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

Theorem oecan 7629
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 7627 . . . . . . 7 ((𝐶 ∈ On ∧ 𝐴 ∈ (On ∖ 2𝑜)) → (𝐵𝐶 → (𝐴𝑜 𝐵) ∈ (𝐴𝑜 𝐶)))
21ancoms 469 . . . . . 6 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐶 ∈ On) → (𝐵𝐶 → (𝐴𝑜 𝐵) ∈ (𝐴𝑜 𝐶)))
323adant2 1078 . . . . 5 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝐶 → (𝐴𝑜 𝐵) ∈ (𝐴𝑜 𝐶)))
4 oeordi 7627 . . . . . . 7 ((𝐵 ∈ On ∧ 𝐴 ∈ (On ∖ 2𝑜)) → (𝐶𝐵 → (𝐴𝑜 𝐶) ∈ (𝐴𝑜 𝐵)))
54ancoms 469 . . . . . 6 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On) → (𝐶𝐵 → (𝐴𝑜 𝐶) ∈ (𝐴𝑜 𝐵)))
653adant3 1079 . . . . 5 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐶𝐵 → (𝐴𝑜 𝐶) ∈ (𝐴𝑜 𝐵)))
73, 6orim12d 882 . . . 4 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵𝐶𝐶𝐵) → ((𝐴𝑜 𝐵) ∈ (𝐴𝑜 𝐶) ∨ (𝐴𝑜 𝐶) ∈ (𝐴𝑜 𝐵))))
87con3d 148 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (¬ ((𝐴𝑜 𝐵) ∈ (𝐴𝑜 𝐶) ∨ (𝐴𝑜 𝐶) ∈ (𝐴𝑜 𝐵)) → ¬ (𝐵𝐶𝐶𝐵)))
9 eldifi 3716 . . . . . 6 (𝐴 ∈ (On ∖ 2𝑜) → 𝐴 ∈ On)
1093ad2ant1 1080 . . . . 5 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐴 ∈ On)
11 simp2 1060 . . . . 5 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐵 ∈ On)
12 oecl 7577 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 𝐵) ∈ On)
1310, 11, 12syl2anc 692 . . . 4 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝑜 𝐵) ∈ On)
14 simp3 1061 . . . . 5 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐶 ∈ On)
15 oecl 7577 . . . . 5 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝑜 𝐶) ∈ On)
1610, 14, 15syl2anc 692 . . . 4 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝑜 𝐶) ∈ On)
17 eloni 5702 . . . . 5 ((𝐴𝑜 𝐵) ∈ On → Ord (𝐴𝑜 𝐵))
18 eloni 5702 . . . . 5 ((𝐴𝑜 𝐶) ∈ On → Ord (𝐴𝑜 𝐶))
19 ordtri3 5728 . . . . 5 ((Ord (𝐴𝑜 𝐵) ∧ Ord (𝐴𝑜 𝐶)) → ((𝐴𝑜 𝐵) = (𝐴𝑜 𝐶) ↔ ¬ ((𝐴𝑜 𝐵) ∈ (𝐴𝑜 𝐶) ∨ (𝐴𝑜 𝐶) ∈ (𝐴𝑜 𝐵))))
2017, 18, 19syl2an 494 . . . 4 (((𝐴𝑜 𝐵) ∈ On ∧ (𝐴𝑜 𝐶) ∈ On) → ((𝐴𝑜 𝐵) = (𝐴𝑜 𝐶) ↔ ¬ ((𝐴𝑜 𝐵) ∈ (𝐴𝑜 𝐶) ∨ (𝐴𝑜 𝐶) ∈ (𝐴𝑜 𝐵))))
2113, 16, 20syl2anc 692 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝑜 𝐵) = (𝐴𝑜 𝐶) ↔ ¬ ((𝐴𝑜 𝐵) ∈ (𝐴𝑜 𝐶) ∨ (𝐴𝑜 𝐶) ∈ (𝐴𝑜 𝐵))))
22 eloni 5702 . . . . 5 (𝐵 ∈ On → Ord 𝐵)
23 eloni 5702 . . . . 5 (𝐶 ∈ On → Ord 𝐶)
24 ordtri3 5728 . . . . 5 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝐶𝐶𝐵)))
2522, 23, 24syl2an 494 . . . 4 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝐶𝐶𝐵)))
26253adant1 1077 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝐶𝐶𝐵)))
278, 21, 263imtr4d 283 . 2 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝑜 𝐵) = (𝐴𝑜 𝐶) → 𝐵 = 𝐶))
28 oveq2 6623 . 2 (𝐵 = 𝐶 → (𝐴𝑜 𝐵) = (𝐴𝑜 𝐶))
2927, 28impbid1 215 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 196  wo 383  w3a 1036   = wceq 1480  wcel 1987  cdif 3557  Ord word 5691  Oncon0 5692  (class class class)co 6615  2𝑜c2o 7514  𝑜 coe 7519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-2o 7521  df-oadd 7524  df-omul 7525  df-oexp 7526
This theorem is referenced by:  oeword  7630  infxpenc2lem1  8802
  Copyright terms: Public domain W3C validator