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Theorem om1r 7788
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 6817 . . 3 (𝑥 = ∅ → (1𝑜 ·𝑜 𝑥) = (1𝑜 ·𝑜 ∅))
2 id 22 . . 3 (𝑥 = ∅ → 𝑥 = ∅)
31, 2eqeq12d 2771 . 2 (𝑥 = ∅ → ((1𝑜 ·𝑜 𝑥) = 𝑥 ↔ (1𝑜 ·𝑜 ∅) = ∅))
4 oveq2 6817 . . 3 (𝑥 = 𝑦 → (1𝑜 ·𝑜 𝑥) = (1𝑜 ·𝑜 𝑦))
5 id 22 . . 3 (𝑥 = 𝑦𝑥 = 𝑦)
64, 5eqeq12d 2771 . 2 (𝑥 = 𝑦 → ((1𝑜 ·𝑜 𝑥) = 𝑥 ↔ (1𝑜 ·𝑜 𝑦) = 𝑦))
7 oveq2 6817 . . 3 (𝑥 = suc 𝑦 → (1𝑜 ·𝑜 𝑥) = (1𝑜 ·𝑜 suc 𝑦))
8 id 22 . . 3 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
97, 8eqeq12d 2771 . 2 (𝑥 = suc 𝑦 → ((1𝑜 ·𝑜 𝑥) = 𝑥 ↔ (1𝑜 ·𝑜 suc 𝑦) = suc 𝑦))
10 oveq2 6817 . . 3 (𝑥 = 𝐴 → (1𝑜 ·𝑜 𝑥) = (1𝑜 ·𝑜 𝐴))
11 id 22 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
1210, 11eqeq12d 2771 . 2 (𝑥 = 𝐴 → ((1𝑜 ·𝑜 𝑥) = 𝑥 ↔ (1𝑜 ·𝑜 𝐴) = 𝐴))
13 om0x 7764 . 2 (1𝑜 ·𝑜 ∅) = ∅
14 1on 7732 . . . . . 6 1𝑜 ∈ On
15 omsuc 7771 . . . . . 6 ((1𝑜 ∈ On ∧ 𝑦 ∈ On) → (1𝑜 ·𝑜 suc 𝑦) = ((1𝑜 ·𝑜 𝑦) +𝑜 1𝑜))
1614, 15mpan 708 . . . . 5 (𝑦 ∈ On → (1𝑜 ·𝑜 suc 𝑦) = ((1𝑜 ·𝑜 𝑦) +𝑜 1𝑜))
17 oveq1 6816 . . . . 5 ((1𝑜 ·𝑜 𝑦) = 𝑦 → ((1𝑜 ·𝑜 𝑦) +𝑜 1𝑜) = (𝑦 +𝑜 1𝑜))
1816, 17sylan9eq 2810 . . . 4 ((𝑦 ∈ On ∧ (1𝑜 ·𝑜 𝑦) = 𝑦) → (1𝑜 ·𝑜 suc 𝑦) = (𝑦 +𝑜 1𝑜))
19 oa1suc 7776 . . . . 5 (𝑦 ∈ On → (𝑦 +𝑜 1𝑜) = suc 𝑦)
2019adantr 472 . . . 4 ((𝑦 ∈ On ∧ (1𝑜 ·𝑜 𝑦) = 𝑦) → (𝑦 +𝑜 1𝑜) = suc 𝑦)
2118, 20eqtrd 2790 . . 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 4685 . . . 4 (∀𝑦𝑥 (1𝑜 ·𝑜 𝑦) = 𝑦 𝑦𝑥 (1𝑜 ·𝑜 𝑦) = 𝑦𝑥 𝑦)
24 uniiun 4721 . . . 4 𝑥 = 𝑦𝑥 𝑦
2523, 24syl6eqr 2808 . . 3 (∀𝑦𝑥 (1𝑜 ·𝑜 𝑦) = 𝑦 𝑦𝑥 (1𝑜 ·𝑜 𝑦) = 𝑥)
26 vex 3339 . . . . 5 𝑥 ∈ V
27 omlim 7778 . . . . . 6 ((1𝑜 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (1𝑜 ·𝑜 𝑥) = 𝑦𝑥 (1𝑜 ·𝑜 𝑦))
2814, 27mpan 708 . . . . 5 ((𝑥 ∈ V ∧ Lim 𝑥) → (1𝑜 ·𝑜 𝑥) = 𝑦𝑥 (1𝑜 ·𝑜 𝑦))
2926, 28mpan 708 . . . 4 (Lim 𝑥 → (1𝑜 ·𝑜 𝑥) = 𝑦𝑥 (1𝑜 ·𝑜 𝑦))
30 limuni 5942 . . . 4 (Lim 𝑥𝑥 = 𝑥)
3129, 30eqeq12d 2771 . . 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 7220 1 (𝐴 ∈ On → (1𝑜 ·𝑜 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1628  wcel 2135  wral 3046  Vcvv 3336  c0 4054   cuni 4584   ciun 4668  Oncon0 5880  Lim wlim 5881  suc csuc 5882  (class class class)co 6809  1𝑜c1o 7718   +𝑜 coa 7722   ·𝑜 comu 7723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-rep 4919  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-reu 3053  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-pss 3727  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4585  df-iun 4670  df-br 4801  df-opab 4861  df-mpt 4878  df-tr 4901  df-id 5170  df-eprel 5175  df-po 5183  df-so 5184  df-fr 5221  df-we 5223  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-pred 5837  df-ord 5883  df-on 5884  df-lim 5885  df-suc 5886  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053  df-ov 6812  df-oprab 6813  df-mpt2 6814  df-om 7227  df-1st 7329  df-2nd 7330  df-wrecs 7572  df-recs 7633  df-rdg 7671  df-1o 7725  df-oadd 7729  df-omul 7730
This theorem is referenced by:  oe1  7789  omword2  7819
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