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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.
StepHypRef Expression
1 oveq2 7440 . . 3 (𝑥 = ∅ → (1o ·o 𝑥) = (1o ·o ∅))
2 id 22 . . 3 (𝑥 = ∅ → 𝑥 = ∅)
31, 2eqeq12d 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 = 𝑦𝑥 = 𝑦)
64, 5eqeq12d 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 𝑦𝑥 = suc 𝑦)
97, 8eqeq12d 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 = 𝐴𝑥 = 𝐴)
1210, 11eqeq12d 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 ∅) = ∅)
1513, 14ax-mp 5 . 2 (1o ·o ∅) = ∅
16 omsuc 8565 . . . . . 6 ((1o ∈ On ∧ 𝑦 ∈ On) → (1o ·o suc 𝑦) = ((1o ·o 𝑦) +o 1o))
1713, 16mpan 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))
1917, 18sylan9eq 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)
2120adantr 480 . . . 4 ((𝑦 ∈ On ∧ (1o ·o 𝑦) = 𝑦) → (𝑦 +o 1o) = suc 𝑦)
2219, 21eqtrd 2776 . . 3 ((𝑦 ∈ On ∧ (1o ·o 𝑦) = 𝑦) → (1o ·o suc 𝑦) = suc 𝑦)
2322ex 412 . 2 (𝑦 ∈ On → ((1o ·o 𝑦) = 𝑦 → (1o ·o suc 𝑦) = suc 𝑦))
24 iuneq2 5010 . . . 4 (∀𝑦𝑥 (1o ·o 𝑦) = 𝑦 𝑦𝑥 (1o ·o 𝑦) = 𝑦𝑥 𝑦)
25 uniiun 5057 . . . 4 𝑥 = 𝑦𝑥 𝑦
2624, 25eqtr4di 2794 . . 3 (∀𝑦𝑥 (1o ·o 𝑦) = 𝑦 𝑦𝑥 (1o ·o 𝑦) = 𝑥)
27 vex 3483 . . . . 5 𝑥 ∈ V
28 omlim 8572 . . . . . 6 ((1o ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (1o ·o 𝑥) = 𝑦𝑥 (1o ·o 𝑦))
2913, 28mpan 690 . . . . 5 ((𝑥 ∈ V ∧ Lim 𝑥) → (1o ·o 𝑥) = 𝑦𝑥 (1o ·o 𝑦))
3027, 29mpan 690 . . . 4 (Lim 𝑥 → (1o ·o 𝑥) = 𝑦𝑥 (1o ·o 𝑦))
31 limuni 6444 . . . 4 (Lim 𝑥𝑥 = 𝑥)
3230, 31eqeq12d 2752 . . 3 (Lim 𝑥 → ((1o ·o 𝑥) = 𝑥 𝑦𝑥 (1o ·o 𝑦) = 𝑥))
3326, 32imbitrrid 246 . 2 (Lim 𝑥 → (∀𝑦𝑥 (1o ·o 𝑦) = 𝑦 → (1o ·o 𝑥) = 𝑥))
343, 6, 9, 12, 15, 23, 33tfinds 7882 1 (𝐴 ∈ On → (1o ·o 𝐴) = 𝐴)
Colors of variables: wff setvar class
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  1oc1o 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
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