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Theorem om0 7549
Description: Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
om0 (𝐴 ∈ On → (𝐴 ·𝑜 ∅) = ∅)

Proof of Theorem om0
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 0elon 5742 . . 3 ∅ ∈ On
2 omv 7544 . . 3 ((𝐴 ∈ On ∧ ∅ ∈ On) → (𝐴 ·𝑜 ∅) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘∅))
31, 2mpan2 706 . 2 (𝐴 ∈ On → (𝐴 ·𝑜 ∅) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘∅))
4 0ex 4755 . . 3 ∅ ∈ V
54rdg0 7469 . 2 (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘∅) = ∅
63, 5syl6eq 2671 1 (𝐴 ∈ On → (𝐴 ·𝑜 ∅) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  Vcvv 3189  c0 3896  cmpt 4678  Oncon0 5687  cfv 5852  (class class class)co 6610  reccrdg 7457   +𝑜 coa 7509   ·𝑜 comu 7510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-omul 7517
This theorem is referenced by:  om0x  7551  oesuclem  7557  omcl  7568  om1  7574  omwordri  7604  om00  7607  odi  7611  omass  7612  oen0  7618  oeoa  7629  oeoelem  7630  oeeui  7634  nnm0  7637  cantnfle  8520  cantnfp1  8530
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