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Theorem om0r 8147
 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 7143 . . 3 (𝑥 = ∅ → (∅ ·o 𝑥) = (∅ ·o ∅))
21eqeq1d 2800 . 2 (𝑥 = ∅ → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o ∅) = ∅))
3 oveq2 7143 . . 3 (𝑥 = 𝑦 → (∅ ·o 𝑥) = (∅ ·o 𝑦))
43eqeq1d 2800 . 2 (𝑥 = 𝑦 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o 𝑦) = ∅))
5 oveq2 7143 . . 3 (𝑥 = suc 𝑦 → (∅ ·o 𝑥) = (∅ ·o suc 𝑦))
65eqeq1d 2800 . 2 (𝑥 = suc 𝑦 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o suc 𝑦) = ∅))
7 oveq2 7143 . . 3 (𝑥 = 𝐴 → (∅ ·o 𝑥) = (∅ ·o 𝐴))
87eqeq1d 2800 . 2 (𝑥 = 𝐴 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o 𝐴) = ∅))
9 0elon 6212 . . 3 ∅ ∈ On
10 om0 8125 . . 3 (∅ ∈ On → (∅ ·o ∅) = ∅)
119, 10ax-mp 5 . 2 (∅ ·o ∅) = ∅
12 oveq1 7142 . . 3 ((∅ ·o 𝑦) = ∅ → ((∅ ·o 𝑦) +o ∅) = (∅ +o ∅))
13 omsuc 8134 . . . . 5 ((∅ ∈ On ∧ 𝑦 ∈ On) → (∅ ·o suc 𝑦) = ((∅ ·o 𝑦) +o ∅))
149, 13mpan 689 . . . 4 (𝑦 ∈ On → (∅ ·o suc 𝑦) = ((∅ ·o 𝑦) +o ∅))
15 oa0 8124 . . . . . . 7 (∅ ∈ On → (∅ +o ∅) = ∅)
169, 15ax-mp 5 . . . . . 6 (∅ +o ∅) = ∅
1716eqcomi 2807 . . . . 5 ∅ = (∅ +o ∅)
1817a1i 11 . . . 4 (𝑦 ∈ On → ∅ = (∅ +o ∅))
1914, 18eqeq12d 2814 . . 3 (𝑦 ∈ On → ((∅ ·o suc 𝑦) = ∅ ↔ ((∅ ·o 𝑦) +o ∅) = (∅ +o ∅)))
2012, 19syl5ibr 249 . 2 (𝑦 ∈ On → ((∅ ·o 𝑦) = ∅ → (∅ ·o suc 𝑦) = ∅))
21 iuneq2 4900 . . . 4 (∀𝑦𝑥 (∅ ·o 𝑦) = ∅ → 𝑦𝑥 (∅ ·o 𝑦) = 𝑦𝑥 ∅)
22 iun0 4948 . . . 4 𝑦𝑥 ∅ = ∅
2321, 22eqtrdi 2849 . . 3 (∀𝑦𝑥 (∅ ·o 𝑦) = ∅ → 𝑦𝑥 (∅ ·o 𝑦) = ∅)
24 vex 3444 . . . . 5 𝑥 ∈ V
25 omlim 8141 . . . . . 6 ((∅ ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∅ ·o 𝑥) = 𝑦𝑥 (∅ ·o 𝑦))
269, 25mpan 689 . . . . 5 ((𝑥 ∈ V ∧ Lim 𝑥) → (∅ ·o 𝑥) = 𝑦𝑥 (∅ ·o 𝑦))
2724, 26mpan 689 . . . 4 (Lim 𝑥 → (∅ ·o 𝑥) = 𝑦𝑥 (∅ ·o 𝑦))
2827eqeq1d 2800 . . 3 (Lim 𝑥 → ((∅ ·o 𝑥) = ∅ ↔ 𝑦𝑥 (∅ ·o 𝑦) = ∅))
2923, 28syl5ibr 249 . 2 (Lim 𝑥 → (∀𝑦𝑥 (∅ ·o 𝑦) = ∅ → (∅ ·o 𝑥) = ∅))
302, 4, 6, 8, 11, 20, 29tfinds 7554 1 (𝐴 ∈ On → (∅ ·o 𝐴) = ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  Vcvv 3441  ∅c0 4243  ∪ ciun 4881  Oncon0 6159  Lim wlim 6160  suc csuc 6161  (class class class)co 7135   +o coa 8082   ·o comu 8083 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-oadd 8089  df-omul 8090 This theorem is referenced by:  omord  8177  omwordi  8180  om00  8184  odi  8188  omass  8189  oeoa  8206  omxpenlem  8601
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