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Theorem oe1m 8575
Description: Ordinal exponentiation with a base of 1. Proposition 8.31(3) of [TakeutiZaring] p. 67. Lemma 2.17 of [Schloeder] p. 6. (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 7432 . . 3 (𝑥 = ∅ → (1oo 𝑥) = (1oo ∅))
21eqeq1d 2728 . 2 (𝑥 = ∅ → ((1oo 𝑥) = 1o ↔ (1oo ∅) = 1o))
3 oveq2 7432 . . 3 (𝑥 = 𝑦 → (1oo 𝑥) = (1oo 𝑦))
43eqeq1d 2728 . 2 (𝑥 = 𝑦 → ((1oo 𝑥) = 1o ↔ (1oo 𝑦) = 1o))
5 oveq2 7432 . . 3 (𝑥 = suc 𝑦 → (1oo 𝑥) = (1oo suc 𝑦))
65eqeq1d 2728 . 2 (𝑥 = suc 𝑦 → ((1oo 𝑥) = 1o ↔ (1oo suc 𝑦) = 1o))
7 oveq2 7432 . . 3 (𝑥 = 𝐴 → (1oo 𝑥) = (1oo 𝐴))
87eqeq1d 2728 . 2 (𝑥 = 𝐴 → ((1oo 𝑥) = 1o ↔ (1oo 𝐴) = 1o))
9 1on 8508 . . 3 1o ∈ On
10 oe0 8552 . . 3 (1o ∈ On → (1oo ∅) = 1o)
119, 10ax-mp 5 . 2 (1oo ∅) = 1o
12 oesuc 8557 . . . . 5 ((1o ∈ On ∧ 𝑦 ∈ On) → (1oo suc 𝑦) = ((1oo 𝑦) ·o 1o))
139, 12mpan 688 . . . 4 (𝑦 ∈ On → (1oo suc 𝑦) = ((1oo 𝑦) ·o 1o))
14 oveq1 7431 . . . . 5 ((1oo 𝑦) = 1o → ((1oo 𝑦) ·o 1o) = (1o ·o 1o))
15 om1 8572 . . . . . 6 (1o ∈ On → (1o ·o 1o) = 1o)
169, 15ax-mp 5 . . . . 5 (1o ·o 1o) = 1o
1714, 16eqtrdi 2782 . . . 4 ((1oo 𝑦) = 1o → ((1oo 𝑦) ·o 1o) = 1o)
1813, 17sylan9eq 2786 . . 3 ((𝑦 ∈ On ∧ (1oo 𝑦) = 1o) → (1oo suc 𝑦) = 1o)
1918ex 411 . 2 (𝑦 ∈ On → ((1oo 𝑦) = 1o → (1oo suc 𝑦) = 1o))
20 iuneq2 5020 . . 3 (∀𝑦𝑥 (1oo 𝑦) = 1o 𝑦𝑥 (1oo 𝑦) = 𝑦𝑥 1o)
21 vex 3466 . . . . . 6 𝑥 ∈ V
22 0lt1o 8534 . . . . . . . 8 ∅ ∈ 1o
23 oelim 8564 . . . . . . . 8 (((1o ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 1o) → (1oo 𝑥) = 𝑦𝑥 (1oo 𝑦))
2422, 23mpan2 689 . . . . . . 7 ((1o ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (1oo 𝑥) = 𝑦𝑥 (1oo 𝑦))
259, 24mpan 688 . . . . . 6 ((𝑥 ∈ V ∧ Lim 𝑥) → (1oo 𝑥) = 𝑦𝑥 (1oo 𝑦))
2621, 25mpan 688 . . . . 5 (Lim 𝑥 → (1oo 𝑥) = 𝑦𝑥 (1oo 𝑦))
2726eqeq1d 2728 . . . 4 (Lim 𝑥 → ((1oo 𝑥) = 1o 𝑦𝑥 (1oo 𝑦) = 1o))
28 0ellim 6439 . . . . . 6 (Lim 𝑥 → ∅ ∈ 𝑥)
29 ne0i 4337 . . . . . 6 (∅ ∈ 𝑥𝑥 ≠ ∅)
30 iunconst 5010 . . . . . 6 (𝑥 ≠ ∅ → 𝑦𝑥 1o = 1o)
3128, 29, 303syl 18 . . . . 5 (Lim 𝑥 𝑦𝑥 1o = 1o)
3231eqeq2d 2737 . . . 4 (Lim 𝑥 → ( 𝑦𝑥 (1oo 𝑦) = 𝑦𝑥 1o 𝑦𝑥 (1oo 𝑦) = 1o))
3327, 32bitr4d 281 . . 3 (Lim 𝑥 → ((1oo 𝑥) = 1o 𝑦𝑥 (1oo 𝑦) = 𝑦𝑥 1o))
3420, 33imbitrrid 245 . 2 (Lim 𝑥 → (∀𝑦𝑥 (1oo 𝑦) = 1o → (1oo 𝑥) = 1o))
352, 4, 6, 8, 11, 19, 34tfinds 7870 1 (𝐴 ∈ On → (1oo 𝐴) = 1o)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  wne 2930  wral 3051  Vcvv 3462  c0 4325   ciun 5001  Oncon0 6376  Lim wlim 6377  suc csuc 6378  (class class class)co 7424  1oc1o 8489   ·o comu 8494  o coe 8495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-oadd 8500  df-omul 8501  df-oexp 8502
This theorem is referenced by:  oewordi  8621  oeoe  8629  cantnflem2  9733
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