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Theorem om0r 7506
 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 6557 . . 3 (𝑥 = ∅ → (∅ ·𝑜 𝑥) = (∅ ·𝑜 ∅))
21eqeq1d 2612 . 2 (𝑥 = ∅ → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 ∅) = ∅))
3 oveq2 6557 . . 3 (𝑥 = 𝑦 → (∅ ·𝑜 𝑥) = (∅ ·𝑜 𝑦))
43eqeq1d 2612 . 2 (𝑥 = 𝑦 → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 𝑦) = ∅))
5 oveq2 6557 . . 3 (𝑥 = suc 𝑦 → (∅ ·𝑜 𝑥) = (∅ ·𝑜 suc 𝑦))
65eqeq1d 2612 . 2 (𝑥 = suc 𝑦 → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 suc 𝑦) = ∅))
7 oveq2 6557 . . 3 (𝑥 = 𝐴 → (∅ ·𝑜 𝑥) = (∅ ·𝑜 𝐴))
87eqeq1d 2612 . 2 (𝑥 = 𝐴 → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 𝐴) = ∅))
9 om0x 7486 . 2 (∅ ·𝑜 ∅) = ∅
10 oveq1 6556 . . 3 ((∅ ·𝑜 𝑦) = ∅ → ((∅ ·𝑜 𝑦) +𝑜 ∅) = (∅ +𝑜 ∅))
11 0elon 5695 . . . . 5 ∅ ∈ On
12 omsuc 7493 . . . . 5 ((∅ ∈ On ∧ 𝑦 ∈ On) → (∅ ·𝑜 suc 𝑦) = ((∅ ·𝑜 𝑦) +𝑜 ∅))
1311, 12mpan 702 . . . 4 (𝑦 ∈ On → (∅ ·𝑜 suc 𝑦) = ((∅ ·𝑜 𝑦) +𝑜 ∅))
14 oa0 7483 . . . . . . 7 (∅ ∈ On → (∅ +𝑜 ∅) = ∅)
1511, 14ax-mp 5 . . . . . 6 (∅ +𝑜 ∅) = ∅
1615eqcomi 2619 . . . . 5 ∅ = (∅ +𝑜 ∅)
1716a1i 11 . . . 4 (𝑦 ∈ On → ∅ = (∅ +𝑜 ∅))
1813, 17eqeq12d 2625 . . 3 (𝑦 ∈ On → ((∅ ·𝑜 suc 𝑦) = ∅ ↔ ((∅ ·𝑜 𝑦) +𝑜 ∅) = (∅ +𝑜 ∅)))
1910, 18syl5ibr 235 . 2 (𝑦 ∈ On → ((∅ ·𝑜 𝑦) = ∅ → (∅ ·𝑜 suc 𝑦) = ∅))
20 iuneq2 4473 . . . 4 (∀𝑦𝑥 (∅ ·𝑜 𝑦) = ∅ → 𝑦𝑥 (∅ ·𝑜 𝑦) = 𝑦𝑥 ∅)
21 iun0 4512 . . . 4 𝑦𝑥 ∅ = ∅
2220, 21syl6eq 2660 . . 3 (∀𝑦𝑥 (∅ ·𝑜 𝑦) = ∅ → 𝑦𝑥 (∅ ·𝑜 𝑦) = ∅)
23 vex 3176 . . . . 5 𝑥 ∈ V
24 omlim 7500 . . . . . 6 ((∅ ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∅ ·𝑜 𝑥) = 𝑦𝑥 (∅ ·𝑜 𝑦))
2511, 24mpan 702 . . . . 5 ((𝑥 ∈ V ∧ Lim 𝑥) → (∅ ·𝑜 𝑥) = 𝑦𝑥 (∅ ·𝑜 𝑦))
2623, 25mpan 702 . . . 4 (Lim 𝑥 → (∅ ·𝑜 𝑥) = 𝑦𝑥 (∅ ·𝑜 𝑦))
2726eqeq1d 2612 . . 3 (Lim 𝑥 → ((∅ ·𝑜 𝑥) = ∅ ↔ 𝑦𝑥 (∅ ·𝑜 𝑦) = ∅))
2822, 27syl5ibr 235 . 2 (Lim 𝑥 → (∀𝑦𝑥 (∅ ·𝑜 𝑦) = ∅ → (∅ ·𝑜 𝑥) = ∅))
292, 4, 6, 8, 9, 19, 28tfinds 6951 1 (𝐴 ∈ On → (∅ ·𝑜 𝐴) = ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  ∅c0 3874  ∪ ciun 4455  Oncon0 5640  Lim wlim 5641  suc csuc 5642  (class class class)co 6549   +𝑜 coa 7444   ·𝑜 comu 7445 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-omul 7452 This theorem is referenced by:  omord  7535  omwordi  7538  om00  7542  odi  7546  omass  7547  oeoa  7564  omxpenlem  7946
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