Theorem om1r 8582

Description: Ordinal multiplication with 1. Proposition 8.18(2) of [TakeutiZaring] p. 63. Lemma 2.15 of [Schloeder] p. 5. (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 7440	. . 3 ⊢ (𝑥 = ∅ → (1o ·o 𝑥) = (1o ·o ∅))
2		id 22	. . 3 ⊢ (𝑥 = ∅ → 𝑥 = ∅)
3	1, 2	eqeq12d 2752	. 2 ⊢ (𝑥 = ∅ → ((1o ·o 𝑥) = 𝑥 ↔ (1o ·o ∅) = ∅))
4		oveq2 7440	. . 3 ⊢ (𝑥 = 𝑦 → (1o ·o 𝑥) = (1o ·o 𝑦))
5		id 22	. . 3 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
6	4, 5	eqeq12d 2752	. 2 ⊢ (𝑥 = 𝑦 → ((1o ·o 𝑥) = 𝑥 ↔ (1o ·o 𝑦) = 𝑦))
7		oveq2 7440	. . 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 2752	. 2 ⊢ (𝑥 = suc 𝑦 → ((1o ·o 𝑥) = 𝑥 ↔ (1o ·o suc 𝑦) = suc 𝑦))
10		oveq2 7440	. . 3 ⊢ (𝑥 = 𝐴 → (1o ·o 𝑥) = (1o ·o 𝐴))
11		id 22	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
12	10, 11	eqeq12d 2752	. 2 ⊢ (𝑥 = 𝐴 → ((1o ·o 𝑥) = 𝑥 ↔ (1o ·o 𝐴) = 𝐴))
13		1on 8519	. . 3 ⊢ 1o ∈ On
14		om0 8556	. . 3 ⊢ (1o ∈ On → (1o ·o ∅) = ∅)
15	13, 14	ax-mp 5	. 2 ⊢ (1o ·o ∅) = ∅
16		omsuc 8565	. . . . . 6 ⊢ ((1o ∈ On ∧ 𝑦 ∈ On) → (1o ·o suc 𝑦) = ((1o ·o 𝑦) +o 1o))
17	13, 16	mpan 690	. . . . 5 ⊢ (𝑦 ∈ On → (1o ·o suc 𝑦) = ((1o ·o 𝑦) +o 1o))
18		oveq1 7439	. . . . 5 ⊢ ((1o ·o 𝑦) = 𝑦 → ((1o ·o 𝑦) +o 1o) = (𝑦 +o 1o))
19	17, 18	sylan9eq 2796	. . . 4 ⊢ ((𝑦 ∈ On ∧ (1o ·o 𝑦) = 𝑦) → (1o ·o suc 𝑦) = (𝑦 +o 1o))
20		oa1suc 8570	. . . . 5 ⊢ (𝑦 ∈ On → (𝑦 +o 1o) = suc 𝑦)
21	20	adantr 480	. . . 4 ⊢ ((𝑦 ∈ On ∧ (1o ·o 𝑦) = 𝑦) → (𝑦 +o 1o) = suc 𝑦)
22	19, 21	eqtrd 2776	. . 3 ⊢ ((𝑦 ∈ On ∧ (1o ·o 𝑦) = 𝑦) → (1o ·o suc 𝑦) = suc 𝑦)
23	22	ex 412	. 2 ⊢ (𝑦 ∈ On → ((1o ·o 𝑦) = 𝑦 → (1o ·o suc 𝑦) = suc 𝑦))
24		iuneq2 5010	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 (1o ·o 𝑦) = 𝑦 → ∪ 𝑦 ∈ 𝑥 (1o ·o 𝑦) = ∪ 𝑦 ∈ 𝑥 𝑦)
25		uniiun 5057	. . . 4 ⊢ ∪ 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦
26	24, 25	eqtr4di 2794	. . 3 ⊢ (∀𝑦 ∈ 𝑥 (1o ·o 𝑦) = 𝑦 → ∪ 𝑦 ∈ 𝑥 (1o ·o 𝑦) = ∪ 𝑥)
27		vex 3483	. . . . 5 ⊢ 𝑥 ∈ V
28		omlim 8572	. . . . . 6 ⊢ ((1o ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (1o ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (1o ·o 𝑦))
29	13, 28	mpan 690	. . . . 5 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (1o ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (1o ·o 𝑦))
30	27, 29	mpan 690	. . . 4 ⊢ (Lim 𝑥 → (1o ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (1o ·o 𝑦))
31		limuni 6444	. . . 4 ⊢ (Lim 𝑥 → 𝑥 = ∪ 𝑥)
32	30, 31	eqeq12d 2752	. . 3 ⊢ (Lim 𝑥 → ((1o ·o 𝑥) = 𝑥 ↔ ∪ 𝑦 ∈ 𝑥 (1o ·o 𝑦) = ∪ 𝑥))
33	26, 32	imbitrrid 246	. 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (1o ·o 𝑦) = 𝑦 → (1o ·o 𝑥) = 𝑥))
34	3, 6, 9, 12, 15, 23, 33	tfinds 7882	1 ⊢ (𝐴 ∈ On → (1o ·o 𝐴) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∀wral 3060  Vcvv 3479  ∅c0 4332  ∪ cuni 4906  ∪ ciun 4990  Oncon0 6383  Lim wlim 6384  suc csuc 6385  (class class class)co 7432  1_oc1o 8500   +_o coa 8504   ·_o comu 8505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-oadd 8511  df-omul 8512

This theorem is referenced by:  oe1  8583  omword2  8613  om1om1r  43302  omabs2  43350  omcl2  43351