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Theorem oe1m 8492
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 7365 . . 3 (𝑥 = ∅ → (1oo 𝑥) = (1oo ∅))
21eqeq1d 2738 . 2 (𝑥 = ∅ → ((1oo 𝑥) = 1o ↔ (1oo ∅) = 1o))
3 oveq2 7365 . . 3 (𝑥 = 𝑦 → (1oo 𝑥) = (1oo 𝑦))
43eqeq1d 2738 . 2 (𝑥 = 𝑦 → ((1oo 𝑥) = 1o ↔ (1oo 𝑦) = 1o))
5 oveq2 7365 . . 3 (𝑥 = suc 𝑦 → (1oo 𝑥) = (1oo suc 𝑦))
65eqeq1d 2738 . 2 (𝑥 = suc 𝑦 → ((1oo 𝑥) = 1o ↔ (1oo suc 𝑦) = 1o))
7 oveq2 7365 . . 3 (𝑥 = 𝐴 → (1oo 𝑥) = (1oo 𝐴))
87eqeq1d 2738 . 2 (𝑥 = 𝐴 → ((1oo 𝑥) = 1o ↔ (1oo 𝐴) = 1o))
9 1on 8424 . . 3 1o ∈ On
10 oe0 8468 . . 3 (1o ∈ On → (1oo ∅) = 1o)
119, 10ax-mp 5 . 2 (1oo ∅) = 1o
12 oesuc 8473 . . . . 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 7364 . . . . 5 ((1oo 𝑦) = 1o → ((1oo 𝑦) ·o 1o) = (1o ·o 1o))
15 om1 8489 . . . . . 6 (1o ∈ On → (1o ·o 1o) = 1o)
169, 15ax-mp 5 . . . . 5 (1o ·o 1o) = 1o
1714, 16eqtrdi 2792 . . . 4 ((1oo 𝑦) = 1o → ((1oo 𝑦) ·o 1o) = 1o)
1813, 17sylan9eq 2796 . . 3 ((𝑦 ∈ On ∧ (1oo 𝑦) = 1o) → (1oo suc 𝑦) = 1o)
1918ex 413 . 2 (𝑦 ∈ On → ((1oo 𝑦) = 1o → (1oo suc 𝑦) = 1o))
20 iuneq2 4973 . . 3 (∀𝑦𝑥 (1oo 𝑦) = 1o 𝑦𝑥 (1oo 𝑦) = 𝑦𝑥 1o)
21 vex 3449 . . . . . 6 𝑥 ∈ V
22 0lt1o 8450 . . . . . . . 8 ∅ ∈ 1o
23 oelim 8480 . . . . . . . 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 2738 . . . 4 (Lim 𝑥 → ((1oo 𝑥) = 1o 𝑦𝑥 (1oo 𝑦) = 1o))
28 0ellim 6380 . . . . . 6 (Lim 𝑥 → ∅ ∈ 𝑥)
29 ne0i 4294 . . . . . 6 (∅ ∈ 𝑥𝑥 ≠ ∅)
30 iunconst 4963 . . . . . 6 (𝑥 ≠ ∅ → 𝑦𝑥 1o = 1o)
3128, 29, 303syl 18 . . . . 5 (Lim 𝑥 𝑦𝑥 1o = 1o)
3231eqeq2d 2747 . . . 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, 33syl5ibr 245 . 2 (Lim 𝑥 → (∀𝑦𝑥 (1oo 𝑦) = 1o → (1oo 𝑥) = 1o))
352, 4, 6, 8, 11, 19, 34tfinds 7796 1 (𝐴 ∈ On → (1oo 𝐴) = 1o)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wne 2943  wral 3064  Vcvv 3445  c0 4282   ciun 4954  Oncon0 6317  Lim wlim 6318  suc csuc 6319  (class class class)co 7357  1oc1o 8405   ·o comu 8410  o coe 8411
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-omul 8417  df-oexp 8418
This theorem is referenced by:  oewordi  8538  oeoe  8546  cantnflem2  9626
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