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

Theorem om0r 8477
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 7359 . . 3 (𝑥 = ∅ → (∅ ·o 𝑥) = (∅ ·o ∅))
21eqeq1d 2739 . 2 (𝑥 = ∅ → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o ∅) = ∅))
3 oveq2 7359 . . 3 (𝑥 = 𝑦 → (∅ ·o 𝑥) = (∅ ·o 𝑦))
43eqeq1d 2739 . 2 (𝑥 = 𝑦 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o 𝑦) = ∅))
5 oveq2 7359 . . 3 (𝑥 = suc 𝑦 → (∅ ·o 𝑥) = (∅ ·o suc 𝑦))
65eqeq1d 2739 . 2 (𝑥 = suc 𝑦 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o suc 𝑦) = ∅))
7 oveq2 7359 . . 3 (𝑥 = 𝐴 → (∅ ·o 𝑥) = (∅ ·o 𝐴))
87eqeq1d 2739 . 2 (𝑥 = 𝐴 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o 𝐴) = ∅))
9 0elon 6369 . . 3 ∅ ∈ On
10 om0 8455 . . 3 (∅ ∈ On → (∅ ·o ∅) = ∅)
119, 10ax-mp 5 . 2 (∅ ·o ∅) = ∅
12 oveq1 7358 . . 3 ((∅ ·o 𝑦) = ∅ → ((∅ ·o 𝑦) +o ∅) = (∅ +o ∅))
13 omsuc 8464 . . . . 5 ((∅ ∈ On ∧ 𝑦 ∈ On) → (∅ ·o suc 𝑦) = ((∅ ·o 𝑦) +o ∅))
149, 13mpan 688 . . . 4 (𝑦 ∈ On → (∅ ·o suc 𝑦) = ((∅ ·o 𝑦) +o ∅))
15 oa0 8454 . . . . . . 7 (∅ ∈ On → (∅ +o ∅) = ∅)
169, 15ax-mp 5 . . . . . 6 (∅ +o ∅) = ∅
1716eqcomi 2746 . . . . 5 ∅ = (∅ +o ∅)
1817a1i 11 . . . 4 (𝑦 ∈ On → ∅ = (∅ +o ∅))
1914, 18eqeq12d 2753 . . 3 (𝑦 ∈ On → ((∅ ·o suc 𝑦) = ∅ ↔ ((∅ ·o 𝑦) +o ∅) = (∅ +o ∅)))
2012, 19syl5ibr 245 . 2 (𝑦 ∈ On → ((∅ ·o 𝑦) = ∅ → (∅ ·o suc 𝑦) = ∅))
21 iuneq2 4971 . . . 4 (∀𝑦𝑥 (∅ ·o 𝑦) = ∅ → 𝑦𝑥 (∅ ·o 𝑦) = 𝑦𝑥 ∅)
22 iun0 5020 . . . 4 𝑦𝑥 ∅ = ∅
2321, 22eqtrdi 2793 . . 3 (∀𝑦𝑥 (∅ ·o 𝑦) = ∅ → 𝑦𝑥 (∅ ·o 𝑦) = ∅)
24 vex 3447 . . . . 5 𝑥 ∈ V
25 omlim 8471 . . . . . 6 ((∅ ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∅ ·o 𝑥) = 𝑦𝑥 (∅ ·o 𝑦))
269, 25mpan 688 . . . . 5 ((𝑥 ∈ V ∧ Lim 𝑥) → (∅ ·o 𝑥) = 𝑦𝑥 (∅ ·o 𝑦))
2724, 26mpan 688 . . . 4 (Lim 𝑥 → (∅ ·o 𝑥) = 𝑦𝑥 (∅ ·o 𝑦))
2827eqeq1d 2739 . . 3 (Lim 𝑥 → ((∅ ·o 𝑥) = ∅ ↔ 𝑦𝑥 (∅ ·o 𝑦) = ∅))
2923, 28syl5ibr 245 . 2 (Lim 𝑥 → (∀𝑦𝑥 (∅ ·o 𝑦) = ∅ → (∅ ·o 𝑥) = ∅))
302, 4, 6, 8, 11, 20, 29tfinds 7788 1 (𝐴 ∈ On → (∅ ·o 𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3062  Vcvv 3443  c0 4280   ciun 4952  Oncon0 6315  Lim wlim 6316  suc csuc 6317  (class class class)co 7351   +o coa 8401   ·o comu 8402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-om 7795  df-2nd 7914  df-frecs 8204  df-wrecs 8235  df-recs 8309  df-rdg 8348  df-oadd 8408  df-omul 8409
This theorem is referenced by:  omord  8507  omwordi  8510  om00  8514  odi  8518  omass  8519  oeoa  8536  omxpenlem  8975  omcl2  41573  omcl3g  41574
  Copyright terms: Public domain W3C validator