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Theorem oe1m 8158
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 7147 . . 3 (𝑥 = ∅ → (1oo 𝑥) = (1oo ∅))
21eqeq1d 2803 . 2 (𝑥 = ∅ → ((1oo 𝑥) = 1o ↔ (1oo ∅) = 1o))
3 oveq2 7147 . . 3 (𝑥 = 𝑦 → (1oo 𝑥) = (1oo 𝑦))
43eqeq1d 2803 . 2 (𝑥 = 𝑦 → ((1oo 𝑥) = 1o ↔ (1oo 𝑦) = 1o))
5 oveq2 7147 . . 3 (𝑥 = suc 𝑦 → (1oo 𝑥) = (1oo suc 𝑦))
65eqeq1d 2803 . 2 (𝑥 = suc 𝑦 → ((1oo 𝑥) = 1o ↔ (1oo suc 𝑦) = 1o))
7 oveq2 7147 . . 3 (𝑥 = 𝐴 → (1oo 𝑥) = (1oo 𝐴))
87eqeq1d 2803 . 2 (𝑥 = 𝐴 → ((1oo 𝑥) = 1o ↔ (1oo 𝐴) = 1o))
9 1on 8096 . . 3 1o ∈ On
10 oe0 8134 . . 3 (1o ∈ On → (1oo ∅) = 1o)
119, 10ax-mp 5 . 2 (1oo ∅) = 1o
12 oesuc 8139 . . . . 5 ((1o ∈ On ∧ 𝑦 ∈ On) → (1oo suc 𝑦) = ((1oo 𝑦) ·o 1o))
139, 12mpan 689 . . . 4 (𝑦 ∈ On → (1oo suc 𝑦) = ((1oo 𝑦) ·o 1o))
14 oveq1 7146 . . . . 5 ((1oo 𝑦) = 1o → ((1oo 𝑦) ·o 1o) = (1o ·o 1o))
15 om1 8155 . . . . . 6 (1o ∈ On → (1o ·o 1o) = 1o)
169, 15ax-mp 5 . . . . 5 (1o ·o 1o) = 1o
1714, 16eqtrdi 2852 . . . 4 ((1oo 𝑦) = 1o → ((1oo 𝑦) ·o 1o) = 1o)
1813, 17sylan9eq 2856 . . 3 ((𝑦 ∈ On ∧ (1oo 𝑦) = 1o) → (1oo suc 𝑦) = 1o)
1918ex 416 . 2 (𝑦 ∈ On → ((1oo 𝑦) = 1o → (1oo suc 𝑦) = 1o))
20 iuneq2 4903 . . 3 (∀𝑦𝑥 (1oo 𝑦) = 1o 𝑦𝑥 (1oo 𝑦) = 𝑦𝑥 1o)
21 vex 3447 . . . . . 6 𝑥 ∈ V
22 0lt1o 8116 . . . . . . . 8 ∅ ∈ 1o
23 oelim 8146 . . . . . . . 8 (((1o ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 1o) → (1oo 𝑥) = 𝑦𝑥 (1oo 𝑦))
2422, 23mpan2 690 . . . . . . 7 ((1o ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (1oo 𝑥) = 𝑦𝑥 (1oo 𝑦))
259, 24mpan 689 . . . . . 6 ((𝑥 ∈ V ∧ Lim 𝑥) → (1oo 𝑥) = 𝑦𝑥 (1oo 𝑦))
2621, 25mpan 689 . . . . 5 (Lim 𝑥 → (1oo 𝑥) = 𝑦𝑥 (1oo 𝑦))
2726eqeq1d 2803 . . . 4 (Lim 𝑥 → ((1oo 𝑥) = 1o 𝑦𝑥 (1oo 𝑦) = 1o))
28 0ellim 6225 . . . . . 6 (Lim 𝑥 → ∅ ∈ 𝑥)
29 ne0i 4253 . . . . . 6 (∅ ∈ 𝑥𝑥 ≠ ∅)
30 iunconst 4893 . . . . . 6 (𝑥 ≠ ∅ → 𝑦𝑥 1o = 1o)
3128, 29, 303syl 18 . . . . 5 (Lim 𝑥 𝑦𝑥 1o = 1o)
3231eqeq2d 2812 . . . 4 (Lim 𝑥 → ( 𝑦𝑥 (1oo 𝑦) = 𝑦𝑥 1o 𝑦𝑥 (1oo 𝑦) = 1o))
3327, 32bitr4d 285 . . 3 (Lim 𝑥 → ((1oo 𝑥) = 1o 𝑦𝑥 (1oo 𝑦) = 𝑦𝑥 1o))
3420, 33syl5ibr 249 . 2 (Lim 𝑥 → (∀𝑦𝑥 (1oo 𝑦) = 1o → (1oo 𝑥) = 1o))
352, 4, 6, 8, 11, 19, 34tfinds 7558 1 (𝐴 ∈ On → (1oo 𝐴) = 1o)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wne 2990  wral 3109  Vcvv 3444  c0 4246   ciun 4884  Oncon0 6163  Lim wlim 6164  suc csuc 6165  (class class class)co 7139  1oc1o 8082   ·o comu 8087  o coe 8088
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-omul 8094  df-oexp 8095
This theorem is referenced by:  oewordi  8204  oeoe  8212  cantnflem2  9141
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