Theorem om1r 8564

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 7427	. . 3 ⊢ (𝑥 = ∅ → (1o ·o 𝑥) = (1o ·o ∅))
2		id 22	. . 3 ⊢ (𝑥 = ∅ → 𝑥 = ∅)
3	1, 2	eqeq12d 2741	. 2 ⊢ (𝑥 = ∅ → ((1o ·o 𝑥) = 𝑥 ↔ (1o ·o ∅) = ∅))
4		oveq2 7427	. . 3 ⊢ (𝑥 = 𝑦 → (1o ·o 𝑥) = (1o ·o 𝑦))
5		id 22	. . 3 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
6	4, 5	eqeq12d 2741	. 2 ⊢ (𝑥 = 𝑦 → ((1o ·o 𝑥) = 𝑥 ↔ (1o ·o 𝑦) = 𝑦))
7		oveq2 7427	. . 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 2741	. 2 ⊢ (𝑥 = suc 𝑦 → ((1o ·o 𝑥) = 𝑥 ↔ (1o ·o suc 𝑦) = suc 𝑦))
10		oveq2 7427	. . 3 ⊢ (𝑥 = 𝐴 → (1o ·o 𝑥) = (1o ·o 𝐴))
11		id 22	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
12	10, 11	eqeq12d 2741	. 2 ⊢ (𝑥 = 𝐴 → ((1o ·o 𝑥) = 𝑥 ↔ (1o ·o 𝐴) = 𝐴))
13		1on 8499	. . 3 ⊢ 1o ∈ On
14		om0 8538	. . 3 ⊢ (1o ∈ On → (1o ·o ∅) = ∅)
15	13, 14	ax-mp 5	. 2 ⊢ (1o ·o ∅) = ∅
16		omsuc 8547	. . . . . 6 ⊢ ((1o ∈ On ∧ 𝑦 ∈ On) → (1o ·o suc 𝑦) = ((1o ·o 𝑦) +o 1o))
17	13, 16	mpan 688	. . . . 5 ⊢ (𝑦 ∈ On → (1o ·o suc 𝑦) = ((1o ·o 𝑦) +o 1o))
18		oveq1 7426	. . . . 5 ⊢ ((1o ·o 𝑦) = 𝑦 → ((1o ·o 𝑦) +o 1o) = (𝑦 +o 1o))
19	17, 18	sylan9eq 2785	. . . 4 ⊢ ((𝑦 ∈ On ∧ (1o ·o 𝑦) = 𝑦) → (1o ·o suc 𝑦) = (𝑦 +o 1o))
20		oa1suc 8552	. . . . 5 ⊢ (𝑦 ∈ On → (𝑦 +o 1o) = suc 𝑦)
21	20	adantr 479	. . . 4 ⊢ ((𝑦 ∈ On ∧ (1o ·o 𝑦) = 𝑦) → (𝑦 +o 1o) = suc 𝑦)
22	19, 21	eqtrd 2765	. . 3 ⊢ ((𝑦 ∈ On ∧ (1o ·o 𝑦) = 𝑦) → (1o ·o suc 𝑦) = suc 𝑦)
23	22	ex 411	. 2 ⊢ (𝑦 ∈ On → ((1o ·o 𝑦) = 𝑦 → (1o ·o suc 𝑦) = suc 𝑦))
24		iuneq2 5016	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 (1o ·o 𝑦) = 𝑦 → ∪ 𝑦 ∈ 𝑥 (1o ·o 𝑦) = ∪ 𝑦 ∈ 𝑥 𝑦)
25		uniiun 5062	. . . 4 ⊢ ∪ 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦
26	24, 25	eqtr4di 2783	. . 3 ⊢ (∀𝑦 ∈ 𝑥 (1o ·o 𝑦) = 𝑦 → ∪ 𝑦 ∈ 𝑥 (1o ·o 𝑦) = ∪ 𝑥)
27		vex 3465	. . . . 5 ⊢ 𝑥 ∈ V
28		omlim 8554	. . . . . 6 ⊢ ((1o ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (1o ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (1o ·o 𝑦))
29	13, 28	mpan 688	. . . . 5 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (1o ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (1o ·o 𝑦))
30	27, 29	mpan 688	. . . 4 ⊢ (Lim 𝑥 → (1o ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (1o ·o 𝑦))
31		limuni 6432	. . . 4 ⊢ (Lim 𝑥 → 𝑥 = ∪ 𝑥)
32	30, 31	eqeq12d 2741	. . 3 ⊢ (Lim 𝑥 → ((1o ·o 𝑥) = 𝑥 ↔ ∪ 𝑦 ∈ 𝑥 (1o ·o 𝑦) = ∪ 𝑥))
33	26, 32	imbitrrid 245	. 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (1o ·o 𝑦) = 𝑦 → (1o ·o 𝑥) = 𝑥))
34	3, 6, 9, 12, 15, 23, 33	tfinds 7865	1 ⊢ (𝐴 ∈ On → (1o ·o 𝐴) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∀wral 3050  Vcvv 3461  ∅c0 4322  ∪ cuni 4909  ∪ ciun 4997  Oncon0 6371  Lim wlim 6372  suc csuc 6373  (class class class)co 7419  1_oc1o 8480   +_o coa 8484   ·_o comu 8485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-oadd 8491  df-omul 8492

This theorem is referenced by:  oe1  8565  omword2  8595  om1om1r  42860  omabs2  42908  omcl2  42909