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Theorem om0r 7776
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 6809 . . 3 (𝑥 = ∅ → (∅ ·𝑜 𝑥) = (∅ ·𝑜 ∅))
21eqeq1d 2750 . 2 (𝑥 = ∅ → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 ∅) = ∅))
3 oveq2 6809 . . 3 (𝑥 = 𝑦 → (∅ ·𝑜 𝑥) = (∅ ·𝑜 𝑦))
43eqeq1d 2750 . 2 (𝑥 = 𝑦 → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 𝑦) = ∅))
5 oveq2 6809 . . 3 (𝑥 = suc 𝑦 → (∅ ·𝑜 𝑥) = (∅ ·𝑜 suc 𝑦))
65eqeq1d 2750 . 2 (𝑥 = suc 𝑦 → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 suc 𝑦) = ∅))
7 oveq2 6809 . . 3 (𝑥 = 𝐴 → (∅ ·𝑜 𝑥) = (∅ ·𝑜 𝐴))
87eqeq1d 2750 . 2 (𝑥 = 𝐴 → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 𝐴) = ∅))
9 om0x 7756 . 2 (∅ ·𝑜 ∅) = ∅
10 oveq1 6808 . . 3 ((∅ ·𝑜 𝑦) = ∅ → ((∅ ·𝑜 𝑦) +𝑜 ∅) = (∅ +𝑜 ∅))
11 0elon 5927 . . . . 5 ∅ ∈ On
12 omsuc 7763 . . . . 5 ((∅ ∈ On ∧ 𝑦 ∈ On) → (∅ ·𝑜 suc 𝑦) = ((∅ ·𝑜 𝑦) +𝑜 ∅))
1311, 12mpan 708 . . . 4 (𝑦 ∈ On → (∅ ·𝑜 suc 𝑦) = ((∅ ·𝑜 𝑦) +𝑜 ∅))
14 oa0 7753 . . . . . . 7 (∅ ∈ On → (∅ +𝑜 ∅) = ∅)
1511, 14ax-mp 5 . . . . . 6 (∅ +𝑜 ∅) = ∅
1615eqcomi 2757 . . . . 5 ∅ = (∅ +𝑜 ∅)
1716a1i 11 . . . 4 (𝑦 ∈ On → ∅ = (∅ +𝑜 ∅))
1813, 17eqeq12d 2763 . . 3 (𝑦 ∈ On → ((∅ ·𝑜 suc 𝑦) = ∅ ↔ ((∅ ·𝑜 𝑦) +𝑜 ∅) = (∅ +𝑜 ∅)))
1910, 18syl5ibr 236 . 2 (𝑦 ∈ On → ((∅ ·𝑜 𝑦) = ∅ → (∅ ·𝑜 suc 𝑦) = ∅))
20 iuneq2 4677 . . . 4 (∀𝑦𝑥 (∅ ·𝑜 𝑦) = ∅ → 𝑦𝑥 (∅ ·𝑜 𝑦) = 𝑦𝑥 ∅)
21 iun0 4716 . . . 4 𝑦𝑥 ∅ = ∅
2220, 21syl6eq 2798 . . 3 (∀𝑦𝑥 (∅ ·𝑜 𝑦) = ∅ → 𝑦𝑥 (∅ ·𝑜 𝑦) = ∅)
23 vex 3331 . . . . 5 𝑥 ∈ V
24 omlim 7770 . . . . . 6 ((∅ ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∅ ·𝑜 𝑥) = 𝑦𝑥 (∅ ·𝑜 𝑦))
2511, 24mpan 708 . . . . 5 ((𝑥 ∈ V ∧ Lim 𝑥) → (∅ ·𝑜 𝑥) = 𝑦𝑥 (∅ ·𝑜 𝑦))
2623, 25mpan 708 . . . 4 (Lim 𝑥 → (∅ ·𝑜 𝑥) = 𝑦𝑥 (∅ ·𝑜 𝑦))
2726eqeq1d 2750 . . 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 7212 1 (𝐴 ∈ On → (∅ ·𝑜 𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1620  wcel 2127  wral 3038  Vcvv 3328  c0 4046   ciun 4660  Oncon0 5872  Lim wlim 5873  suc csuc 5874  (class class class)co 6801   +𝑜 coa 7714   ·𝑜 comu 7715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-pss 3719  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-tp 4314  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-tr 4893  df-id 5162  df-eprel 5167  df-po 5175  df-so 5176  df-fr 5213  df-we 5215  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-pred 5829  df-ord 5875  df-on 5876  df-lim 5877  df-suc 5878  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-om 7219  df-1st 7321  df-2nd 7322  df-wrecs 7564  df-recs 7625  df-rdg 7663  df-oadd 7721  df-omul 7722
This theorem is referenced by:  omord  7805  omwordi  7808  om00  7812  odi  7816  omass  7817  oeoa  7834  omxpenlem  8214
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