Theorem om0r 8474

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.

Step	Hyp	Ref	Expression
1		oveq2 7375	. . 3 ⊢ (𝑥 = ∅ → (∅ ·o 𝑥) = (∅ ·o ∅))
2	1	eqeq1d 2739	. 2 ⊢ (𝑥 = ∅ → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o ∅) = ∅))
3		oveq2 7375	. . 3 ⊢ (𝑥 = 𝑦 → (∅ ·o 𝑥) = (∅ ·o 𝑦))
4	3	eqeq1d 2739	. 2 ⊢ (𝑥 = 𝑦 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o 𝑦) = ∅))
5		oveq2 7375	. . 3 ⊢ (𝑥 = suc 𝑦 → (∅ ·o 𝑥) = (∅ ·o suc 𝑦))
6	5	eqeq1d 2739	. 2 ⊢ (𝑥 = suc 𝑦 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o suc 𝑦) = ∅))
7		oveq2 7375	. . 3 ⊢ (𝑥 = 𝐴 → (∅ ·o 𝑥) = (∅ ·o 𝐴))
8	7	eqeq1d 2739	. 2 ⊢ (𝑥 = 𝐴 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o 𝐴) = ∅))
9		0elon 6379	. . 3 ⊢ ∅ ∈ On
10		om0 8452	. . 3 ⊢ (∅ ∈ On → (∅ ·o ∅) = ∅)
11	9, 10	ax-mp 5	. 2 ⊢ (∅ ·o ∅) = ∅
12		oveq1 7374	. . 3 ⊢ ((∅ ·o 𝑦) = ∅ → ((∅ ·o 𝑦) +o ∅) = (∅ +o ∅))
13		omsuc 8461	. . . . 5 ⊢ ((∅ ∈ On ∧ 𝑦 ∈ On) → (∅ ·o suc 𝑦) = ((∅ ·o 𝑦) +o ∅))
14	9, 13	mpan 691	. . . 4 ⊢ (𝑦 ∈ On → (∅ ·o suc 𝑦) = ((∅ ·o 𝑦) +o ∅))
15		oa0 8451	. . . . . . 7 ⊢ (∅ ∈ On → (∅ +o ∅) = ∅)
16	9, 15	ax-mp 5	. . . . . 6 ⊢ (∅ +o ∅) = ∅
17	16	eqcomi 2746	. . . . 5 ⊢ ∅ = (∅ +o ∅)
18	17	a1i 11	. . . 4 ⊢ (𝑦 ∈ On → ∅ = (∅ +o ∅))
19	14, 18	eqeq12d 2753	. . 3 ⊢ (𝑦 ∈ On → ((∅ ·o suc 𝑦) = ∅ ↔ ((∅ ·o 𝑦) +o ∅) = (∅ +o ∅)))
20	12, 19	imbitrrid 246	. 2 ⊢ (𝑦 ∈ On → ((∅ ·o 𝑦) = ∅ → (∅ ·o suc 𝑦) = ∅))
21		iuneq2 4954	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 (∅ ·o 𝑦) = ∅ → ∪ 𝑦 ∈ 𝑥 (∅ ·o 𝑦) = ∪ 𝑦 ∈ 𝑥 ∅)
22		iun0 5005	. . . 4 ⊢ ∪ 𝑦 ∈ 𝑥 ∅ = ∅
23	21, 22	eqtrdi 2788	. . 3 ⊢ (∀𝑦 ∈ 𝑥 (∅ ·o 𝑦) = ∅ → ∪ 𝑦 ∈ 𝑥 (∅ ·o 𝑦) = ∅)
24		vex 3434	. . . . 5 ⊢ 𝑥 ∈ V
25		omlim 8468	. . . . . 6 ⊢ ((∅ ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∅ ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ ·o 𝑦))
26	9, 25	mpan 691	. . . . 5 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (∅ ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ ·o 𝑦))
27	24, 26	mpan 691	. . . 4 ⊢ (Lim 𝑥 → (∅ ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ ·o 𝑦))
28	27	eqeq1d 2739	. . 3 ⊢ (Lim 𝑥 → ((∅ ·o 𝑥) = ∅ ↔ ∪ 𝑦 ∈ 𝑥 (∅ ·o 𝑦) = ∅))
29	23, 28	imbitrrid 246	. 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (∅ ·o 𝑦) = ∅ → (∅ ·o 𝑥) = ∅))
30	2, 4, 6, 8, 11, 20, 29	tfinds 7811	1 ⊢ (𝐴 ∈ On → (∅ ·o 𝐴) = ∅)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3430  ∅c0 4274  ∪ ciun 4934  Oncon0 6324  Lim wlim 6325  suc csuc 6326  (class class class)co 7367   +_o coa 8402   ·_o comu 8403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pr 5376  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6266  df-ord 6327  df-on 6328  df-lim 6329  df-suc 6330  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-oadd 8409  df-omul 8410

This theorem is referenced by:  omord  8503  omwordi  8506  om00  8510  odi  8514  omass  8515  oeoa  8533  omxpenlem  9016  onmcl  43759  omcl2  43761  omcl3g  43762