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Theorem om1r 8446
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 7346 . . 3 (𝑥 = ∅ → (1o ·o 𝑥) = (1o ·o ∅))
2 id 22 . . 3 (𝑥 = ∅ → 𝑥 = ∅)
31, 2eqeq12d 2752 . 2 (𝑥 = ∅ → ((1o ·o 𝑥) = 𝑥 ↔ (1o ·o ∅) = ∅))
4 oveq2 7346 . . 3 (𝑥 = 𝑦 → (1o ·o 𝑥) = (1o ·o 𝑦))
5 id 22 . . 3 (𝑥 = 𝑦𝑥 = 𝑦)
64, 5eqeq12d 2752 . 2 (𝑥 = 𝑦 → ((1o ·o 𝑥) = 𝑥 ↔ (1o ·o 𝑦) = 𝑦))
7 oveq2 7346 . . 3 (𝑥 = suc 𝑦 → (1o ·o 𝑥) = (1o ·o suc 𝑦))
8 id 22 . . 3 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
97, 8eqeq12d 2752 . 2 (𝑥 = suc 𝑦 → ((1o ·o 𝑥) = 𝑥 ↔ (1o ·o suc 𝑦) = suc 𝑦))
10 oveq2 7346 . . 3 (𝑥 = 𝐴 → (1o ·o 𝑥) = (1o ·o 𝐴))
11 id 22 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
1210, 11eqeq12d 2752 . 2 (𝑥 = 𝐴 → ((1o ·o 𝑥) = 𝑥 ↔ (1o ·o 𝐴) = 𝐴))
13 1on 8380 . . 3 1o ∈ On
14 om0 8419 . . 3 (1o ∈ On → (1o ·o ∅) = ∅)
1513, 14ax-mp 5 . 2 (1o ·o ∅) = ∅
16 omsuc 8428 . . . . . 6 ((1o ∈ On ∧ 𝑦 ∈ On) → (1o ·o suc 𝑦) = ((1o ·o 𝑦) +o 1o))
1713, 16mpan 687 . . . . 5 (𝑦 ∈ On → (1o ·o suc 𝑦) = ((1o ·o 𝑦) +o 1o))
18 oveq1 7345 . . . . 5 ((1o ·o 𝑦) = 𝑦 → ((1o ·o 𝑦) +o 1o) = (𝑦 +o 1o))
1917, 18sylan9eq 2796 . . . 4 ((𝑦 ∈ On ∧ (1o ·o 𝑦) = 𝑦) → (1o ·o suc 𝑦) = (𝑦 +o 1o))
20 oa1suc 8433 . . . . 5 (𝑦 ∈ On → (𝑦 +o 1o) = suc 𝑦)
2120adantr 481 . . . 4 ((𝑦 ∈ On ∧ (1o ·o 𝑦) = 𝑦) → (𝑦 +o 1o) = suc 𝑦)
2219, 21eqtrd 2776 . . 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 4961 . . . 4 (∀𝑦𝑥 (1o ·o 𝑦) = 𝑦 𝑦𝑥 (1o ·o 𝑦) = 𝑦𝑥 𝑦)
25 uniiun 5006 . . . 4 𝑥 = 𝑦𝑥 𝑦
2624, 25eqtr4di 2794 . . 3 (∀𝑦𝑥 (1o ·o 𝑦) = 𝑦 𝑦𝑥 (1o ·o 𝑦) = 𝑥)
27 vex 3445 . . . . 5 𝑥 ∈ V
28 omlim 8435 . . . . . 6 ((1o ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (1o ·o 𝑥) = 𝑦𝑥 (1o ·o 𝑦))
2913, 28mpan 687 . . . . 5 ((𝑥 ∈ V ∧ Lim 𝑥) → (1o ·o 𝑥) = 𝑦𝑥 (1o ·o 𝑦))
3027, 29mpan 687 . . . 4 (Lim 𝑥 → (1o ·o 𝑥) = 𝑦𝑥 (1o ·o 𝑦))
31 limuni 6363 . . . 4 (Lim 𝑥𝑥 = 𝑥)
3230, 31eqeq12d 2752 . . 3 (Lim 𝑥 → ((1o ·o 𝑥) = 𝑥 𝑦𝑥 (1o ·o 𝑦) = 𝑥))
3326, 32syl5ibr 245 . 2 (Lim 𝑥 → (∀𝑦𝑥 (1o ·o 𝑦) = 𝑦 → (1o ·o 𝑥) = 𝑥))
343, 6, 9, 12, 15, 23, 33tfinds 7775 1 (𝐴 ∈ On → (1o ·o 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1540  wcel 2105  wral 3061  Vcvv 3441  c0 4270   cuni 4853   ciun 4942  Oncon0 6303  Lim wlim 6304  suc csuc 6305  (class class class)co 7338  1oc1o 8361   +o coa 8365   ·o comu 8366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5230  ax-sep 5244  ax-nul 5251  ax-pr 5373  ax-un 7651
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4271  df-if 4475  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4854  df-iun 4944  df-br 5094  df-opab 5156  df-mpt 5177  df-tr 5211  df-id 5519  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5576  df-we 5578  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6239  df-ord 6306  df-on 6307  df-lim 6308  df-suc 6309  df-iota 6432  df-fun 6482  df-fn 6483  df-f 6484  df-f1 6485  df-fo 6486  df-f1o 6487  df-fv 6488  df-ov 7341  df-oprab 7342  df-mpo 7343  df-om 7782  df-2nd 7901  df-frecs 8168  df-wrecs 8199  df-recs 8273  df-rdg 8312  df-1o 8368  df-oadd 8372  df-omul 8373
This theorem is referenced by:  oe1  8447  omword2  8477  omabs2  41369  omcl2  41370
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