Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  oe1m Structured version   Visualization version   GIF version

Theorem oe1m 7670
 Description: Ordinal exponentiation with a mantissa of 1. Proposition 8.31(3) of [TakeutiZaring] p. 67. (Contributed by NM, 2-Jan-2005.)
Assertion
Ref Expression
oe1m (𝐴 ∈ On → (1𝑜𝑜 𝐴) = 1𝑜)

Proof of Theorem oe1m
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6698 . . 3 (𝑥 = ∅ → (1𝑜𝑜 𝑥) = (1𝑜𝑜 ∅))
21eqeq1d 2653 . 2 (𝑥 = ∅ → ((1𝑜𝑜 𝑥) = 1𝑜 ↔ (1𝑜𝑜 ∅) = 1𝑜))
3 oveq2 6698 . . 3 (𝑥 = 𝑦 → (1𝑜𝑜 𝑥) = (1𝑜𝑜 𝑦))
43eqeq1d 2653 . 2 (𝑥 = 𝑦 → ((1𝑜𝑜 𝑥) = 1𝑜 ↔ (1𝑜𝑜 𝑦) = 1𝑜))
5 oveq2 6698 . . 3 (𝑥 = suc 𝑦 → (1𝑜𝑜 𝑥) = (1𝑜𝑜 suc 𝑦))
65eqeq1d 2653 . 2 (𝑥 = suc 𝑦 → ((1𝑜𝑜 𝑥) = 1𝑜 ↔ (1𝑜𝑜 suc 𝑦) = 1𝑜))
7 oveq2 6698 . . 3 (𝑥 = 𝐴 → (1𝑜𝑜 𝑥) = (1𝑜𝑜 𝐴))
87eqeq1d 2653 . 2 (𝑥 = 𝐴 → ((1𝑜𝑜 𝑥) = 1𝑜 ↔ (1𝑜𝑜 𝐴) = 1𝑜))
9 1on 7612 . . 3 1𝑜 ∈ On
10 oe0 7647 . . 3 (1𝑜 ∈ On → (1𝑜𝑜 ∅) = 1𝑜)
119, 10ax-mp 5 . 2 (1𝑜𝑜 ∅) = 1𝑜
12 oesuc 7652 . . . . 5 ((1𝑜 ∈ On ∧ 𝑦 ∈ On) → (1𝑜𝑜 suc 𝑦) = ((1𝑜𝑜 𝑦) ·𝑜 1𝑜))
139, 12mpan 706 . . . 4 (𝑦 ∈ On → (1𝑜𝑜 suc 𝑦) = ((1𝑜𝑜 𝑦) ·𝑜 1𝑜))
14 oveq1 6697 . . . . 5 ((1𝑜𝑜 𝑦) = 1𝑜 → ((1𝑜𝑜 𝑦) ·𝑜 1𝑜) = (1𝑜 ·𝑜 1𝑜))
15 om1 7667 . . . . . 6 (1𝑜 ∈ On → (1𝑜 ·𝑜 1𝑜) = 1𝑜)
169, 15ax-mp 5 . . . . 5 (1𝑜 ·𝑜 1𝑜) = 1𝑜
1714, 16syl6eq 2701 . . . 4 ((1𝑜𝑜 𝑦) = 1𝑜 → ((1𝑜𝑜 𝑦) ·𝑜 1𝑜) = 1𝑜)
1813, 17sylan9eq 2705 . . 3 ((𝑦 ∈ On ∧ (1𝑜𝑜 𝑦) = 1𝑜) → (1𝑜𝑜 suc 𝑦) = 1𝑜)
1918ex 449 . 2 (𝑦 ∈ On → ((1𝑜𝑜 𝑦) = 1𝑜 → (1𝑜𝑜 suc 𝑦) = 1𝑜))
20 iuneq2 4569 . . 3 (∀𝑦𝑥 (1𝑜𝑜 𝑦) = 1𝑜 𝑦𝑥 (1𝑜𝑜 𝑦) = 𝑦𝑥 1𝑜)
21 vex 3234 . . . . . 6 𝑥 ∈ V
22 0lt1o 7629 . . . . . . . 8 ∅ ∈ 1𝑜
23 oelim 7659 . . . . . . . 8 (((1𝑜 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 1𝑜) → (1𝑜𝑜 𝑥) = 𝑦𝑥 (1𝑜𝑜 𝑦))
2422, 23mpan2 707 . . . . . . 7 ((1𝑜 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (1𝑜𝑜 𝑥) = 𝑦𝑥 (1𝑜𝑜 𝑦))
259, 24mpan 706 . . . . . 6 ((𝑥 ∈ V ∧ Lim 𝑥) → (1𝑜𝑜 𝑥) = 𝑦𝑥 (1𝑜𝑜 𝑦))
2621, 25mpan 706 . . . . 5 (Lim 𝑥 → (1𝑜𝑜 𝑥) = 𝑦𝑥 (1𝑜𝑜 𝑦))
2726eqeq1d 2653 . . . 4 (Lim 𝑥 → ((1𝑜𝑜 𝑥) = 1𝑜 𝑦𝑥 (1𝑜𝑜 𝑦) = 1𝑜))
28 0ellim 5825 . . . . . 6 (Lim 𝑥 → ∅ ∈ 𝑥)
29 ne0i 3954 . . . . . 6 (∅ ∈ 𝑥𝑥 ≠ ∅)
30 iunconst 4561 . . . . . 6 (𝑥 ≠ ∅ → 𝑦𝑥 1𝑜 = 1𝑜)
3128, 29, 303syl 18 . . . . 5 (Lim 𝑥 𝑦𝑥 1𝑜 = 1𝑜)
3231eqeq2d 2661 . . . 4 (Lim 𝑥 → ( 𝑦𝑥 (1𝑜𝑜 𝑦) = 𝑦𝑥 1𝑜 𝑦𝑥 (1𝑜𝑜 𝑦) = 1𝑜))
3327, 32bitr4d 271 . . 3 (Lim 𝑥 → ((1𝑜𝑜 𝑥) = 1𝑜 𝑦𝑥 (1𝑜𝑜 𝑦) = 𝑦𝑥 1𝑜))
3420, 33syl5ibr 236 . 2 (Lim 𝑥 → (∀𝑦𝑥 (1𝑜𝑜 𝑦) = 1𝑜 → (1𝑜𝑜 𝑥) = 1𝑜))
352, 4, 6, 8, 11, 19, 34tfinds 7101 1 (𝐴 ∈ On → (1𝑜𝑜 𝐴) = 1𝑜)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  ∀wral 2941  Vcvv 3231  ∅c0 3948  ∪ ciun 4552  Oncon0 5761  Lim wlim 5762  suc csuc 5763  (class class class)co 6690  1𝑜c1o 7598   ·𝑜 comu 7603   ↑𝑜 coe 7604 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610  df-oexp 7611 This theorem is referenced by:  oewordi  7716  oeoe  7724  cantnflem2  8625
 Copyright terms: Public domain W3C validator