MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  om0r Structured version   Visualization version   GIF version

Theorem om0r 8018
Description: Ordinal multiplication with zero. Proposition 8.18(1) of [TakeutiZaring] p. 63. (Contributed by NM, 3-Aug-2004.)
Assertion
Ref Expression
om0r (𝐴 ∈ On → (∅ ·o 𝐴) = ∅)

Proof of Theorem om0r
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 7027 . . 3 (𝑥 = ∅ → (∅ ·o 𝑥) = (∅ ·o ∅))
21eqeq1d 2796 . 2 (𝑥 = ∅ → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o ∅) = ∅))
3 oveq2 7027 . . 3 (𝑥 = 𝑦 → (∅ ·o 𝑥) = (∅ ·o 𝑦))
43eqeq1d 2796 . 2 (𝑥 = 𝑦 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o 𝑦) = ∅))
5 oveq2 7027 . . 3 (𝑥 = suc 𝑦 → (∅ ·o 𝑥) = (∅ ·o suc 𝑦))
65eqeq1d 2796 . 2 (𝑥 = suc 𝑦 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o suc 𝑦) = ∅))
7 oveq2 7027 . . 3 (𝑥 = 𝐴 → (∅ ·o 𝑥) = (∅ ·o 𝐴))
87eqeq1d 2796 . 2 (𝑥 = 𝐴 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o 𝐴) = ∅))
9 0elon 6122 . . 3 ∅ ∈ On
10 om0 7996 . . 3 (∅ ∈ On → (∅ ·o ∅) = ∅)
119, 10ax-mp 5 . 2 (∅ ·o ∅) = ∅
12 oveq1 7026 . . 3 ((∅ ·o 𝑦) = ∅ → ((∅ ·o 𝑦) +o ∅) = (∅ +o ∅))
13 omsuc 8005 . . . . 5 ((∅ ∈ On ∧ 𝑦 ∈ On) → (∅ ·o suc 𝑦) = ((∅ ·o 𝑦) +o ∅))
149, 13mpan 686 . . . 4 (𝑦 ∈ On → (∅ ·o suc 𝑦) = ((∅ ·o 𝑦) +o ∅))
15 oa0 7995 . . . . . . 7 (∅ ∈ On → (∅ +o ∅) = ∅)
169, 15ax-mp 5 . . . . . 6 (∅ +o ∅) = ∅
1716eqcomi 2803 . . . . 5 ∅ = (∅ +o ∅)
1817a1i 11 . . . 4 (𝑦 ∈ On → ∅ = (∅ +o ∅))
1914, 18eqeq12d 2809 . . 3 (𝑦 ∈ On → ((∅ ·o suc 𝑦) = ∅ ↔ ((∅ ·o 𝑦) +o ∅) = (∅ +o ∅)))
2012, 19syl5ibr 247 . 2 (𝑦 ∈ On → ((∅ ·o 𝑦) = ∅ → (∅ ·o suc 𝑦) = ∅))
21 iuneq2 4845 . . . 4 (∀𝑦𝑥 (∅ ·o 𝑦) = ∅ → 𝑦𝑥 (∅ ·o 𝑦) = 𝑦𝑥 ∅)
22 iun0 4886 . . . 4 𝑦𝑥 ∅ = ∅
2321, 22syl6eq 2846 . . 3 (∀𝑦𝑥 (∅ ·o 𝑦) = ∅ → 𝑦𝑥 (∅ ·o 𝑦) = ∅)
24 vex 3439 . . . . 5 𝑥 ∈ V
25 omlim 8012 . . . . . 6 ((∅ ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∅ ·o 𝑥) = 𝑦𝑥 (∅ ·o 𝑦))
269, 25mpan 686 . . . . 5 ((𝑥 ∈ V ∧ Lim 𝑥) → (∅ ·o 𝑥) = 𝑦𝑥 (∅ ·o 𝑦))
2724, 26mpan 686 . . . 4 (Lim 𝑥 → (∅ ·o 𝑥) = 𝑦𝑥 (∅ ·o 𝑦))
2827eqeq1d 2796 . . 3 (Lim 𝑥 → ((∅ ·o 𝑥) = ∅ ↔ 𝑦𝑥 (∅ ·o 𝑦) = ∅))
2923, 28syl5ibr 247 . 2 (Lim 𝑥 → (∀𝑦𝑥 (∅ ·o 𝑦) = ∅ → (∅ ·o 𝑥) = ∅))
302, 4, 6, 8, 11, 20, 29tfinds 7433 1 (𝐴 ∈ On → (∅ ·o 𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1522  wcel 2080  wral 3104  Vcvv 3436  c0 4213   ciun 4827  Oncon0 6069  Lim wlim 6070  suc csuc 6071  (class class class)co 7019   +o coa 7953   ·o comu 7954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-rep 5084  ax-sep 5097  ax-nul 5104  ax-pow 5160  ax-pr 5224  ax-un 7322
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ne 2984  df-ral 3109  df-rex 3110  df-reu 3111  df-rab 3113  df-v 3438  df-sbc 3708  df-csb 3814  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-pss 3878  df-nul 4214  df-if 4384  df-pw 4457  df-sn 4475  df-pr 4477  df-tp 4479  df-op 4481  df-uni 4748  df-iun 4829  df-br 4965  df-opab 5027  df-mpt 5044  df-tr 5067  df-id 5351  df-eprel 5356  df-po 5365  df-so 5366  df-fr 5405  df-we 5407  df-xp 5452  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-rn 5457  df-res 5458  df-ima 5459  df-pred 6026  df-ord 6072  df-on 6073  df-lim 6074  df-suc 6075  df-iota 6192  df-fun 6230  df-fn 6231  df-f 6232  df-f1 6233  df-fo 6234  df-f1o 6235  df-fv 6236  df-ov 7022  df-oprab 7023  df-mpo 7024  df-om 7440  df-wrecs 7801  df-recs 7863  df-rdg 7901  df-oadd 7960  df-omul 7961
This theorem is referenced by:  omord  8047  omwordi  8050  om00  8054  odi  8058  omass  8059  oeoa  8076  omxpenlem  8468
  Copyright terms: Public domain W3C validator