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

Theorem omcan 7509
Description: Left cancellation law for ordinal multiplication. Proposition 8.20 of [TakeutiZaring] p. 63 and its converse. (Contributed by NM, 14-Dec-2004.)
Assertion
Ref Expression
omcan (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐶) ↔ 𝐵 = 𝐶))

Proof of Theorem omcan
StepHypRef Expression
1 omordi 7506 . . . . . . . . 9 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐵𝐶 → (𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 𝐶)))
21ex 448 . . . . . . . 8 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (∅ ∈ 𝐴 → (𝐵𝐶 → (𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 𝐶))))
32ancoms 467 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐴 → (𝐵𝐶 → (𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 𝐶))))
433adant2 1072 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐴 → (𝐵𝐶 → (𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 𝐶))))
54imp 443 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐵𝐶 → (𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 𝐶)))
6 omordi 7506 . . . . . . . . 9 (((𝐵 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐶𝐵 → (𝐴 ·𝑜 𝐶) ∈ (𝐴 ·𝑜 𝐵)))
76ex 448 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (∅ ∈ 𝐴 → (𝐶𝐵 → (𝐴 ·𝑜 𝐶) ∈ (𝐴 ·𝑜 𝐵))))
87ancoms 467 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐴 → (𝐶𝐵 → (𝐴 ·𝑜 𝐶) ∈ (𝐴 ·𝑜 𝐵))))
983adant3 1073 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐴 → (𝐶𝐵 → (𝐴 ·𝑜 𝐶) ∈ (𝐴 ·𝑜 𝐵))))
109imp 443 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐶𝐵 → (𝐴 ·𝑜 𝐶) ∈ (𝐴 ·𝑜 𝐵)))
115, 10orim12d 878 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐵𝐶𝐶𝐵) → ((𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 𝐶) ∨ (𝐴 ·𝑜 𝐶) ∈ (𝐴 ·𝑜 𝐵))))
1211con3d 146 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → (¬ ((𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 𝐶) ∨ (𝐴 ·𝑜 𝐶) ∈ (𝐴 ·𝑜 𝐵)) → ¬ (𝐵𝐶𝐶𝐵)))
13 omcl 7476 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On)
14 eloni 5632 . . . . . . 7 ((𝐴 ·𝑜 𝐵) ∈ On → Ord (𝐴 ·𝑜 𝐵))
1513, 14syl 17 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴 ·𝑜 𝐵))
16 omcl 7476 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ·𝑜 𝐶) ∈ On)
17 eloni 5632 . . . . . . 7 ((𝐴 ·𝑜 𝐶) ∈ On → Ord (𝐴 ·𝑜 𝐶))
1816, 17syl 17 . . . . . 6 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → Ord (𝐴 ·𝑜 𝐶))
19 ordtri3 5658 . . . . . 6 ((Ord (𝐴 ·𝑜 𝐵) ∧ Ord (𝐴 ·𝑜 𝐶)) → ((𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐶) ↔ ¬ ((𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 𝐶) ∨ (𝐴 ·𝑜 𝐶) ∈ (𝐴 ·𝑜 𝐵))))
2015, 18, 19syl2an 492 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ∈ On ∧ 𝐶 ∈ On)) → ((𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐶) ↔ ¬ ((𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 𝐶) ∨ (𝐴 ·𝑜 𝐶) ∈ (𝐴 ·𝑜 𝐵))))
21203impdi 1372 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐶) ↔ ¬ ((𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 𝐶) ∨ (𝐴 ·𝑜 𝐶) ∈ (𝐴 ·𝑜 𝐵))))
2221adantr 479 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐶) ↔ ¬ ((𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 𝐶) ∨ (𝐴 ·𝑜 𝐶) ∈ (𝐴 ·𝑜 𝐵))))
23 eloni 5632 . . . . . 6 (𝐵 ∈ On → Ord 𝐵)
24 eloni 5632 . . . . . 6 (𝐶 ∈ On → Ord 𝐶)
25 ordtri3 5658 . . . . . 6 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝐶𝐶𝐵)))
2623, 24, 25syl2an 492 . . . . 5 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝐶𝐶𝐵)))
27263adant1 1071 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝐶𝐶𝐵)))
2827adantr 479 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝐶𝐶𝐵)))
2912, 22, 283imtr4d 281 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐶) → 𝐵 = 𝐶))
30 oveq2 6531 . 2 (𝐵 = 𝐶 → (𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐶))
3129, 30impbid1 213 1 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐶) ↔ 𝐵 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382  w3a 1030   = wceq 1474  wcel 1975  c0 3869  Ord word 5621  Oncon0 5622  (class class class)co 6523   ·𝑜 comu 7418
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-oadd 7424  df-omul 7425
This theorem is referenced by:  omword  7510  fin1a2lem4  9081
  Copyright terms: Public domain W3C validator