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Theorem om1r 7668
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.
StepHypRef Expression
1 oveq2 6698 . . 3 (𝑥 = ∅ → (1𝑜 ·𝑜 𝑥) = (1𝑜 ·𝑜 ∅))
2 id 22 . . 3 (𝑥 = ∅ → 𝑥 = ∅)
31, 2eqeq12d 2666 . 2 (𝑥 = ∅ → ((1𝑜 ·𝑜 𝑥) = 𝑥 ↔ (1𝑜 ·𝑜 ∅) = ∅))
4 oveq2 6698 . . 3 (𝑥 = 𝑦 → (1𝑜 ·𝑜 𝑥) = (1𝑜 ·𝑜 𝑦))
5 id 22 . . 3 (𝑥 = 𝑦𝑥 = 𝑦)
64, 5eqeq12d 2666 . 2 (𝑥 = 𝑦 → ((1𝑜 ·𝑜 𝑥) = 𝑥 ↔ (1𝑜 ·𝑜 𝑦) = 𝑦))
7 oveq2 6698 . . 3 (𝑥 = suc 𝑦 → (1𝑜 ·𝑜 𝑥) = (1𝑜 ·𝑜 suc 𝑦))
8 id 22 . . 3 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
97, 8eqeq12d 2666 . 2 (𝑥 = suc 𝑦 → ((1𝑜 ·𝑜 𝑥) = 𝑥 ↔ (1𝑜 ·𝑜 suc 𝑦) = suc 𝑦))
10 oveq2 6698 . . 3 (𝑥 = 𝐴 → (1𝑜 ·𝑜 𝑥) = (1𝑜 ·𝑜 𝐴))
11 id 22 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
1210, 11eqeq12d 2666 . 2 (𝑥 = 𝐴 → ((1𝑜 ·𝑜 𝑥) = 𝑥 ↔ (1𝑜 ·𝑜 𝐴) = 𝐴))
13 om0x 7644 . 2 (1𝑜 ·𝑜 ∅) = ∅
14 1on 7612 . . . . . 6 1𝑜 ∈ On
15 omsuc 7651 . . . . . 6 ((1𝑜 ∈ On ∧ 𝑦 ∈ On) → (1𝑜 ·𝑜 suc 𝑦) = ((1𝑜 ·𝑜 𝑦) +𝑜 1𝑜))
1614, 15mpan 706 . . . . 5 (𝑦 ∈ On → (1𝑜 ·𝑜 suc 𝑦) = ((1𝑜 ·𝑜 𝑦) +𝑜 1𝑜))
17 oveq1 6697 . . . . 5 ((1𝑜 ·𝑜 𝑦) = 𝑦 → ((1𝑜 ·𝑜 𝑦) +𝑜 1𝑜) = (𝑦 +𝑜 1𝑜))
1816, 17sylan9eq 2705 . . . 4 ((𝑦 ∈ On ∧ (1𝑜 ·𝑜 𝑦) = 𝑦) → (1𝑜 ·𝑜 suc 𝑦) = (𝑦 +𝑜 1𝑜))
19 oa1suc 7656 . . . . 5 (𝑦 ∈ On → (𝑦 +𝑜 1𝑜) = suc 𝑦)
2019adantr 480 . . . 4 ((𝑦 ∈ On ∧ (1𝑜 ·𝑜 𝑦) = 𝑦) → (𝑦 +𝑜 1𝑜) = suc 𝑦)
2118, 20eqtrd 2685 . . 3 ((𝑦 ∈ On ∧ (1𝑜 ·𝑜 𝑦) = 𝑦) → (1𝑜 ·𝑜 suc 𝑦) = suc 𝑦)
2221ex 449 . 2 (𝑦 ∈ On → ((1𝑜 ·𝑜 𝑦) = 𝑦 → (1𝑜 ·𝑜 suc 𝑦) = suc 𝑦))
23 iuneq2 4569 . . . 4 (∀𝑦𝑥 (1𝑜 ·𝑜 𝑦) = 𝑦 𝑦𝑥 (1𝑜 ·𝑜 𝑦) = 𝑦𝑥 𝑦)
24 uniiun 4605 . . . 4 𝑥 = 𝑦𝑥 𝑦
2523, 24syl6eqr 2703 . . 3 (∀𝑦𝑥 (1𝑜 ·𝑜 𝑦) = 𝑦 𝑦𝑥 (1𝑜 ·𝑜 𝑦) = 𝑥)
26 vex 3234 . . . . 5 𝑥 ∈ V
27 omlim 7658 . . . . . 6 ((1𝑜 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (1𝑜 ·𝑜 𝑥) = 𝑦𝑥 (1𝑜 ·𝑜 𝑦))
2814, 27mpan 706 . . . . 5 ((𝑥 ∈ V ∧ Lim 𝑥) → (1𝑜 ·𝑜 𝑥) = 𝑦𝑥 (1𝑜 ·𝑜 𝑦))
2926, 28mpan 706 . . . 4 (Lim 𝑥 → (1𝑜 ·𝑜 𝑥) = 𝑦𝑥 (1𝑜 ·𝑜 𝑦))
30 limuni 5823 . . . 4 (Lim 𝑥𝑥 = 𝑥)
3129, 30eqeq12d 2666 . . 3 (Lim 𝑥 → ((1𝑜 ·𝑜 𝑥) = 𝑥 𝑦𝑥 (1𝑜 ·𝑜 𝑦) = 𝑥))
3225, 31syl5ibr 236 . 2 (Lim 𝑥 → (∀𝑦𝑥 (1𝑜 ·𝑜 𝑦) = 𝑦 → (1𝑜 ·𝑜 𝑥) = 𝑥))
333, 6, 9, 12, 13, 22, 32tfinds 7101 1 (𝐴 ∈ On → (1𝑜 ·𝑜 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  c0 3948   cuni 4468   ciun 4552  Oncon0 5761  Lim wlim 5762  suc csuc 5763  (class class class)co 6690  1𝑜c1o 7598   +𝑜 coa 7602   ·𝑜 comu 7603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610
This theorem is referenced by:  oe1  7669  omword2  7699
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