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Theorem oe1m 8582
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 7439 . . 3 (𝑥 = ∅ → (1oo 𝑥) = (1oo ∅))
21eqeq1d 2737 . 2 (𝑥 = ∅ → ((1oo 𝑥) = 1o ↔ (1oo ∅) = 1o))
3 oveq2 7439 . . 3 (𝑥 = 𝑦 → (1oo 𝑥) = (1oo 𝑦))
43eqeq1d 2737 . 2 (𝑥 = 𝑦 → ((1oo 𝑥) = 1o ↔ (1oo 𝑦) = 1o))
5 oveq2 7439 . . 3 (𝑥 = suc 𝑦 → (1oo 𝑥) = (1oo suc 𝑦))
65eqeq1d 2737 . 2 (𝑥 = suc 𝑦 → ((1oo 𝑥) = 1o ↔ (1oo suc 𝑦) = 1o))
7 oveq2 7439 . . 3 (𝑥 = 𝐴 → (1oo 𝑥) = (1oo 𝐴))
87eqeq1d 2737 . 2 (𝑥 = 𝐴 → ((1oo 𝑥) = 1o ↔ (1oo 𝐴) = 1o))
9 1on 8517 . . 3 1o ∈ On
10 oe0 8559 . . 3 (1o ∈ On → (1oo ∅) = 1o)
119, 10ax-mp 5 . 2 (1oo ∅) = 1o
12 oesuc 8564 . . . . 5 ((1o ∈ On ∧ 𝑦 ∈ On) → (1oo suc 𝑦) = ((1oo 𝑦) ·o 1o))
139, 12mpan 690 . . . 4 (𝑦 ∈ On → (1oo suc 𝑦) = ((1oo 𝑦) ·o 1o))
14 oveq1 7438 . . . . 5 ((1oo 𝑦) = 1o → ((1oo 𝑦) ·o 1o) = (1o ·o 1o))
15 om1 8579 . . . . . 6 (1o ∈ On → (1o ·o 1o) = 1o)
169, 15ax-mp 5 . . . . 5 (1o ·o 1o) = 1o
1714, 16eqtrdi 2791 . . . 4 ((1oo 𝑦) = 1o → ((1oo 𝑦) ·o 1o) = 1o)
1813, 17sylan9eq 2795 . . 3 ((𝑦 ∈ On ∧ (1oo 𝑦) = 1o) → (1oo suc 𝑦) = 1o)
1918ex 412 . 2 (𝑦 ∈ On → ((1oo 𝑦) = 1o → (1oo suc 𝑦) = 1o))
20 iuneq2 5016 . . 3 (∀𝑦𝑥 (1oo 𝑦) = 1o 𝑦𝑥 (1oo 𝑦) = 𝑦𝑥 1o)
21 vex 3482 . . . . . 6 𝑥 ∈ V
22 0lt1o 8541 . . . . . . . 8 ∅ ∈ 1o
23 oelim 8571 . . . . . . . 8 (((1o ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 1o) → (1oo 𝑥) = 𝑦𝑥 (1oo 𝑦))
2422, 23mpan2 691 . . . . . . 7 ((1o ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (1oo 𝑥) = 𝑦𝑥 (1oo 𝑦))
259, 24mpan 690 . . . . . 6 ((𝑥 ∈ V ∧ Lim 𝑥) → (1oo 𝑥) = 𝑦𝑥 (1oo 𝑦))
2621, 25mpan 690 . . . . 5 (Lim 𝑥 → (1oo 𝑥) = 𝑦𝑥 (1oo 𝑦))
2726eqeq1d 2737 . . . 4 (Lim 𝑥 → ((1oo 𝑥) = 1o 𝑦𝑥 (1oo 𝑦) = 1o))
28 0ellim 6449 . . . . . 6 (Lim 𝑥 → ∅ ∈ 𝑥)
29 ne0i 4347 . . . . . 6 (∅ ∈ 𝑥𝑥 ≠ ∅)
30 iunconst 5006 . . . . . 6 (𝑥 ≠ ∅ → 𝑦𝑥 1o = 1o)
3128, 29, 303syl 18 . . . . 5 (Lim 𝑥 𝑦𝑥 1o = 1o)
3231eqeq2d 2746 . . . 4 (Lim 𝑥 → ( 𝑦𝑥 (1oo 𝑦) = 𝑦𝑥 1o 𝑦𝑥 (1oo 𝑦) = 1o))
3327, 32bitr4d 282 . . 3 (Lim 𝑥 → ((1oo 𝑥) = 1o 𝑦𝑥 (1oo 𝑦) = 𝑦𝑥 1o))
3420, 33imbitrrid 246 . 2 (Lim 𝑥 → (∀𝑦𝑥 (1oo 𝑦) = 1o → (1oo 𝑥) = 1o))
352, 4, 6, 8, 11, 19, 34tfinds 7881 1 (𝐴 ∈ On → (1oo 𝐴) = 1o)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wne 2938  wral 3059  Vcvv 3478  c0 4339   ciun 4996  Oncon0 6386  Lim wlim 6387  suc csuc 6388  (class class class)co 7431  1oc1o 8498   ·o comu 8503  o coe 8504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-omul 8510  df-oexp 8511
This theorem is referenced by:  oewordi  8628  oeoe  8636  cantnflem2  9728
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