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.

Step	Hyp	Ref	Expression
1		oveq2 6823	. . 3 ⊢ (𝑥 = ∅ → (∅ ·𝑜 𝑥) = (∅ ·𝑜 ∅))
2	1	eqeq1d 2763	. 2 ⊢ (𝑥 = ∅ → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 ∅) = ∅))
3		oveq2 6823	. . 3 ⊢ (𝑥 = 𝑦 → (∅ ·𝑜 𝑥) = (∅ ·𝑜 𝑦))
4	3	eqeq1d 2763	. 2 ⊢ (𝑥 = 𝑦 → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 𝑦) = ∅))
5		oveq2 6823	. . 3 ⊢ (𝑥 = suc 𝑦 → (∅ ·𝑜 𝑥) = (∅ ·𝑜 suc 𝑦))
6	5	eqeq1d 2763	. 2 ⊢ (𝑥 = suc 𝑦 → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 suc 𝑦) = ∅))
7		oveq2 6823	. . 3 ⊢ (𝑥 = 𝐴 → (∅ ·𝑜 𝑥) = (∅ ·𝑜 𝐴))
8	7	eqeq1d 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 ∅))
13	11, 12	mpan 708	. . . 4 ⊢ (𝑦 ∈ On → (∅ ·𝑜 suc 𝑦) = ((∅ ·𝑜 𝑦) +𝑜 ∅))
14		oa0 7768	. . . . . . 7 ⊢ (∅ ∈ On → (∅ +𝑜 ∅) = ∅)
15	11, 14	ax-mp 5	. . . . . 6 ⊢ (∅ +𝑜 ∅) = ∅
16	15	eqcomi 2770	. . . . 5 ⊢ ∅ = (∅ +𝑜 ∅)
17	16	a1i 11	. . . 4 ⊢ (𝑦 ∈ On → ∅ = (∅ +𝑜 ∅))
18	13, 17	eqeq12d 2776	. . 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 4690	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 (∅ ·𝑜 𝑦) = ∅ → ∪ 𝑦 ∈ 𝑥 (∅ ·𝑜 𝑦) = ∪ 𝑦 ∈ 𝑥 ∅)
21		iun0 4729	. . . 4 ⊢ ∪ 𝑦 ∈ 𝑥 ∅ = ∅
22	20, 21	syl6eq 2811	. . 3 ⊢ (∀𝑦 ∈ 𝑥 (∅ ·𝑜 𝑦) = ∅ → ∪ 𝑦 ∈ 𝑥 (∅ ·𝑜 𝑦) = ∅)
23		vex 3344	. . . . 5 ⊢ 𝑥 ∈ V
24		omlim 7785	. . . . . 6 ⊢ ((∅ ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∅ ·𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ ·𝑜 𝑦))
25	11, 24	mpan 708	. . . . 5 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (∅ ·𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ ·𝑜 𝑦))
26	23, 25	mpan 708	. . . 4 ⊢ (Lim 𝑥 → (∅ ·𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ ·𝑜 𝑦))
27	26	eqeq1d 2763	. . 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 7226	1 ⊢ (𝐴 ∈ On → (∅ ·𝑜 𝐴) = ∅)

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