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Theorem omcan 8608
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 8605 . . . . . . . . 9 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐵𝐶 → (𝐴 ·o 𝐵) ∈ (𝐴 ·o 𝐶)))
21ex 412 . . . . . . . 8 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (∅ ∈ 𝐴 → (𝐵𝐶 → (𝐴 ·o 𝐵) ∈ (𝐴 ·o 𝐶))))
32ancoms 458 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐴 → (𝐵𝐶 → (𝐴 ·o 𝐵) ∈ (𝐴 ·o 𝐶))))
433adant2 1131 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐴 → (𝐵𝐶 → (𝐴 ·o 𝐵) ∈ (𝐴 ·o 𝐶))))
54imp 406 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐵𝐶 → (𝐴 ·o 𝐵) ∈ (𝐴 ·o 𝐶)))
6 omordi 8605 . . . . . . . . 9 (((𝐵 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐶𝐵 → (𝐴 ·o 𝐶) ∈ (𝐴 ·o 𝐵)))
76ex 412 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (∅ ∈ 𝐴 → (𝐶𝐵 → (𝐴 ·o 𝐶) ∈ (𝐴 ·o 𝐵))))
87ancoms 458 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐴 → (𝐶𝐵 → (𝐴 ·o 𝐶) ∈ (𝐴 ·o 𝐵))))
983adant3 1132 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐴 → (𝐶𝐵 → (𝐴 ·o 𝐶) ∈ (𝐴 ·o 𝐵))))
109imp 406 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐶𝐵 → (𝐴 ·o 𝐶) ∈ (𝐴 ·o 𝐵)))
115, 10orim12d 966 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐵𝐶𝐶𝐵) → ((𝐴 ·o 𝐵) ∈ (𝐴 ·o 𝐶) ∨ (𝐴 ·o 𝐶) ∈ (𝐴 ·o 𝐵))))
1211con3d 152 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → (¬ ((𝐴 ·o 𝐵) ∈ (𝐴 ·o 𝐶) ∨ (𝐴 ·o 𝐶) ∈ (𝐴 ·o 𝐵)) → ¬ (𝐵𝐶𝐶𝐵)))
13 omcl 8575 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
14 eloni 6393 . . . . . . 7 ((𝐴 ·o 𝐵) ∈ On → Ord (𝐴 ·o 𝐵))
1513, 14syl 17 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴 ·o 𝐵))
16 omcl 8575 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ·o 𝐶) ∈ On)
17 eloni 6393 . . . . . . 7 ((𝐴 ·o 𝐶) ∈ On → Ord (𝐴 ·o 𝐶))
1816, 17syl 17 . . . . . 6 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → Ord (𝐴 ·o 𝐶))
19 ordtri3 6419 . . . . . 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 596 . . . . 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 1350 . . . 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 480 . . 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 6393 . . . . . 6 (𝐵 ∈ On → Ord 𝐵)
24 eloni 6393 . . . . . 6 (𝐶 ∈ On → Ord 𝐶)
25 ordtri3 6419 . . . . . 6 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝐶𝐶𝐵)))
2623, 24, 25syl2an 596 . . . . 5 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝐶𝐶𝐵)))
27263adant1 1130 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝐶𝐶𝐵)))
2827adantr 480 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝐶𝐶𝐵)))
2912, 22, 283imtr4d 294 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶))
30 oveq2 7440 . 2 (𝐵 = 𝐶 → (𝐴 ·o 𝐵) = (𝐴 ·o 𝐶))
3129, 30impbid1 225 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 206  wa 395  wo 847  w3a 1086   = wceq 1539  wcel 2107  c0 4332  Ord word 6382  Oncon0 6383  (class class class)co 7432   ·o comu 8505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-oadd 8511  df-omul 8512
This theorem is referenced by:  omword  8609  fin1a2lem4  10444
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