Theorem omsuc 8444

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 8346	. . 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 7746	. . 3 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On)
4		omv 8430	. . 3 ⊢ ((𝐴 ∈ On ∧ suc 𝐵 ∈ On) → (𝐴 ·o suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘suc 𝐵))
5	3, 4	sylan2 593	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘suc 𝐵))
6		ovex 7382	. . . 4 ⊢ (𝐴 ·o 𝐵) ∈ V
7		oveq1 7356	. . . . 5 ⊢ (𝑥 = (𝐴 ·o 𝐵) → (𝑥 +o 𝐴) = ((𝐴 ·o 𝐵) +o 𝐴))
8		eqid 2729	. . . . 5 ⊢ (𝑥 ∈ V ↦ (𝑥 +o 𝐴)) = (𝑥 ∈ V ↦ (𝑥 +o 𝐴))
9		ovex 7382	. . . . 5 ⊢ ((𝐴 ·o 𝐵) +o 𝐴) ∈ V
10	7, 8, 9	fvmpt 6930	. . . 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 8430	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵))
13	12	fveq2d 6826	. . 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 2778	. 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 2774	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o suc 𝐵) = ((𝐴 ·o 𝐵) +o 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3436  ∅c0 4284   ↦ cmpt 5173  Oncon0 6307  suc csuc 6309  ‘cfv 6482  (class class class)co 7349  reccrdg 8331   +_o coa 8385   ·_o comu 8386

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-rep 5218  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

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 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  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 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-omul 8393

This theorem is referenced by:  omcl  8454  om0r  8457  om1r  8461  omordi  8484  omwordri  8490  omlimcl  8496  odi  8497  omass  8498  oneo  8499  omeulem1  8500  omeulem2  8501  oeoelem  8516  oaabs2  8567  omxpenlem  8995  cantnflt  9568  cantnflem1d  9584  infxpenc  9912  onexomgt  43224  omlimcl2  43225  onexoegt  43227  om0suclim  43259  oaomoencom  43300  omabs2  43315  naddwordnexlem0  43379  naddwordnexlem3  43382  om2  43387