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Theorem om0r 8577
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 7439 . . 3 (𝑥 = ∅ → (∅ ·o 𝑥) = (∅ ·o ∅))
21eqeq1d 2739 . 2 (𝑥 = ∅ → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o ∅) = ∅))
3 oveq2 7439 . . 3 (𝑥 = 𝑦 → (∅ ·o 𝑥) = (∅ ·o 𝑦))
43eqeq1d 2739 . 2 (𝑥 = 𝑦 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o 𝑦) = ∅))
5 oveq2 7439 . . 3 (𝑥 = suc 𝑦 → (∅ ·o 𝑥) = (∅ ·o suc 𝑦))
65eqeq1d 2739 . 2 (𝑥 = suc 𝑦 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o suc 𝑦) = ∅))
7 oveq2 7439 . . 3 (𝑥 = 𝐴 → (∅ ·o 𝑥) = (∅ ·o 𝐴))
87eqeq1d 2739 . 2 (𝑥 = 𝐴 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o 𝐴) = ∅))
9 0elon 6438 . . 3 ∅ ∈ On
10 om0 8555 . . 3 (∅ ∈ On → (∅ ·o ∅) = ∅)
119, 10ax-mp 5 . 2 (∅ ·o ∅) = ∅
12 oveq1 7438 . . 3 ((∅ ·o 𝑦) = ∅ → ((∅ ·o 𝑦) +o ∅) = (∅ +o ∅))
13 omsuc 8564 . . . . 5 ((∅ ∈ On ∧ 𝑦 ∈ On) → (∅ ·o suc 𝑦) = ((∅ ·o 𝑦) +o ∅))
149, 13mpan 690 . . . 4 (𝑦 ∈ On → (∅ ·o suc 𝑦) = ((∅ ·o 𝑦) +o ∅))
15 oa0 8554 . . . . . . 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, 19imbitrrid 246 . 2 (𝑦 ∈ On → ((∅ ·o 𝑦) = ∅ → (∅ ·o suc 𝑦) = ∅))
21 iuneq2 5011 . . . 4 (∀𝑦𝑥 (∅ ·o 𝑦) = ∅ → 𝑦𝑥 (∅ ·o 𝑦) = 𝑦𝑥 ∅)
22 iun0 5062 . . . 4 𝑦𝑥 ∅ = ∅
2321, 22eqtrdi 2793 . . 3 (∀𝑦𝑥 (∅ ·o 𝑦) = ∅ → 𝑦𝑥 (∅ ·o 𝑦) = ∅)
24 vex 3484 . . . . 5 𝑥 ∈ V
25 omlim 8571 . . . . . 6 ((∅ ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∅ ·o 𝑥) = 𝑦𝑥 (∅ ·o 𝑦))
269, 25mpan 690 . . . . 5 ((𝑥 ∈ V ∧ Lim 𝑥) → (∅ ·o 𝑥) = 𝑦𝑥 (∅ ·o 𝑦))
2724, 26mpan 690 . . . 4 (Lim 𝑥 → (∅ ·o 𝑥) = 𝑦𝑥 (∅ ·o 𝑦))
2827eqeq1d 2739 . . 3 (Lim 𝑥 → ((∅ ·o 𝑥) = ∅ ↔ 𝑦𝑥 (∅ ·o 𝑦) = ∅))
2923, 28imbitrrid 246 . 2 (Lim 𝑥 → (∀𝑦𝑥 (∅ ·o 𝑦) = ∅ → (∅ ·o 𝑥) = ∅))
302, 4, 6, 8, 11, 20, 29tfinds 7881 1 (𝐴 ∈ On → (∅ ·o 𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  c0 4333   ciun 4991  Oncon0 6384  Lim wlim 6385  suc csuc 6386  (class class class)co 7431   +o coa 8503   ·o comu 8504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-oadd 8510  df-omul 8511
This theorem is referenced by:  omord  8606  omwordi  8609  om00  8613  odi  8617  omass  8618  oeoa  8635  omxpenlem  9113  onmcl  43344  omcl2  43346  omcl3g  43347
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