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Theorem oe1m 7485
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 6531 . . 3 (𝑥 = ∅ → (1𝑜𝑜 𝑥) = (1𝑜𝑜 ∅))
21eqeq1d 2607 . 2 (𝑥 = ∅ → ((1𝑜𝑜 𝑥) = 1𝑜 ↔ (1𝑜𝑜 ∅) = 1𝑜))
3 oveq2 6531 . . 3 (𝑥 = 𝑦 → (1𝑜𝑜 𝑥) = (1𝑜𝑜 𝑦))
43eqeq1d 2607 . 2 (𝑥 = 𝑦 → ((1𝑜𝑜 𝑥) = 1𝑜 ↔ (1𝑜𝑜 𝑦) = 1𝑜))
5 oveq2 6531 . . 3 (𝑥 = suc 𝑦 → (1𝑜𝑜 𝑥) = (1𝑜𝑜 suc 𝑦))
65eqeq1d 2607 . 2 (𝑥 = suc 𝑦 → ((1𝑜𝑜 𝑥) = 1𝑜 ↔ (1𝑜𝑜 suc 𝑦) = 1𝑜))
7 oveq2 6531 . . 3 (𝑥 = 𝐴 → (1𝑜𝑜 𝑥) = (1𝑜𝑜 𝐴))
87eqeq1d 2607 . 2 (𝑥 = 𝐴 → ((1𝑜𝑜 𝑥) = 1𝑜 ↔ (1𝑜𝑜 𝐴) = 1𝑜))
9 1on 7427 . . 3 1𝑜 ∈ On
10 oe0 7462 . . 3 (1𝑜 ∈ On → (1𝑜𝑜 ∅) = 1𝑜)
119, 10ax-mp 5 . 2 (1𝑜𝑜 ∅) = 1𝑜
12 oesuc 7467 . . . . 5 ((1𝑜 ∈ On ∧ 𝑦 ∈ On) → (1𝑜𝑜 suc 𝑦) = ((1𝑜𝑜 𝑦) ·𝑜 1𝑜))
139, 12mpan 701 . . . 4 (𝑦 ∈ On → (1𝑜𝑜 suc 𝑦) = ((1𝑜𝑜 𝑦) ·𝑜 1𝑜))
14 oveq1 6530 . . . . 5 ((1𝑜𝑜 𝑦) = 1𝑜 → ((1𝑜𝑜 𝑦) ·𝑜 1𝑜) = (1𝑜 ·𝑜 1𝑜))
15 om1 7482 . . . . . 6 (1𝑜 ∈ On → (1𝑜 ·𝑜 1𝑜) = 1𝑜)
169, 15ax-mp 5 . . . . 5 (1𝑜 ·𝑜 1𝑜) = 1𝑜
1714, 16syl6eq 2655 . . . 4 ((1𝑜𝑜 𝑦) = 1𝑜 → ((1𝑜𝑜 𝑦) ·𝑜 1𝑜) = 1𝑜)
1813, 17sylan9eq 2659 . . 3 ((𝑦 ∈ On ∧ (1𝑜𝑜 𝑦) = 1𝑜) → (1𝑜𝑜 suc 𝑦) = 1𝑜)
1918ex 448 . 2 (𝑦 ∈ On → ((1𝑜𝑜 𝑦) = 1𝑜 → (1𝑜𝑜 suc 𝑦) = 1𝑜))
20 iuneq2 4463 . . 3 (∀𝑦𝑥 (1𝑜𝑜 𝑦) = 1𝑜 𝑦𝑥 (1𝑜𝑜 𝑦) = 𝑦𝑥 1𝑜)
21 vex 3171 . . . . . 6 𝑥 ∈ V
22 0lt1o 7444 . . . . . . . 8 ∅ ∈ 1𝑜
23 oelim 7474 . . . . . . . 8 (((1𝑜 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 1𝑜) → (1𝑜𝑜 𝑥) = 𝑦𝑥 (1𝑜𝑜 𝑦))
2422, 23mpan2 702 . . . . . . 7 ((1𝑜 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (1𝑜𝑜 𝑥) = 𝑦𝑥 (1𝑜𝑜 𝑦))
259, 24mpan 701 . . . . . 6 ((𝑥 ∈ V ∧ Lim 𝑥) → (1𝑜𝑜 𝑥) = 𝑦𝑥 (1𝑜𝑜 𝑦))
2621, 25mpan 701 . . . . 5 (Lim 𝑥 → (1𝑜𝑜 𝑥) = 𝑦𝑥 (1𝑜𝑜 𝑦))
2726eqeq1d 2607 . . . 4 (Lim 𝑥 → ((1𝑜𝑜 𝑥) = 1𝑜 𝑦𝑥 (1𝑜𝑜 𝑦) = 1𝑜))
28 0ellim 5686 . . . . . 6 (Lim 𝑥 → ∅ ∈ 𝑥)
29 ne0i 3875 . . . . . 6 (∅ ∈ 𝑥𝑥 ≠ ∅)
30 iunconst 4455 . . . . . 6 (𝑥 ≠ ∅ → 𝑦𝑥 1𝑜 = 1𝑜)
3128, 29, 303syl 18 . . . . 5 (Lim 𝑥 𝑦𝑥 1𝑜 = 1𝑜)
3231eqeq2d 2615 . . . 4 (Lim 𝑥 → ( 𝑦𝑥 (1𝑜𝑜 𝑦) = 𝑦𝑥 1𝑜 𝑦𝑥 (1𝑜𝑜 𝑦) = 1𝑜))
3327, 32bitr4d 269 . . 3 (Lim 𝑥 → ((1𝑜𝑜 𝑥) = 1𝑜 𝑦𝑥 (1𝑜𝑜 𝑦) = 𝑦𝑥 1𝑜))
3420, 33syl5ibr 234 . 2 (Lim 𝑥 → (∀𝑦𝑥 (1𝑜𝑜 𝑦) = 1𝑜 → (1𝑜𝑜 𝑥) = 1𝑜))
352, 4, 6, 8, 11, 19, 34tfinds 6924 1 (𝐴 ∈ On → (1𝑜𝑜 𝐴) = 1𝑜)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1975  wne 2775  wral 2891  Vcvv 3168  c0 3869   ciun 4445  Oncon0 5622  Lim wlim 5623  suc csuc 5624  (class class class)co 6523  1𝑜c1o 7413   ·𝑜 comu 7418  𝑜 coe 7419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-om 6931  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-oadd 7424  df-omul 7425  df-oexp 7426
This theorem is referenced by:  oewordi  7531  oeoe  7539  cantnflem2  8443
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