Theorem omsuc 7651

Description: Multiplication with successor. Definition 8.15 of [TakeutiZaring] p. 62. (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 7565	. . 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		suceloni 7055	. . 3 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On)
4		omv 7637	. . 3 ⊢ ((𝐴 ∈ On ∧ suc 𝐵 ∈ On) → (𝐴 ·𝑜 suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘suc 𝐵))
5	3, 4	sylan2 490	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘suc 𝐵))
6		ovex 6718	. . . 4 ⊢ (𝐴 ·𝑜 𝐵) ∈ V
7		oveq1 6697	. . . . 5 ⊢ (𝑥 = (𝐴 ·𝑜 𝐵) → (𝑥 +𝑜 𝐴) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))
8		eqid 2651	. . . . 5 ⊢ (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)) = (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))
9		ovex 6718	. . . . 5 ⊢ ((𝐴 ·𝑜 𝐵) +𝑜 𝐴) ∈ V
10	7, 8, 9	fvmpt 6321	. . . 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 7637	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵))
13	12	fveq2d 6233	. . 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	syl5eqr 2699	. 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 2695	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 suc 𝐵) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  Vcvv 3231  ∅c0 3948   ↦ cmpt 4762  Oncon0 5761  suc csuc 5763  ‘cfv 5926  (class class class)co 6690  reccrdg 7550   +_𝑜 coa 7602   ·_𝑜 comu 7603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-omul 7610

This theorem is referenced by:  omcl  7661  om0r  7664  om1r  7668  omordi  7691  omwordri  7697  omlimcl  7703  odi  7704  omass  7705  oneo  7706  omeulem1  7707  omeulem2  7708  oeoelem  7723  oaabs2  7770  omxpenlem  8102  cantnflt  8607  cantnflem1d  8623  infxpenc  8879