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Theorem om1r 8580
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 7439 . . 3 (𝑥 = ∅ → (1o ·o 𝑥) = (1o ·o ∅))
2 id 22 . . 3 (𝑥 = ∅ → 𝑥 = ∅)
31, 2eqeq12d 2751 . 2 (𝑥 = ∅ → ((1o ·o 𝑥) = 𝑥 ↔ (1o ·o ∅) = ∅))
4 oveq2 7439 . . 3 (𝑥 = 𝑦 → (1o ·o 𝑥) = (1o ·o 𝑦))
5 id 22 . . 3 (𝑥 = 𝑦𝑥 = 𝑦)
64, 5eqeq12d 2751 . 2 (𝑥 = 𝑦 → ((1o ·o 𝑥) = 𝑥 ↔ (1o ·o 𝑦) = 𝑦))
7 oveq2 7439 . . 3 (𝑥 = suc 𝑦 → (1o ·o 𝑥) = (1o ·o suc 𝑦))
8 id 22 . . 3 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
97, 8eqeq12d 2751 . 2 (𝑥 = suc 𝑦 → ((1o ·o 𝑥) = 𝑥 ↔ (1o ·o suc 𝑦) = suc 𝑦))
10 oveq2 7439 . . 3 (𝑥 = 𝐴 → (1o ·o 𝑥) = (1o ·o 𝐴))
11 id 22 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
1210, 11eqeq12d 2751 . 2 (𝑥 = 𝐴 → ((1o ·o 𝑥) = 𝑥 ↔ (1o ·o 𝐴) = 𝐴))
13 1on 8517 . . 3 1o ∈ On
14 om0 8554 . . 3 (1o ∈ On → (1o ·o ∅) = ∅)
1513, 14ax-mp 5 . 2 (1o ·o ∅) = ∅
16 omsuc 8563 . . . . . 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 7438 . . . . 5 ((1o ·o 𝑦) = 𝑦 → ((1o ·o 𝑦) +o 1o) = (𝑦 +o 1o))
1917, 18sylan9eq 2795 . . . 4 ((𝑦 ∈ On ∧ (1o ·o 𝑦) = 𝑦) → (1o ·o suc 𝑦) = (𝑦 +o 1o))
20 oa1suc 8568 . . . . 5 (𝑦 ∈ On → (𝑦 +o 1o) = suc 𝑦)
2120adantr 480 . . . 4 ((𝑦 ∈ On ∧ (1o ·o 𝑦) = 𝑦) → (𝑦 +o 1o) = suc 𝑦)
2219, 21eqtrd 2775 . . 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 5016 . . . 4 (∀𝑦𝑥 (1o ·o 𝑦) = 𝑦 𝑦𝑥 (1o ·o 𝑦) = 𝑦𝑥 𝑦)
25 uniiun 5063 . . . 4 𝑥 = 𝑦𝑥 𝑦
2624, 25eqtr4di 2793 . . 3 (∀𝑦𝑥 (1o ·o 𝑦) = 𝑦 𝑦𝑥 (1o ·o 𝑦) = 𝑥)
27 vex 3482 . . . . 5 𝑥 ∈ V
28 omlim 8570 . . . . . 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 6447 . . . 4 (Lim 𝑥𝑥 = 𝑥)
3230, 31eqeq12d 2751 . . 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 7881 1 (𝐴 ∈ On → (1o ·o 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  c0 4339   cuni 4912   ciun 4996  Oncon0 6386  Lim wlim 6387  suc csuc 6388  (class class class)co 7431  1oc1o 8498   +o coa 8502   ·o comu 8503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-omul 8510
This theorem is referenced by:  oe1  8581  omword2  8611  om1om1r  43274  omabs2  43322  omcl2  43323
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