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Theorem om1r 8159
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 7156 . . 3 (𝑥 = ∅ → (1o ·o 𝑥) = (1o ·o ∅))
2 id 22 . . 3 (𝑥 = ∅ → 𝑥 = ∅)
31, 2eqeq12d 2842 . 2 (𝑥 = ∅ → ((1o ·o 𝑥) = 𝑥 ↔ (1o ·o ∅) = ∅))
4 oveq2 7156 . . 3 (𝑥 = 𝑦 → (1o ·o 𝑥) = (1o ·o 𝑦))
5 id 22 . . 3 (𝑥 = 𝑦𝑥 = 𝑦)
64, 5eqeq12d 2842 . 2 (𝑥 = 𝑦 → ((1o ·o 𝑥) = 𝑥 ↔ (1o ·o 𝑦) = 𝑦))
7 oveq2 7156 . . 3 (𝑥 = suc 𝑦 → (1o ·o 𝑥) = (1o ·o suc 𝑦))
8 id 22 . . 3 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
97, 8eqeq12d 2842 . 2 (𝑥 = suc 𝑦 → ((1o ·o 𝑥) = 𝑥 ↔ (1o ·o suc 𝑦) = suc 𝑦))
10 oveq2 7156 . . 3 (𝑥 = 𝐴 → (1o ·o 𝑥) = (1o ·o 𝐴))
11 id 22 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
1210, 11eqeq12d 2842 . 2 (𝑥 = 𝐴 → ((1o ·o 𝑥) = 𝑥 ↔ (1o ·o 𝐴) = 𝐴))
13 1on 8100 . . 3 1o ∈ On
14 om0 8133 . . 3 (1o ∈ On → (1o ·o ∅) = ∅)
1513, 14ax-mp 5 . 2 (1o ·o ∅) = ∅
16 omsuc 8142 . . . . . 6 ((1o ∈ On ∧ 𝑦 ∈ On) → (1o ·o suc 𝑦) = ((1o ·o 𝑦) +o 1o))
1713, 16mpan 686 . . . . 5 (𝑦 ∈ On → (1o ·o suc 𝑦) = ((1o ·o 𝑦) +o 1o))
18 oveq1 7155 . . . . 5 ((1o ·o 𝑦) = 𝑦 → ((1o ·o 𝑦) +o 1o) = (𝑦 +o 1o))
1917, 18sylan9eq 2881 . . . 4 ((𝑦 ∈ On ∧ (1o ·o 𝑦) = 𝑦) → (1o ·o suc 𝑦) = (𝑦 +o 1o))
20 oa1suc 8147 . . . . 5 (𝑦 ∈ On → (𝑦 +o 1o) = suc 𝑦)
2120adantr 481 . . . 4 ((𝑦 ∈ On ∧ (1o ·o 𝑦) = 𝑦) → (𝑦 +o 1o) = suc 𝑦)
2219, 21eqtrd 2861 . . 3 ((𝑦 ∈ On ∧ (1o ·o 𝑦) = 𝑦) → (1o ·o suc 𝑦) = suc 𝑦)
2322ex 413 . 2 (𝑦 ∈ On → ((1o ·o 𝑦) = 𝑦 → (1o ·o suc 𝑦) = suc 𝑦))
24 iuneq2 4935 . . . 4 (∀𝑦𝑥 (1o ·o 𝑦) = 𝑦 𝑦𝑥 (1o ·o 𝑦) = 𝑦𝑥 𝑦)
25 uniiun 4979 . . . 4 𝑥 = 𝑦𝑥 𝑦
2624, 25syl6eqr 2879 . . 3 (∀𝑦𝑥 (1o ·o 𝑦) = 𝑦 𝑦𝑥 (1o ·o 𝑦) = 𝑥)
27 vex 3503 . . . . 5 𝑥 ∈ V
28 omlim 8149 . . . . . 6 ((1o ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (1o ·o 𝑥) = 𝑦𝑥 (1o ·o 𝑦))
2913, 28mpan 686 . . . . 5 ((𝑥 ∈ V ∧ Lim 𝑥) → (1o ·o 𝑥) = 𝑦𝑥 (1o ·o 𝑦))
3027, 29mpan 686 . . . 4 (Lim 𝑥 → (1o ·o 𝑥) = 𝑦𝑥 (1o ·o 𝑦))
31 limuni 6249 . . . 4 (Lim 𝑥𝑥 = 𝑥)
3230, 31eqeq12d 2842 . . 3 (Lim 𝑥 → ((1o ·o 𝑥) = 𝑥 𝑦𝑥 (1o ·o 𝑦) = 𝑥))
3326, 32syl5ibr 247 . 2 (Lim 𝑥 → (∀𝑦𝑥 (1o ·o 𝑦) = 𝑦 → (1o ·o 𝑥) = 𝑥))
343, 6, 9, 12, 15, 23, 33tfinds 7562 1 (𝐴 ∈ On → (1o ·o 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  wral 3143  Vcvv 3500  c0 4295   cuni 4837   ciun 4917  Oncon0 6189  Lim wlim 6190  suc csuc 6191  (class class class)co 7148  1oc1o 8086   +o coa 8090   ·o comu 8091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-omul 8098
This theorem is referenced by:  oe1  8160  omword2  8190
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