Theorem omsuc 8582

Description: Multiplication with successor. Definition 8.15 of [TakeutiZaring] p. 62. Definition 2.5 of [Schloeder] p. 4. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)

Assertion

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

Proof of Theorem omsuc

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rdgsuc 8480	. . 3 ⊢ (𝐵 ∈ On → (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘suc 𝐵) = ((𝑥 ∈ V ↦ (𝑥 +o 𝐴))‘(rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵)))
2	1	adantl 481	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘suc 𝐵) = ((𝑥 ∈ V ↦ (𝑥 +o 𝐴))‘(rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵)))
3		onsuc 7847	. . 3 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On)
4		omv 8568	. . 3 ⊢ ((𝐴 ∈ On ∧ suc 𝐵 ∈ On) → (𝐴 ·o suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘suc 𝐵))
5	3, 4	sylan2 592	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘suc 𝐵))
6		ovex 7481	. . . 4 ⊢ (𝐴 ·o 𝐵) ∈ V
7		oveq1 7455	. . . . 5 ⊢ (𝑥 = (𝐴 ·o 𝐵) → (𝑥 +o 𝐴) = ((𝐴 ·o 𝐵) +o 𝐴))
8		eqid 2740	. . . . 5 ⊢ (𝑥 ∈ V ↦ (𝑥 +o 𝐴)) = (𝑥 ∈ V ↦ (𝑥 +o 𝐴))
9		ovex 7481	. . . . 5 ⊢ ((𝐴 ·o 𝐵) +o 𝐴) ∈ V
10	7, 8, 9	fvmpt 7029	. . . 4 ⊢ ((𝐴 ·o 𝐵) ∈ V → ((𝑥 ∈ V ↦ (𝑥 +o 𝐴))‘(𝐴 ·o 𝐵)) = ((𝐴 ·o 𝐵) +o 𝐴))
11	6, 10	ax-mp 5	. . 3 ⊢ ((𝑥 ∈ V ↦ (𝑥 +o 𝐴))‘(𝐴 ·o 𝐵)) = ((𝐴 ·o 𝐵) +o 𝐴)
12		omv 8568	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵))
13	12	fveq2d 6924	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 +o 𝐴))‘(𝐴 ·o 𝐵)) = ((𝑥 ∈ V ↦ (𝑥 +o 𝐴))‘(rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵)))
14	11, 13	eqtr3id 2794	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝐵) +o 𝐴) = ((𝑥 ∈ V ↦ (𝑥 +o 𝐴))‘(rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵)))
15	2, 5, 14	3eqtr4d 2790	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o suc 𝐵) = ((𝐴 ·o 𝐵) +o 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488  ∅c0 4352   ↦ cmpt 5249  Oncon0 6395  suc csuc 6397  ‘cfv 6573  (class class class)co 7448  reccrdg 8465   +_o coa 8519   ·_o comu 8520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-omul 8527

This theorem is referenced by:  omcl  8592  om0r  8595  om1r  8599  omordi  8622  omwordri  8628  omlimcl  8634  odi  8635  omass  8636  oneo  8637  omeulem1  8638  omeulem2  8639  oeoelem  8654  oaabs2  8705  omxpenlem  9139  cantnflt  9741  cantnflem1d  9757  infxpenc  10087  onexomgt  43202  omlimcl2  43203  onexoegt  43205  om0suclim  43238  oaomoencom  43279  omabs2  43294  naddwordnexlem0  43358  naddwordnexlem3  43361  om2  43366