Theorem om1r 8505

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 7398	. . 3 ⊢ (𝑥 = ∅ → (1o ·o 𝑥) = (1o ·o ∅))
2		id 22	. . 3 ⊢ (𝑥 = ∅ → 𝑥 = ∅)
3	1, 2	eqeq12d 2777	. 2 ⊢ (𝑥 = ∅ → ((1o ·o 𝑥) = 𝑥 ↔ (1o ·o ∅) = ∅))
4		oveq2 7398	. . 3 ⊢ (𝑥 = 𝑦 → (1o ·o 𝑥) = (1o ·o 𝑦))
5		id 22	. . 3 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
6	4, 5	eqeq12d 2777	. 2 ⊢ (𝑥 = 𝑦 → ((1o ·o 𝑥) = 𝑥 ↔ (1o ·o 𝑦) = 𝑦))
7		oveq2 7398	. . 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 2777	. 2 ⊢ (𝑥 = suc 𝑦 → ((1o ·o 𝑥) = 𝑥 ↔ (1o ·o suc 𝑦) = suc 𝑦))
10		oveq2 7398	. . 3 ⊢ (𝑥 = 𝐴 → (1o ·o 𝑥) = (1o ·o 𝐴))
11		id 22	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
12	10, 11	eqeq12d 2777	. 2 ⊢ (𝑥 = 𝐴 → ((1o ·o 𝑥) = 𝑥 ↔ (1o ·o 𝐴) = 𝐴))
13		1on 8443	. . 3 ⊢ 1o ∈ On
14		om0 8479	. . 3 ⊢ (1o ∈ On → (1o ·o ∅) = ∅)
15	13, 14	ax-mp 5	. 2 ⊢ (1o ·o ∅) = ∅
16		omsuc 8488	. . . . . 6 ⊢ ((1o ∈ On ∧ 𝑦 ∈ On) → (1o ·o suc 𝑦) = ((1o ·o 𝑦) +o 1o))
17	13, 16	mpan 700	. . . . 5 ⊢ (𝑦 ∈ On → (1o ·o suc 𝑦) = ((1o ·o 𝑦) +o 1o))
18		oveq1 7397	. . . . 5 ⊢ ((1o ·o 𝑦) = 𝑦 → ((1o ·o 𝑦) +o 1o) = (𝑦 +o 1o))
19	17, 18	sylan9eq 2816	. . . 4 ⊢ ((𝑦 ∈ On ∧ (1o ·o 𝑦) = 𝑦) → (1o ·o suc 𝑦) = (𝑦 +o 1o))
20		oa1suc 8493	. . . . 5 ⊢ (𝑦 ∈ On → (𝑦 +o 1o) = suc 𝑦)
21	20	adantr 484	. . . 4 ⊢ ((𝑦 ∈ On ∧ (1o ·o 𝑦) = 𝑦) → (𝑦 +o 1o) = suc 𝑦)
22	19, 21	eqtrd 2796	. . 3 ⊢ ((𝑦 ∈ On ∧ (1o ·o 𝑦) = 𝑦) → (1o ·o suc 𝑦) = suc 𝑦)
23	22	ex 416	. 2 ⊢ (𝑦 ∈ On → ((1o ·o 𝑦) = 𝑦 → (1o ·o suc 𝑦) = suc 𝑦))
24		iuneq2 4966	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 (1o ·o 𝑦) = 𝑦 → ∪ 𝑦 ∈ 𝑥 (1o ·o 𝑦) = ∪ 𝑦 ∈ 𝑥 𝑦)
25		uniiun 5013	. . . 4 ⊢ ∪ 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦
26	24, 25	eqtr4di 2814	. . 3 ⊢ (∀𝑦 ∈ 𝑥 (1o ·o 𝑦) = 𝑦 → ∪ 𝑦 ∈ 𝑥 (1o ·o 𝑦) = ∪ 𝑥)
27		vex 3457	. . . . 5 ⊢ 𝑥 ∈ V
28		omlim 8495	. . . . . 6 ⊢ ((1o ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (1o ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (1o ·o 𝑦))
29	13, 28	mpan 700	. . . . 5 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (1o ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (1o ·o 𝑦))
30	27, 29	mpan 700	. . . 4 ⊢ (Lim 𝑥 → (1o ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (1o ·o 𝑦))
31		limuni 6402	. . . 4 ⊢ (Lim 𝑥 → 𝑥 = ∪ 𝑥)
32	30, 31	eqeq12d 2777	. . 3 ⊢ (Lim 𝑥 → ((1o ·o 𝑥) = 𝑥 ↔ ∪ 𝑦 ∈ 𝑥 (1o ·o 𝑦) = ∪ 𝑥))
33	26, 32	imbitrrid 248	. 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (1o ·o 𝑦) = 𝑦 → (1o ·o 𝑥) = 𝑥))
34	3, 6, 9, 12, 15, 23, 33	tfinds 7834	1 ⊢ (𝐴 ∈ On → (1o ·o 𝐴) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  Vcvv 3453  ∅c0 4283  ∪ cuni 4862  ∪ ciun 4946  Oncon0 6340  Lim wlim 6341  suc csuc 6342  (class class class)co 7390  1_oc1o 8423   +_o coa 8427   ·_o comu 8428

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pr 5387  ax-un 7712

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6282  df-ord 6343  df-on 6344  df-lim 6345  df-suc 6346  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7841  df-2nd 7965  df-frecs 8255  df-wrecs 8286  df-recs 8335  df-rdg 8374  df-1o 8430  df-oadd 8434  df-omul 8435

This theorem is referenced by:  oe1  8506  omword2  8536  om1om1r  43821  omabs2  43869  omcl2  43870