Theorem om0r 8547

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 7428	. . 3 ⊢ (𝑥 = ∅ → (∅ ·o 𝑥) = (∅ ·o ∅))
2	1	eqeq1d 2763	. 2 ⊢ (𝑥 = ∅ → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o ∅) = ∅))
3		oveq2 7428	. . 3 ⊢ (𝑥 = 𝑦 → (∅ ·o 𝑥) = (∅ ·o 𝑦))
4	3	eqeq1d 2763	. 2 ⊢ (𝑥 = 𝑦 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o 𝑦) = ∅))
5		oveq2 7428	. . 3 ⊢ (𝑥 = suc 𝑦 → (∅ ·o 𝑥) = (∅ ·o suc 𝑦))
6	5	eqeq1d 2763	. 2 ⊢ (𝑥 = suc 𝑦 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o suc 𝑦) = ∅))
7		oveq2 7428	. . 3 ⊢ (𝑥 = 𝐴 → (∅ ·o 𝑥) = (∅ ·o 𝐴))
8	7	eqeq1d 2763	. 2 ⊢ (𝑥 = 𝐴 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o 𝐴) = ∅))
9		0elon 6418	. . 3 ⊢ ∅ ∈ On
10		om0 8525	. . 3 ⊢ (∅ ∈ On → (∅ ·o ∅) = ∅)
11	9, 10	ax-mp 5	. 2 ⊢ (∅ ·o ∅) = ∅
12		oveq1 7427	. . 3 ⊢ ((∅ ·o 𝑦) = ∅ → ((∅ ·o 𝑦) +o ∅) = (∅ +o ∅))
13		omsuc 8534	. . . . 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 8524	. . . . . . 7 ⊢ (∅ ∈ On → (∅ +o ∅) = ∅)
16	9, 15	ax-mp 5	. . . . . 6 ⊢ (∅ +o ∅) = ∅
17	16	eqcomi 2770	. . . . 5 ⊢ ∅ = (∅ +o ∅)
18	17	a1i 11	. . . 4 ⊢ (𝑦 ∈ On → ∅ = (∅ +o ∅))
19	14, 18	eqeq12d 2777	. . 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 4971	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 (∅ ·o 𝑦) = ∅ → ∪ 𝑦 ∈ 𝑥 (∅ ·o 𝑦) = ∪ 𝑦 ∈ 𝑥 ∅)
22		iun0 5020	. . . 4 ⊢ ∪ 𝑦 ∈ 𝑥 ∅ = ∅
23	21, 22	eqtrdi 2812	. . 3 ⊢ (∀𝑦 ∈ 𝑥 (∅ ·o 𝑦) = ∅ → ∪ 𝑦 ∈ 𝑥 (∅ ·o 𝑦) = ∅)
24		vex 3455	. . . . 5 ⊢ 𝑥 ∈ V
25		omlim 8541	. . . . . 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 2763	. . 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 7871	1 ⊢ (𝐴 ∈ On → (∅ ·o 𝐴) = ∅)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  Vcvv 3451  ∅c0 4279  ∪ ciun 4951  Oncon0 6362  Lim wlim 6363  suc csuc 6364  (class class class)co 7420   +_o coa 8473   ·_o comu 8474

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 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7751

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-oadd 8480  df-omul 8481

This theorem is used by:  omord  8576  omwordi  8579  om00  8583  odi  8587  omass  8588  oeoa  8606  omxpenlem  9097  onmcl  44332  omcl2  44334  omcl3g  44335