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Theorem om1r 8490
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 7365 . . 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 7365 . . 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 7365 . . 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 7365 . . 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 8424 . . 3 1o ∈ On
14 om0 8463 . . 3 (1o ∈ On → (1o ·o ∅) = ∅)
1513, 14ax-mp 5 . 2 (1o ·o ∅) = ∅
16 omsuc 8472 . . . . . 6 ((1o ∈ On ∧ 𝑦 ∈ On) → (1o ·o suc 𝑦) = ((1o ·o 𝑦) +o 1o))
1713, 16mpan 688 . . . . 5 (𝑦 ∈ On → (1o ·o suc 𝑦) = ((1o ·o 𝑦) +o 1o))
18 oveq1 7364 . . . . 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 8477 . . . . 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 4973 . . . 4 (∀𝑦𝑥 (1o ·o 𝑦) = 𝑦 𝑦𝑥 (1o ·o 𝑦) = 𝑦𝑥 𝑦)
25 uniiun 5018 . . . 4 𝑥 = 𝑦𝑥 𝑦
2624, 25eqtr4di 2794 . . 3 (∀𝑦𝑥 (1o ·o 𝑦) = 𝑦 𝑦𝑥 (1o ·o 𝑦) = 𝑥)
27 vex 3449 . . . . 5 𝑥 ∈ V
28 omlim 8479 . . . . . 6 ((1o ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (1o ·o 𝑥) = 𝑦𝑥 (1o ·o 𝑦))
2913, 28mpan 688 . . . . 5 ((𝑥 ∈ V ∧ Lim 𝑥) → (1o ·o 𝑥) = 𝑦𝑥 (1o ·o 𝑦))
3027, 29mpan 688 . . . 4 (Lim 𝑥 → (1o ·o 𝑥) = 𝑦𝑥 (1o ·o 𝑦))
31 limuni 6378 . . . 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 7796 1 (𝐴 ∈ On → (1o ·o 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3064  Vcvv 3445  c0 4282   cuni 4865   ciun 4954  Oncon0 6317  Lim wlim 6318  suc csuc 6319  (class class class)co 7357  1oc1o 8405   +o coa 8409   ·o comu 8410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-omul 8417
This theorem is referenced by:  oe1  8491  omword2  8521  om1om1r  41605  omabs2  41651  omcl2  41652
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