Theorem om1r 7510

Description: Ordinal multiplication with 1. Proposition 8.18(2) of [TakeutiZaring] p. 63. (Contributed by NM, 3-Aug-2004.)

Assertion

Ref	Expression
om1r	⊢ (𝐴 ∈ On → (1𝑜 ·𝑜 𝐴) = 𝐴)

Proof of Theorem om1r

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6557	. . 3 ⊢ (𝑥 = ∅ → (1𝑜 ·𝑜 𝑥) = (1𝑜 ·𝑜 ∅))
2		id 22	. . 3 ⊢ (𝑥 = ∅ → 𝑥 = ∅)
3	1, 2	eqeq12d 2625	. 2 ⊢ (𝑥 = ∅ → ((1𝑜 ·𝑜 𝑥) = 𝑥 ↔ (1𝑜 ·𝑜 ∅) = ∅))
4		oveq2 6557	. . 3 ⊢ (𝑥 = 𝑦 → (1𝑜 ·𝑜 𝑥) = (1𝑜 ·𝑜 𝑦))
5		id 22	. . 3 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
6	4, 5	eqeq12d 2625	. 2 ⊢ (𝑥 = 𝑦 → ((1𝑜 ·𝑜 𝑥) = 𝑥 ↔ (1𝑜 ·𝑜 𝑦) = 𝑦))
7		oveq2 6557	. . 3 ⊢ (𝑥 = suc 𝑦 → (1𝑜 ·𝑜 𝑥) = (1𝑜 ·𝑜 suc 𝑦))
8		id 22	. . 3 ⊢ (𝑥 = suc 𝑦 → 𝑥 = suc 𝑦)
9	7, 8	eqeq12d 2625	. 2 ⊢ (𝑥 = suc 𝑦 → ((1𝑜 ·𝑜 𝑥) = 𝑥 ↔ (1𝑜 ·𝑜 suc 𝑦) = suc 𝑦))
10		oveq2 6557	. . 3 ⊢ (𝑥 = 𝐴 → (1𝑜 ·𝑜 𝑥) = (1𝑜 ·𝑜 𝐴))
11		id 22	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
12	10, 11	eqeq12d 2625	. 2 ⊢ (𝑥 = 𝐴 → ((1𝑜 ·𝑜 𝑥) = 𝑥 ↔ (1𝑜 ·𝑜 𝐴) = 𝐴))
13		om0x 7486	. 2 ⊢ (1𝑜 ·𝑜 ∅) = ∅
14		1on 7454	. . . . . 6 ⊢ 1𝑜 ∈ On
15		omsuc 7493	. . . . . 6 ⊢ ((1𝑜 ∈ On ∧ 𝑦 ∈ On) → (1𝑜 ·𝑜 suc 𝑦) = ((1𝑜 ·𝑜 𝑦) +𝑜 1𝑜))
16	14, 15	mpan 702	. . . . 5 ⊢ (𝑦 ∈ On → (1𝑜 ·𝑜 suc 𝑦) = ((1𝑜 ·𝑜 𝑦) +𝑜 1𝑜))
17		oveq1 6556	. . . . 5 ⊢ ((1𝑜 ·𝑜 𝑦) = 𝑦 → ((1𝑜 ·𝑜 𝑦) +𝑜 1𝑜) = (𝑦 +𝑜 1𝑜))
18	16, 17	sylan9eq 2664	. . . 4 ⊢ ((𝑦 ∈ On ∧ (1𝑜 ·𝑜 𝑦) = 𝑦) → (1𝑜 ·𝑜 suc 𝑦) = (𝑦 +𝑜 1𝑜))
19		oa1suc 7498	. . . . 5 ⊢ (𝑦 ∈ On → (𝑦 +𝑜 1𝑜) = suc 𝑦)
20	19	adantr 480	. . . 4 ⊢ ((𝑦 ∈ On ∧ (1𝑜 ·𝑜 𝑦) = 𝑦) → (𝑦 +𝑜 1𝑜) = suc 𝑦)
21	18, 20	eqtrd 2644	. . 3 ⊢ ((𝑦 ∈ On ∧ (1𝑜 ·𝑜 𝑦) = 𝑦) → (1𝑜 ·𝑜 suc 𝑦) = suc 𝑦)
22	21	ex 449	. 2 ⊢ (𝑦 ∈ On → ((1𝑜 ·𝑜 𝑦) = 𝑦 → (1𝑜 ·𝑜 suc 𝑦) = suc 𝑦))
23		iuneq2 4473	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 (1𝑜 ·𝑜 𝑦) = 𝑦 → ∪ 𝑦 ∈ 𝑥 (1𝑜 ·𝑜 𝑦) = ∪ 𝑦 ∈ 𝑥 𝑦)
24		uniiun 4509	. . . 4 ⊢ ∪ 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦
25	23, 24	syl6eqr 2662	. . 3 ⊢ (∀𝑦 ∈ 𝑥 (1𝑜 ·𝑜 𝑦) = 𝑦 → ∪ 𝑦 ∈ 𝑥 (1𝑜 ·𝑜 𝑦) = ∪ 𝑥)
26		vex 3176	. . . . 5 ⊢ 𝑥 ∈ V
27		omlim 7500	. . . . . 6 ⊢ ((1𝑜 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (1𝑜 ·𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (1𝑜 ·𝑜 𝑦))
28	14, 27	mpan 702	. . . . 5 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (1𝑜 ·𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (1𝑜 ·𝑜 𝑦))
29	26, 28	mpan 702	. . . 4 ⊢ (Lim 𝑥 → (1𝑜 ·𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (1𝑜 ·𝑜 𝑦))
30		limuni 5702	. . . 4 ⊢ (Lim 𝑥 → 𝑥 = ∪ 𝑥)
31	29, 30	eqeq12d 2625	. . 3 ⊢ (Lim 𝑥 → ((1𝑜 ·𝑜 𝑥) = 𝑥 ↔ ∪ 𝑦 ∈ 𝑥 (1𝑜 ·𝑜 𝑦) = ∪ 𝑥))
32	25, 31	syl5ibr 235	. 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (1𝑜 ·𝑜 𝑦) = 𝑦 → (1𝑜 ·𝑜 𝑥) = 𝑥))
33	3, 6, 9, 12, 13, 22, 32	tfinds 6951	1 ⊢ (𝐴 ∈ On → (1𝑜 ·𝑜 𝐴) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  ∅c0 3874  ∪ cuni 4372  ∪ ciun 4455  Oncon0 5640  Lim wlim 5641  suc csuc 5642  (class class class)co 6549  1_𝑜c1o 7440   +_𝑜 coa 7444   ·_𝑜 comu 7445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452

This theorem is referenced by:  oe1  7511  omword2  7541