Theorem om0r 8525

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 7416	. . 3 ⊢ (𝑥 = ∅ → (∅ ·o 𝑥) = (∅ ·o ∅))
2	1	eqeq1d 2762	. 2 ⊢ (𝑥 = ∅ → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o ∅) = ∅))
3		oveq2 7416	. . 3 ⊢ (𝑥 = 𝑦 → (∅ ·o 𝑥) = (∅ ·o 𝑦))
4	3	eqeq1d 2762	. 2 ⊢ (𝑥 = 𝑦 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o 𝑦) = ∅))
5		oveq2 7416	. . 3 ⊢ (𝑥 = suc 𝑦 → (∅ ·o 𝑥) = (∅ ·o suc 𝑦))
6	5	eqeq1d 2762	. 2 ⊢ (𝑥 = suc 𝑦 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o suc 𝑦) = ∅))
7		oveq2 7416	. . 3 ⊢ (𝑥 = 𝐴 → (∅ ·o 𝑥) = (∅ ·o 𝐴))
8	7	eqeq1d 2762	. 2 ⊢ (𝑥 = 𝐴 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o 𝐴) = ∅))
9		0elon 6407	. . 3 ⊢ ∅ ∈ On
10		om0 8503	. . 3 ⊢ (∅ ∈ On → (∅ ·o ∅) = ∅)
11	9, 10	ax-mp 5	. 2 ⊢ (∅ ·o ∅) = ∅
12		oveq1 7415	. . 3 ⊢ ((∅ ·o 𝑦) = ∅ → ((∅ ·o 𝑦) +o ∅) = (∅ +o ∅))
13		omsuc 8512	. . . . 5 ⊢ ((∅ ∈ On ∧ 𝑦 ∈ On) → (∅ ·o suc 𝑦) = ((∅ ·o 𝑦) +o ∅))
14	9, 13	mpan 703	. . . 4 ⊢ (𝑦 ∈ On → (∅ ·o suc 𝑦) = ((∅ ·o 𝑦) +o ∅))
15		oa0 8502	. . . . . . 7 ⊢ (∅ ∈ On → (∅ +o ∅) = ∅)
16	9, 15	ax-mp 5	. . . . . 6 ⊢ (∅ +o ∅) = ∅
17	16	eqcomi 2769	. . . . 5 ⊢ ∅ = (∅ +o ∅)
18	17	a1i 11	. . . 4 ⊢ (𝑦 ∈ On → ∅ = (∅ +o ∅))
19	14, 18	eqeq12d 2776	. . 3 ⊢ (𝑦 ∈ On → ((∅ ·o suc 𝑦) = ∅ ↔ ((∅ ·o 𝑦) +o ∅) = (∅ +o ∅)))
20	12, 19	imbitrrid 249	. 2 ⊢ (𝑦 ∈ On → ((∅ ·o 𝑦) = ∅ → (∅ ·o suc 𝑦) = ∅))
21		iuneq2 4970	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 (∅ ·o 𝑦) = ∅ → ∪ 𝑦 ∈ 𝑥 (∅ ·o 𝑦) = ∪ 𝑦 ∈ 𝑥 ∅)
22		iun0 5019	. . . 4 ⊢ ∪ 𝑦 ∈ 𝑥 ∅ = ∅
23	21, 22	eqtrdi 2811	. . 3 ⊢ (∀𝑦 ∈ 𝑥 (∅ ·o 𝑦) = ∅ → ∪ 𝑦 ∈ 𝑥 (∅ ·o 𝑦) = ∅)
24		vex 3454	. . . . 5 ⊢ 𝑥 ∈ V
25		omlim 8519	. . . . . 6 ⊢ ((∅ ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∅ ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ ·o 𝑦))
26	9, 25	mpan 703	. . . . 5 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (∅ ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ ·o 𝑦))
27	24, 26	mpan 703	. . . 4 ⊢ (Lim 𝑥 → (∅ ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ ·o 𝑦))
28	27	eqeq1d 2762	. . 3 ⊢ (Lim 𝑥 → ((∅ ·o 𝑥) = ∅ ↔ ∪ 𝑦 ∈ 𝑥 (∅ ·o 𝑦) = ∅))
29	23, 28	imbitrrid 249	. 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (∅ ·o 𝑦) = ∅ → (∅ ·o 𝑥) = ∅))
30	2, 4, 6, 8, 11, 20, 29	tfinds 7854	1 ⊢ (𝐴 ∈ On → (∅ ·o 𝐴) = ∅)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  Vcvv 3450  ∅c0 4278  ∪ ciun 4950  Oncon0 6351  Lim wlim 6352  suc csuc 6353  (class class class)co 7408   +_o coa 8451   ·_o comu 8452

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-oadd 8458  df-omul 8459

This theorem is used by:  omord  8554  omwordi  8557  om00  8561  odi  8565  omass  8566  oeoa  8584  omxpenlem  9075  onmcl  44276  omcl2  44278  omcl3g  44279