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Theorem om0r 7565
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 6613 . . 3 (𝑥 = ∅ → (∅ ·𝑜 𝑥) = (∅ ·𝑜 ∅))
21eqeq1d 2628 . 2 (𝑥 = ∅ → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 ∅) = ∅))
3 oveq2 6613 . . 3 (𝑥 = 𝑦 → (∅ ·𝑜 𝑥) = (∅ ·𝑜 𝑦))
43eqeq1d 2628 . 2 (𝑥 = 𝑦 → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 𝑦) = ∅))
5 oveq2 6613 . . 3 (𝑥 = suc 𝑦 → (∅ ·𝑜 𝑥) = (∅ ·𝑜 suc 𝑦))
65eqeq1d 2628 . 2 (𝑥 = suc 𝑦 → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 suc 𝑦) = ∅))
7 oveq2 6613 . . 3 (𝑥 = 𝐴 → (∅ ·𝑜 𝑥) = (∅ ·𝑜 𝐴))
87eqeq1d 2628 . 2 (𝑥 = 𝐴 → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 𝐴) = ∅))
9 om0x 7545 . 2 (∅ ·𝑜 ∅) = ∅
10 oveq1 6612 . . 3 ((∅ ·𝑜 𝑦) = ∅ → ((∅ ·𝑜 𝑦) +𝑜 ∅) = (∅ +𝑜 ∅))
11 0elon 5740 . . . . 5 ∅ ∈ On
12 omsuc 7552 . . . . 5 ((∅ ∈ On ∧ 𝑦 ∈ On) → (∅ ·𝑜 suc 𝑦) = ((∅ ·𝑜 𝑦) +𝑜 ∅))
1311, 12mpan 705 . . . 4 (𝑦 ∈ On → (∅ ·𝑜 suc 𝑦) = ((∅ ·𝑜 𝑦) +𝑜 ∅))
14 oa0 7542 . . . . . . 7 (∅ ∈ On → (∅ +𝑜 ∅) = ∅)
1511, 14ax-mp 5 . . . . . 6 (∅ +𝑜 ∅) = ∅
1615eqcomi 2635 . . . . 5 ∅ = (∅ +𝑜 ∅)
1716a1i 11 . . . 4 (𝑦 ∈ On → ∅ = (∅ +𝑜 ∅))
1813, 17eqeq12d 2641 . . 3 (𝑦 ∈ On → ((∅ ·𝑜 suc 𝑦) = ∅ ↔ ((∅ ·𝑜 𝑦) +𝑜 ∅) = (∅ +𝑜 ∅)))
1910, 18syl5ibr 236 . 2 (𝑦 ∈ On → ((∅ ·𝑜 𝑦) = ∅ → (∅ ·𝑜 suc 𝑦) = ∅))
20 iuneq2 4508 . . . 4 (∀𝑦𝑥 (∅ ·𝑜 𝑦) = ∅ → 𝑦𝑥 (∅ ·𝑜 𝑦) = 𝑦𝑥 ∅)
21 iun0 4547 . . . 4 𝑦𝑥 ∅ = ∅
2220, 21syl6eq 2676 . . 3 (∀𝑦𝑥 (∅ ·𝑜 𝑦) = ∅ → 𝑦𝑥 (∅ ·𝑜 𝑦) = ∅)
23 vex 3194 . . . . 5 𝑥 ∈ V
24 omlim 7559 . . . . . 6 ((∅ ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∅ ·𝑜 𝑥) = 𝑦𝑥 (∅ ·𝑜 𝑦))
2511, 24mpan 705 . . . . 5 ((𝑥 ∈ V ∧ Lim 𝑥) → (∅ ·𝑜 𝑥) = 𝑦𝑥 (∅ ·𝑜 𝑦))
2623, 25mpan 705 . . . 4 (Lim 𝑥 → (∅ ·𝑜 𝑥) = 𝑦𝑥 (∅ ·𝑜 𝑦))
2726eqeq1d 2628 . . 3 (Lim 𝑥 → ((∅ ·𝑜 𝑥) = ∅ ↔ 𝑦𝑥 (∅ ·𝑜 𝑦) = ∅))
2822, 27syl5ibr 236 . 2 (Lim 𝑥 → (∀𝑦𝑥 (∅ ·𝑜 𝑦) = ∅ → (∅ ·𝑜 𝑥) = ∅))
292, 4, 6, 8, 9, 19, 28tfinds 7007 1 (𝐴 ∈ On → (∅ ·𝑜 𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  wral 2912  Vcvv 3191  c0 3896   ciun 4490  Oncon0 5685  Lim wlim 5686  suc csuc 5687  (class class class)co 6605   +𝑜 coa 7503   ·𝑜 comu 7504
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  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 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-oadd 7510  df-omul 7511
This theorem is referenced by:  omord  7594  omwordi  7597  om00  7601  odi  7605  omass  7606  oeoa  7623  omxpenlem  8006
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