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Theorem om1r 7861
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 6884 . . 3 (𝑥 = ∅ → (1𝑜 ·𝑜 𝑥) = (1𝑜 ·𝑜 ∅))
2 id 22 . . 3 (𝑥 = ∅ → 𝑥 = ∅)
31, 2eqeq12d 2812 . 2 (𝑥 = ∅ → ((1𝑜 ·𝑜 𝑥) = 𝑥 ↔ (1𝑜 ·𝑜 ∅) = ∅))
4 oveq2 6884 . . 3 (𝑥 = 𝑦 → (1𝑜 ·𝑜 𝑥) = (1𝑜 ·𝑜 𝑦))
5 id 22 . . 3 (𝑥 = 𝑦𝑥 = 𝑦)
64, 5eqeq12d 2812 . 2 (𝑥 = 𝑦 → ((1𝑜 ·𝑜 𝑥) = 𝑥 ↔ (1𝑜 ·𝑜 𝑦) = 𝑦))
7 oveq2 6884 . . 3 (𝑥 = suc 𝑦 → (1𝑜 ·𝑜 𝑥) = (1𝑜 ·𝑜 suc 𝑦))
8 id 22 . . 3 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
97, 8eqeq12d 2812 . 2 (𝑥 = suc 𝑦 → ((1𝑜 ·𝑜 𝑥) = 𝑥 ↔ (1𝑜 ·𝑜 suc 𝑦) = suc 𝑦))
10 oveq2 6884 . . 3 (𝑥 = 𝐴 → (1𝑜 ·𝑜 𝑥) = (1𝑜 ·𝑜 𝐴))
11 id 22 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
1210, 11eqeq12d 2812 . 2 (𝑥 = 𝐴 → ((1𝑜 ·𝑜 𝑥) = 𝑥 ↔ (1𝑜 ·𝑜 𝐴) = 𝐴))
13 1on 7804 . . 3 1𝑜 ∈ On
14 om0 7835 . . 3 (1𝑜 ∈ On → (1𝑜 ·𝑜 ∅) = ∅)
1513, 14ax-mp 5 . 2 (1𝑜 ·𝑜 ∅) = ∅
16 omsuc 7844 . . . . . 6 ((1𝑜 ∈ On ∧ 𝑦 ∈ On) → (1𝑜 ·𝑜 suc 𝑦) = ((1𝑜 ·𝑜 𝑦) +𝑜 1𝑜))
1713, 16mpan 682 . . . . 5 (𝑦 ∈ On → (1𝑜 ·𝑜 suc 𝑦) = ((1𝑜 ·𝑜 𝑦) +𝑜 1𝑜))
18 oveq1 6883 . . . . 5 ((1𝑜 ·𝑜 𝑦) = 𝑦 → ((1𝑜 ·𝑜 𝑦) +𝑜 1𝑜) = (𝑦 +𝑜 1𝑜))
1917, 18sylan9eq 2851 . . . 4 ((𝑦 ∈ On ∧ (1𝑜 ·𝑜 𝑦) = 𝑦) → (1𝑜 ·𝑜 suc 𝑦) = (𝑦 +𝑜 1𝑜))
20 oa1suc 7849 . . . . 5 (𝑦 ∈ On → (𝑦 +𝑜 1𝑜) = suc 𝑦)
2120adantr 473 . . . 4 ((𝑦 ∈ On ∧ (1𝑜 ·𝑜 𝑦) = 𝑦) → (𝑦 +𝑜 1𝑜) = suc 𝑦)
2219, 21eqtrd 2831 . . 3 ((𝑦 ∈ On ∧ (1𝑜 ·𝑜 𝑦) = 𝑦) → (1𝑜 ·𝑜 suc 𝑦) = suc 𝑦)
2322ex 402 . 2 (𝑦 ∈ On → ((1𝑜 ·𝑜 𝑦) = 𝑦 → (1𝑜 ·𝑜 suc 𝑦) = suc 𝑦))
24 iuneq2 4725 . . . 4 (∀𝑦𝑥 (1𝑜 ·𝑜 𝑦) = 𝑦 𝑦𝑥 (1𝑜 ·𝑜 𝑦) = 𝑦𝑥 𝑦)
25 uniiun 4761 . . . 4 𝑥 = 𝑦𝑥 𝑦
2624, 25syl6eqr 2849 . . 3 (∀𝑦𝑥 (1𝑜 ·𝑜 𝑦) = 𝑦 𝑦𝑥 (1𝑜 ·𝑜 𝑦) = 𝑥)
27 vex 3386 . . . . 5 𝑥 ∈ V
28 omlim 7851 . . . . . 6 ((1𝑜 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (1𝑜 ·𝑜 𝑥) = 𝑦𝑥 (1𝑜 ·𝑜 𝑦))
2913, 28mpan 682 . . . . 5 ((𝑥 ∈ V ∧ Lim 𝑥) → (1𝑜 ·𝑜 𝑥) = 𝑦𝑥 (1𝑜 ·𝑜 𝑦))
3027, 29mpan 682 . . . 4 (Lim 𝑥 → (1𝑜 ·𝑜 𝑥) = 𝑦𝑥 (1𝑜 ·𝑜 𝑦))
31 limuni 5999 . . . 4 (Lim 𝑥𝑥 = 𝑥)
3230, 31eqeq12d 2812 . . 3 (Lim 𝑥 → ((1𝑜 ·𝑜 𝑥) = 𝑥 𝑦𝑥 (1𝑜 ·𝑜 𝑦) = 𝑥))
3326, 32syl5ibr 238 . 2 (Lim 𝑥 → (∀𝑦𝑥 (1𝑜 ·𝑜 𝑦) = 𝑦 → (1𝑜 ·𝑜 𝑥) = 𝑥))
343, 6, 9, 12, 15, 23, 33tfinds 7291 1 (𝐴 ∈ On → (1𝑜 ·𝑜 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  wral 3087  Vcvv 3383  c0 4113   cuni 4626   ciun 4708  Oncon0 5939  Lim wlim 5940  suc csuc 5941  (class class class)co 6876  1𝑜c1o 7790   +𝑜 coa 7794   ·𝑜 comu 7795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2375  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-tp 4371  df-op 4373  df-uni 4627  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-tr 4944  df-id 5218  df-eprel 5223  df-po 5231  df-so 5232  df-fr 5269  df-we 5271  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-pred 5896  df-ord 5942  df-on 5943  df-lim 5944  df-suc 5945  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-om 7298  df-wrecs 7643  df-recs 7705  df-rdg 7743  df-1o 7797  df-oadd 7801  df-omul 7802
This theorem is referenced by:  oe1  7862  omword2  7892
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