Theorem om0r 7619

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 6658	. . 3 ⊢ (𝑥 = ∅ → (∅ ·𝑜 𝑥) = (∅ ·𝑜 ∅))
2	1	eqeq1d 2624	. 2 ⊢ (𝑥 = ∅ → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 ∅) = ∅))
3		oveq2 6658	. . 3 ⊢ (𝑥 = 𝑦 → (∅ ·𝑜 𝑥) = (∅ ·𝑜 𝑦))
4	3	eqeq1d 2624	. 2 ⊢ (𝑥 = 𝑦 → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 𝑦) = ∅))
5		oveq2 6658	. . 3 ⊢ (𝑥 = suc 𝑦 → (∅ ·𝑜 𝑥) = (∅ ·𝑜 suc 𝑦))
6	5	eqeq1d 2624	. 2 ⊢ (𝑥 = suc 𝑦 → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 suc 𝑦) = ∅))
7		oveq2 6658	. . 3 ⊢ (𝑥 = 𝐴 → (∅ ·𝑜 𝑥) = (∅ ·𝑜 𝐴))
8	7	eqeq1d 2624	. 2 ⊢ (𝑥 = 𝐴 → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 𝐴) = ∅))
9		om0x 7599	. 2 ⊢ (∅ ·𝑜 ∅) = ∅
10		oveq1 6657	. . 3 ⊢ ((∅ ·𝑜 𝑦) = ∅ → ((∅ ·𝑜 𝑦) +𝑜 ∅) = (∅ +𝑜 ∅))
11		0elon 5778	. . . . 5 ⊢ ∅ ∈ On
12		omsuc 7606	. . . . 5 ⊢ ((∅ ∈ On ∧ 𝑦 ∈ On) → (∅ ·𝑜 suc 𝑦) = ((∅ ·𝑜 𝑦) +𝑜 ∅))
13	11, 12	mpan 706	. . . 4 ⊢ (𝑦 ∈ On → (∅ ·𝑜 suc 𝑦) = ((∅ ·𝑜 𝑦) +𝑜 ∅))
14		oa0 7596	. . . . . . 7 ⊢ (∅ ∈ On → (∅ +𝑜 ∅) = ∅)
15	11, 14	ax-mp 5	. . . . . 6 ⊢ (∅ +𝑜 ∅) = ∅
16	15	eqcomi 2631	. . . . 5 ⊢ ∅ = (∅ +𝑜 ∅)
17	16	a1i 11	. . . 4 ⊢ (𝑦 ∈ On → ∅ = (∅ +𝑜 ∅))
18	13, 17	eqeq12d 2637	. . 3 ⊢ (𝑦 ∈ On → ((∅ ·𝑜 suc 𝑦) = ∅ ↔ ((∅ ·𝑜 𝑦) +𝑜 ∅) = (∅ +𝑜 ∅)))
19	10, 18	syl5ibr 236	. 2 ⊢ (𝑦 ∈ On → ((∅ ·𝑜 𝑦) = ∅ → (∅ ·𝑜 suc 𝑦) = ∅))
20		iuneq2 4537	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 (∅ ·𝑜 𝑦) = ∅ → ∪ 𝑦 ∈ 𝑥 (∅ ·𝑜 𝑦) = ∪ 𝑦 ∈ 𝑥 ∅)
21		iun0 4576	. . . 4 ⊢ ∪ 𝑦 ∈ 𝑥 ∅ = ∅
22	20, 21	syl6eq 2672	. . 3 ⊢ (∀𝑦 ∈ 𝑥 (∅ ·𝑜 𝑦) = ∅ → ∪ 𝑦 ∈ 𝑥 (∅ ·𝑜 𝑦) = ∅)
23		vex 3203	. . . . 5 ⊢ 𝑥 ∈ V
24		omlim 7613	. . . . . 6 ⊢ ((∅ ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∅ ·𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ ·𝑜 𝑦))
25	11, 24	mpan 706	. . . . 5 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (∅ ·𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ ·𝑜 𝑦))
26	23, 25	mpan 706	. . . 4 ⊢ (Lim 𝑥 → (∅ ·𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ ·𝑜 𝑦))
27	26	eqeq1d 2624	. . 3 ⊢ (Lim 𝑥 → ((∅ ·𝑜 𝑥) = ∅ ↔ ∪ 𝑦 ∈ 𝑥 (∅ ·𝑜 𝑦) = ∅))
28	22, 27	syl5ibr 236	. 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (∅ ·𝑜 𝑦) = ∅ → (∅ ·𝑜 𝑥) = ∅))
29	2, 4, 6, 8, 9, 19, 28	tfinds 7059	1 ⊢ (𝐴 ∈ On → (∅ ·𝑜 𝐴) = ∅)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200  ∅c0 3915  ∪ ciun 4520  Oncon0 5723  Lim wlim 5724  suc csuc 5725  (class class class)co 6650   +_𝑜 coa 7557   ·_𝑜 comu 7558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-omul 7565

This theorem is referenced by:  omord  7648  omwordi  7651  om00  7655  odi  7659  omass  7660  oeoa  7677  omxpenlem  8061