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Theorem om0r 7791
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 6823 . . 3 (𝑥 = ∅ → (∅ ·𝑜 𝑥) = (∅ ·𝑜 ∅))
21eqeq1d 2763 . 2 (𝑥 = ∅ → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 ∅) = ∅))
3 oveq2 6823 . . 3 (𝑥 = 𝑦 → (∅ ·𝑜 𝑥) = (∅ ·𝑜 𝑦))
43eqeq1d 2763 . 2 (𝑥 = 𝑦 → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 𝑦) = ∅))
5 oveq2 6823 . . 3 (𝑥 = suc 𝑦 → (∅ ·𝑜 𝑥) = (∅ ·𝑜 suc 𝑦))
65eqeq1d 2763 . 2 (𝑥 = suc 𝑦 → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 suc 𝑦) = ∅))
7 oveq2 6823 . . 3 (𝑥 = 𝐴 → (∅ ·𝑜 𝑥) = (∅ ·𝑜 𝐴))
87eqeq1d 2763 . 2 (𝑥 = 𝐴 → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 𝐴) = ∅))
9 om0x 7771 . 2 (∅ ·𝑜 ∅) = ∅
10 oveq1 6822 . . 3 ((∅ ·𝑜 𝑦) = ∅ → ((∅ ·𝑜 𝑦) +𝑜 ∅) = (∅ +𝑜 ∅))
11 0elon 5940 . . . . 5 ∅ ∈ On
12 omsuc 7778 . . . . 5 ((∅ ∈ On ∧ 𝑦 ∈ On) → (∅ ·𝑜 suc 𝑦) = ((∅ ·𝑜 𝑦) +𝑜 ∅))
1311, 12mpan 708 . . . 4 (𝑦 ∈ On → (∅ ·𝑜 suc 𝑦) = ((∅ ·𝑜 𝑦) +𝑜 ∅))
14 oa0 7768 . . . . . . 7 (∅ ∈ On → (∅ +𝑜 ∅) = ∅)
1511, 14ax-mp 5 . . . . . 6 (∅ +𝑜 ∅) = ∅
1615eqcomi 2770 . . . . 5 ∅ = (∅ +𝑜 ∅)
1716a1i 11 . . . 4 (𝑦 ∈ On → ∅ = (∅ +𝑜 ∅))
1813, 17eqeq12d 2776 . . 3 (𝑦 ∈ On → ((∅ ·𝑜 suc 𝑦) = ∅ ↔ ((∅ ·𝑜 𝑦) +𝑜 ∅) = (∅ +𝑜 ∅)))
1910, 18syl5ibr 236 . 2 (𝑦 ∈ On → ((∅ ·𝑜 𝑦) = ∅ → (∅ ·𝑜 suc 𝑦) = ∅))
20 iuneq2 4690 . . . 4 (∀𝑦𝑥 (∅ ·𝑜 𝑦) = ∅ → 𝑦𝑥 (∅ ·𝑜 𝑦) = 𝑦𝑥 ∅)
21 iun0 4729 . . . 4 𝑦𝑥 ∅ = ∅
2220, 21syl6eq 2811 . . 3 (∀𝑦𝑥 (∅ ·𝑜 𝑦) = ∅ → 𝑦𝑥 (∅ ·𝑜 𝑦) = ∅)
23 vex 3344 . . . . 5 𝑥 ∈ V
24 omlim 7785 . . . . . 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 2763 . . 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 7226 1 (𝐴 ∈ On → (∅ ·𝑜 𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2140  wral 3051  Vcvv 3341  c0 4059   ciun 4673  Oncon0 5885  Lim wlim 5886  suc csuc 5887  (class class class)co 6815   +𝑜 coa 7728   ·𝑜 comu 7729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-tp 4327  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-tr 4906  df-id 5175  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-we 5228  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-pred 5842  df-ord 5888  df-on 5889  df-lim 5890  df-suc 5891  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-om 7233  df-1st 7335  df-2nd 7336  df-wrecs 7578  df-recs 7639  df-rdg 7677  df-oadd 7735  df-omul 7736
This theorem is referenced by:  omord  7820  omwordi  7823  om00  7827  odi  7831  omass  7832  oeoa  7849  omxpenlem  8229
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