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Theorem oe1m 8451
Description: Ordinal exponentiation with a base 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 7349 . . 3 (𝑥 = ∅ → (1oo 𝑥) = (1oo ∅))
21eqeq1d 2739 . 2 (𝑥 = ∅ → ((1oo 𝑥) = 1o ↔ (1oo ∅) = 1o))
3 oveq2 7349 . . 3 (𝑥 = 𝑦 → (1oo 𝑥) = (1oo 𝑦))
43eqeq1d 2739 . 2 (𝑥 = 𝑦 → ((1oo 𝑥) = 1o ↔ (1oo 𝑦) = 1o))
5 oveq2 7349 . . 3 (𝑥 = suc 𝑦 → (1oo 𝑥) = (1oo suc 𝑦))
65eqeq1d 2739 . 2 (𝑥 = suc 𝑦 → ((1oo 𝑥) = 1o ↔ (1oo suc 𝑦) = 1o))
7 oveq2 7349 . . 3 (𝑥 = 𝐴 → (1oo 𝑥) = (1oo 𝐴))
87eqeq1d 2739 . 2 (𝑥 = 𝐴 → ((1oo 𝑥) = 1o ↔ (1oo 𝐴) = 1o))
9 1on 8383 . . 3 1o ∈ On
10 oe0 8427 . . 3 (1o ∈ On → (1oo ∅) = 1o)
119, 10ax-mp 5 . 2 (1oo ∅) = 1o
12 oesuc 8432 . . . . 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 7348 . . . . 5 ((1oo 𝑦) = 1o → ((1oo 𝑦) ·o 1o) = (1o ·o 1o))
15 om1 8448 . . . . . 6 (1o ∈ On → (1o ·o 1o) = 1o)
169, 15ax-mp 5 . . . . 5 (1o ·o 1o) = 1o
1714, 16eqtrdi 2793 . . . 4 ((1oo 𝑦) = 1o → ((1oo 𝑦) ·o 1o) = 1o)
1813, 17sylan9eq 2797 . . 3 ((𝑦 ∈ On ∧ (1oo 𝑦) = 1o) → (1oo suc 𝑦) = 1o)
1918ex 414 . 2 (𝑦 ∈ On → ((1oo 𝑦) = 1o → (1oo suc 𝑦) = 1o))
20 iuneq2 4964 . . 3 (∀𝑦𝑥 (1oo 𝑦) = 1o 𝑦𝑥 (1oo 𝑦) = 𝑦𝑥 1o)
21 vex 3446 . . . . . 6 𝑥 ∈ V
22 0lt1o 8409 . . . . . . . 8 ∅ ∈ 1o
23 oelim 8439 . . . . . . . 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 2739 . . . 4 (Lim 𝑥 → ((1oo 𝑥) = 1o 𝑦𝑥 (1oo 𝑦) = 1o))
28 0ellim 6368 . . . . . 6 (Lim 𝑥 → ∅ ∈ 𝑥)
29 ne0i 4285 . . . . . 6 (∅ ∈ 𝑥𝑥 ≠ ∅)
30 iunconst 4954 . . . . . 6 (𝑥 ≠ ∅ → 𝑦𝑥 1o = 1o)
3128, 29, 303syl 18 . . . . 5 (Lim 𝑥 𝑦𝑥 1o = 1o)
3231eqeq2d 2748 . . . 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, 33syl5ibr 246 . 2 (Lim 𝑥 → (∀𝑦𝑥 (1oo 𝑦) = 1o → (1oo 𝑥) = 1o))
352, 4, 6, 8, 11, 19, 34tfinds 7778 1 (𝐴 ∈ On → (1oo 𝐴) = 1o)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1541  wcel 2106  wne 2941  wral 3062  Vcvv 3442  c0 4273   ciun 4945  Oncon0 6306  Lim wlim 6307  suc csuc 6308  (class class class)co 7341  1oc1o 8364   ·o comu 8369  o coe 8370
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 2708  ax-rep 5233  ax-sep 5247  ax-nul 5254  ax-pr 5376  ax-un 7654
This theorem depends on definitions:  df-bi 206  df-an 398  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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3731  df-csb 3847  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3920  df-nul 4274  df-if 4478  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4857  df-iun 4947  df-br 5097  df-opab 5159  df-mpt 5180  df-tr 5214  df-id 5522  df-eprel 5528  df-po 5536  df-so 5537  df-fr 5579  df-we 5581  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6242  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6435  df-fun 6485  df-fn 6486  df-f 6487  df-f1 6488  df-fo 6489  df-f1o 6490  df-fv 6491  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7785  df-2nd 7904  df-frecs 8171  df-wrecs 8202  df-recs 8276  df-rdg 8315  df-1o 8371  df-oadd 8375  df-omul 8376  df-oexp 8377
This theorem is referenced by:  oewordi  8497  oeoe  8505  cantnflem2  9551
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