Theorem om1r 8168

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 7163	. . 3 ⊢ (𝑥 = ∅ → (1o ·o 𝑥) = (1o ·o ∅))
2		id 22	. . 3 ⊢ (𝑥 = ∅ → 𝑥 = ∅)
3	1, 2	eqeq12d 2837	. 2 ⊢ (𝑥 = ∅ → ((1o ·o 𝑥) = 𝑥 ↔ (1o ·o ∅) = ∅))
4		oveq2 7163	. . 3 ⊢ (𝑥 = 𝑦 → (1o ·o 𝑥) = (1o ·o 𝑦))
5		id 22	. . 3 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
6	4, 5	eqeq12d 2837	. 2 ⊢ (𝑥 = 𝑦 → ((1o ·o 𝑥) = 𝑥 ↔ (1o ·o 𝑦) = 𝑦))
7		oveq2 7163	. . 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 2837	. 2 ⊢ (𝑥 = suc 𝑦 → ((1o ·o 𝑥) = 𝑥 ↔ (1o ·o suc 𝑦) = suc 𝑦))
10		oveq2 7163	. . 3 ⊢ (𝑥 = 𝐴 → (1o ·o 𝑥) = (1o ·o 𝐴))
11		id 22	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
12	10, 11	eqeq12d 2837	. 2 ⊢ (𝑥 = 𝐴 → ((1o ·o 𝑥) = 𝑥 ↔ (1o ·o 𝐴) = 𝐴))
13		1on 8108	. . 3 ⊢ 1o ∈ On
14		om0 8141	. . 3 ⊢ (1o ∈ On → (1o ·o ∅) = ∅)
15	13, 14	ax-mp 5	. 2 ⊢ (1o ·o ∅) = ∅
16		omsuc 8150	. . . . . 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 7162	. . . . 5 ⊢ ((1o ·o 𝑦) = 𝑦 → ((1o ·o 𝑦) +o 1o) = (𝑦 +o 1o))
19	17, 18	sylan9eq 2876	. . . 4 ⊢ ((𝑦 ∈ On ∧ (1o ·o 𝑦) = 𝑦) → (1o ·o suc 𝑦) = (𝑦 +o 1o))
20		oa1suc 8155	. . . . 5 ⊢ (𝑦 ∈ On → (𝑦 +o 1o) = suc 𝑦)
21	20	adantr 483	. . . 4 ⊢ ((𝑦 ∈ On ∧ (1o ·o 𝑦) = 𝑦) → (𝑦 +o 1o) = suc 𝑦)
22	19, 21	eqtrd 2856	. . 3 ⊢ ((𝑦 ∈ On ∧ (1o ·o 𝑦) = 𝑦) → (1o ·o suc 𝑦) = suc 𝑦)
23	22	ex 415	. 2 ⊢ (𝑦 ∈ On → ((1o ·o 𝑦) = 𝑦 → (1o ·o suc 𝑦) = suc 𝑦))
24		iuneq2 4937	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 (1o ·o 𝑦) = 𝑦 → ∪ 𝑦 ∈ 𝑥 (1o ·o 𝑦) = ∪ 𝑦 ∈ 𝑥 𝑦)
25		uniiun 4981	. . . 4 ⊢ ∪ 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦
26	24, 25	syl6eqr 2874	. . 3 ⊢ (∀𝑦 ∈ 𝑥 (1o ·o 𝑦) = 𝑦 → ∪ 𝑦 ∈ 𝑥 (1o ·o 𝑦) = ∪ 𝑥)
27		vex 3497	. . . . 5 ⊢ 𝑥 ∈ V
28		omlim 8157	. . . . . 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 6250	. . . 4 ⊢ (Lim 𝑥 → 𝑥 = ∪ 𝑥)
32	30, 31	eqeq12d 2837	. . 3 ⊢ (Lim 𝑥 → ((1o ·o 𝑥) = 𝑥 ↔ ∪ 𝑦 ∈ 𝑥 (1o ·o 𝑦) = ∪ 𝑥))
33	26, 32	syl5ibr 248	. 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (1o ·o 𝑦) = 𝑦 → (1o ·o 𝑥) = 𝑥))
34	3, 6, 9, 12, 15, 23, 33	tfinds 7573	1 ⊢ (𝐴 ∈ On → (1o ·o 𝐴) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  Vcvv 3494  ∅c0 4290  ∪ cuni 4837  ∪ ciun 4918  Oncon0 6190  Lim wlim 6191  suc csuc 6192  (class class class)co 7155  1_oc1o 8094   +_o coa 8098   ·_o comu 8099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7580  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-1o 8101  df-oadd 8105  df-omul 8106

This theorem is referenced by:  oe1  8169  omword2  8199