Theorem om1r 8530

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 7424	. . 3 ⊢ (𝑥 = ∅ → (1o ·o 𝑥) = (1o ·o ∅))
2		id 23	. . 3 ⊢ (𝑥 = ∅ → 𝑥 = ∅)
3	1, 2	eqeq12d 2781	. 2 ⊢ (𝑥 = ∅ → ((1o ·o 𝑥) = 𝑥 ↔ (1o ·o ∅) = ∅))
4		oveq2 7424	. . 3 ⊢ (𝑥 = 𝑦 → (1o ·o 𝑥) = (1o ·o 𝑦))
5		id 23	. . 3 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
6	4, 5	eqeq12d 2781	. 2 ⊢ (𝑥 = 𝑦 → ((1o ·o 𝑥) = 𝑥 ↔ (1o ·o 𝑦) = 𝑦))
7		oveq2 7424	. . 3 ⊢ (𝑥 = suc 𝑦 → (1o ·o 𝑥) = (1o ·o suc 𝑦))
8		id 23	. . 3 ⊢ (𝑥 = suc 𝑦 → 𝑥 = suc 𝑦)
9	7, 8	eqeq12d 2781	. 2 ⊢ (𝑥 = suc 𝑦 → ((1o ·o 𝑥) = 𝑥 ↔ (1o ·o suc 𝑦) = suc 𝑦))
10		oveq2 7424	. . 3 ⊢ (𝑥 = 𝐴 → (1o ·o 𝑥) = (1o ·o 𝐴))
11		id 23	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
12	10, 11	eqeq12d 2781	. 2 ⊢ (𝑥 = 𝐴 → ((1o ·o 𝑥) = 𝑥 ↔ (1o ·o 𝐴) = 𝐴))
13		1on 8468	. . 3 ⊢ 1o ∈ On
14		om0 8504	. . 3 ⊢ (1o ∈ On → (1o ·o ∅) = ∅)
15	13, 14	ax-mp 5	. 2 ⊢ (1o ·o ∅) = ∅
16		omsuc 8513	. . . . . 6 ⊢ ((1o ∈ On ∧ 𝑦 ∈ On) → (1o ·o suc 𝑦) = ((1o ·o 𝑦) +o 1o))
17	13, 16	mpan 703	. . . . 5 ⊢ (𝑦 ∈ On → (1o ·o suc 𝑦) = ((1o ·o 𝑦) +o 1o))
18		oveq1 7423	. . . . 5 ⊢ ((1o ·o 𝑦) = 𝑦 → ((1o ·o 𝑦) +o 1o) = (𝑦 +o 1o))
19	17, 18	sylan9eq 2820	. . . 4 ⊢ ((𝑦 ∈ On ∧ (1o ·o 𝑦) = 𝑦) → (1o ·o suc 𝑦) = (𝑦 +o 1o))
20		oa1suc 8518	. . . . 5 ⊢ (𝑦 ∈ On → (𝑦 +o 1o) = suc 𝑦)
21	20	adantr 486	. . . 4 ⊢ ((𝑦 ∈ On ∧ (1o ·o 𝑦) = 𝑦) → (𝑦 +o 1o) = suc 𝑦)
22	19, 21	eqtrd 2800	. . 3 ⊢ ((𝑦 ∈ On ∧ (1o ·o 𝑦) = 𝑦) → (1o ·o suc 𝑦) = suc 𝑦)
23	22	ex 418	. 2 ⊢ (𝑦 ∈ On → ((1o ·o 𝑦) = 𝑦 → (1o ·o suc 𝑦) = suc 𝑦))
24		iuneq2 4978	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 (1o ·o 𝑦) = 𝑦 → ∪ 𝑦 ∈ 𝑥 (1o ·o 𝑦) = ∪ 𝑦 ∈ 𝑥 𝑦)
25		uniiun 5025	. . . 4 ⊢ ∪ 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦
26	24, 25	eqtr4di 2818	. . 3 ⊢ (∀𝑦 ∈ 𝑥 (1o ·o 𝑦) = 𝑦 → ∪ 𝑦 ∈ 𝑥 (1o ·o 𝑦) = ∪ 𝑥)
27		vex 3461	. . . . 5 ⊢ 𝑥 ∈ V
28		omlim 8520	. . . . . 6 ⊢ ((1o ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (1o ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (1o ·o 𝑦))
29	13, 28	mpan 703	. . . . 5 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (1o ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (1o ·o 𝑦))
30	27, 29	mpan 703	. . . 4 ⊢ (Lim 𝑥 → (1o ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (1o ·o 𝑦))
31		limuni 6427	. . . 4 ⊢ (Lim 𝑥 → 𝑥 = ∪ 𝑥)
32	30, 31	eqeq12d 2781	. . 3 ⊢ (Lim 𝑥 → ((1o ·o 𝑥) = 𝑥 ↔ ∪ 𝑦 ∈ 𝑥 (1o ·o 𝑦) = ∪ 𝑥))
33	26, 32	imbitrrid 249	. 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (1o ·o 𝑦) = 𝑦 → (1o ·o 𝑥) = 𝑥))
34	3, 6, 9, 12, 15, 23, 33	tfinds 7858	1 ⊢ (𝐴 ∈ On → (1o ·o 𝐴) = 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  Vcvv 3457  ∅c0 4286  ∪ cuni 4874  ∪ ciun 4958  Oncon0 6364  Lim wlim 6365  suc csuc 6366  (class class class)co 7416  1_oc1o 8448   +_o coa 8452   ·_o comu 8453

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7865  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-oadd 8459  df-omul 8460

This theorem is used by:  oe1  8531  omword2  8561  om1om1r  44044  omabs2  44092  omcl2  44093