ↦ cmpt 5225   ↾ cres 5672  Oncon0 6354  suc csuc 6356  ‘cfv 6533  (class class class)co 7394  ωcom 7839  reccrdg 8393   +_o coa 8447   ·_o comu 8448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5293  ax-nul 5300  ax-pr 5421  ax-un 7709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7397  df-oprab 7398  df-mpo 7399  df-om 7840  df-2nd 7960  df-frecs 8250  df-wrecs 8281  df-recs 8355  df-rdg 8394  df-omul 8455

This theorem is referenced by:  om1  8527  nnmsuc  8592  onmcl  41916