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Theorem omcan 8400
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 8397 . . . . . . . . 9 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐵𝐶 → (𝐴 ·o 𝐵) ∈ (𝐴 ·o 𝐶)))
21ex 413 . . . . . . . 8 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (∅ ∈ 𝐴 → (𝐵𝐶 → (𝐴 ·o 𝐵) ∈ (𝐴 ·o 𝐶))))
32ancoms 459 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐴 → (𝐵𝐶 → (𝐴 ·o 𝐵) ∈ (𝐴 ·o 𝐶))))
433adant2 1130 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐴 → (𝐵𝐶 → (𝐴 ·o 𝐵) ∈ (𝐴 ·o 𝐶))))
54imp 407 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐵𝐶 → (𝐴 ·o 𝐵) ∈ (𝐴 ·o 𝐶)))
6 omordi 8397 . . . . . . . . 9 (((𝐵 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐶𝐵 → (𝐴 ·o 𝐶) ∈ (𝐴 ·o 𝐵)))
76ex 413 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (∅ ∈ 𝐴 → (𝐶𝐵 → (𝐴 ·o 𝐶) ∈ (𝐴 ·o 𝐵))))
87ancoms 459 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐴 → (𝐶𝐵 → (𝐴 ·o 𝐶) ∈ (𝐴 ·o 𝐵))))
983adant3 1131 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐴 → (𝐶𝐵 → (𝐴 ·o 𝐶) ∈ (𝐴 ·o 𝐵))))
109imp 407 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐶𝐵 → (𝐴 ·o 𝐶) ∈ (𝐴 ·o 𝐵)))
115, 10orim12d 962 . . . 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 8366 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
14 eloni 6276 . . . . . . 7 ((𝐴 ·o 𝐵) ∈ On → Ord (𝐴 ·o 𝐵))
1513, 14syl 17 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴 ·o 𝐵))
16 omcl 8366 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ·o 𝐶) ∈ On)
17 eloni 6276 . . . . . . 7 ((𝐴 ·o 𝐶) ∈ On → Ord (𝐴 ·o 𝐶))
1816, 17syl 17 . . . . . 6 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → Ord (𝐴 ·o 𝐶))
19 ordtri3 6302 . . . . . 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 1349 . . . 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 481 . . 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 6276 . . . . . 6 (𝐵 ∈ On → Ord 𝐵)
24 eloni 6276 . . . . . 6 (𝐶 ∈ On → Ord 𝐶)
25 ordtri3 6302 . . . . . 6 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝐶𝐶𝐵)))
2623, 24, 25syl2an 596 . . . . 5 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝐶𝐶𝐵)))
27263adant1 1129 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝐶𝐶𝐵)))
2827adantr 481 . . 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 7283 . 2 (𝐵 = 𝐶 → (𝐴 ·o 𝐵) = (𝐴 ·o 𝐶))
3129, 30impbid1 224 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 205  wa 396  wo 844  w3a 1086   = wceq 1539  wcel 2106  c0 4256  Ord word 6265  Oncon0 6266  (class class class)co 7275   ·o comu 8295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-oadd 8301  df-omul 8302
This theorem is referenced by:  omword  8401  fin1a2lem4  10159
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