MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oecan Structured version   Visualization version   GIF version

Theorem oecan 8239
Description: Left cancellation law for ordinal exponentiation. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)
Assertion
Ref Expression
oecan ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴o 𝐵) = (𝐴o 𝐶) ↔ 𝐵 = 𝐶))

Proof of Theorem oecan
StepHypRef Expression
1 oeordi 8237 . . . . . . 7 ((𝐶 ∈ On ∧ 𝐴 ∈ (On ∖ 2o)) → (𝐵𝐶 → (𝐴o 𝐵) ∈ (𝐴o 𝐶)))
21ancoms 462 . . . . . 6 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐶 ∈ On) → (𝐵𝐶 → (𝐴o 𝐵) ∈ (𝐴o 𝐶)))
323adant2 1132 . . . . 5 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝐶 → (𝐴o 𝐵) ∈ (𝐴o 𝐶)))
4 oeordi 8237 . . . . . . 7 ((𝐵 ∈ On ∧ 𝐴 ∈ (On ∖ 2o)) → (𝐶𝐵 → (𝐴o 𝐶) ∈ (𝐴o 𝐵)))
54ancoms 462 . . . . . 6 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → (𝐶𝐵 → (𝐴o 𝐶) ∈ (𝐴o 𝐵)))
653adant3 1133 . . . . 5 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐶𝐵 → (𝐴o 𝐶) ∈ (𝐴o 𝐵)))
73, 6orim12d 964 . . . 4 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵𝐶𝐶𝐵) → ((𝐴o 𝐵) ∈ (𝐴o 𝐶) ∨ (𝐴o 𝐶) ∈ (𝐴o 𝐵))))
87con3d 155 . . 3 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (¬ ((𝐴o 𝐵) ∈ (𝐴o 𝐶) ∨ (𝐴o 𝐶) ∈ (𝐴o 𝐵)) → ¬ (𝐵𝐶𝐶𝐵)))
9 eldifi 4015 . . . . . 6 (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
1093ad2ant1 1134 . . . . 5 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐴 ∈ On)
11 simp2 1138 . . . . 5 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐵 ∈ On)
12 oecl 8186 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) ∈ On)
1310, 11, 12syl2anc 587 . . . 4 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴o 𝐵) ∈ On)
14 simp3 1139 . . . . 5 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐶 ∈ On)
15 oecl 8186 . . . . 5 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴o 𝐶) ∈ On)
1610, 14, 15syl2anc 587 . . . 4 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴o 𝐶) ∈ On)
17 eloni 6176 . . . . 5 ((𝐴o 𝐵) ∈ On → Ord (𝐴o 𝐵))
18 eloni 6176 . . . . 5 ((𝐴o 𝐶) ∈ On → Ord (𝐴o 𝐶))
19 ordtri3 6202 . . . . 5 ((Ord (𝐴o 𝐵) ∧ Ord (𝐴o 𝐶)) → ((𝐴o 𝐵) = (𝐴o 𝐶) ↔ ¬ ((𝐴o 𝐵) ∈ (𝐴o 𝐶) ∨ (𝐴o 𝐶) ∈ (𝐴o 𝐵))))
2017, 18, 19syl2an 599 . . . 4 (((𝐴o 𝐵) ∈ On ∧ (𝐴o 𝐶) ∈ On) → ((𝐴o 𝐵) = (𝐴o 𝐶) ↔ ¬ ((𝐴o 𝐵) ∈ (𝐴o 𝐶) ∨ (𝐴o 𝐶) ∈ (𝐴o 𝐵))))
2113, 16, 20syl2anc 587 . . 3 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴o 𝐵) = (𝐴o 𝐶) ↔ ¬ ((𝐴o 𝐵) ∈ (𝐴o 𝐶) ∨ (𝐴o 𝐶) ∈ (𝐴o 𝐵))))
22 eloni 6176 . . . . 5 (𝐵 ∈ On → Ord 𝐵)
23 eloni 6176 . . . . 5 (𝐶 ∈ On → Ord 𝐶)
24 ordtri3 6202 . . . . 5 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝐶𝐶𝐵)))
2522, 23, 24syl2an 599 . . . 4 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝐶𝐶𝐵)))
26253adant1 1131 . . 3 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝐶𝐶𝐵)))
278, 21, 263imtr4d 297 . 2 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴o 𝐵) = (𝐴o 𝐶) → 𝐵 = 𝐶))
28 oveq2 7172 . 2 (𝐵 = 𝐶 → (𝐴o 𝐵) = (𝐴o 𝐶))
2927, 28impbid1 228 1 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴o 𝐵) = (𝐴o 𝐶) ↔ 𝐵 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wo 846  w3a 1088   = wceq 1542  wcel 2113  cdif 3838  Ord word 6165  Oncon0 6166  (class class class)co 7164  2oc2o 8118  o coe 8123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pr 5293  ax-un 7473
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-pss 3860  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-tp 4518  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-tr 5134  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-ov 7167  df-oprab 7168  df-mpo 7169  df-om 7594  df-wrecs 7969  df-recs 8030  df-rdg 8068  df-1o 8124  df-2o 8125  df-oadd 8128  df-omul 8129  df-oexp 8130
This theorem is referenced by:  oeword  8240  infxpenc2lem1  9512
  Copyright terms: Public domain W3C validator