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Theorem omcan 7807
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 7804 . . . . . . . . 9 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐵𝐶 → (𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 𝐶)))
21ex 397 . . . . . . . 8 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (∅ ∈ 𝐴 → (𝐵𝐶 → (𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 𝐶))))
32ancoms 446 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐴 → (𝐵𝐶 → (𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 𝐶))))
433adant2 1125 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐴 → (𝐵𝐶 → (𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 𝐶))))
54imp 393 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐵𝐶 → (𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 𝐶)))
6 omordi 7804 . . . . . . . . 9 (((𝐵 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐶𝐵 → (𝐴 ·𝑜 𝐶) ∈ (𝐴 ·𝑜 𝐵)))
76ex 397 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (∅ ∈ 𝐴 → (𝐶𝐵 → (𝐴 ·𝑜 𝐶) ∈ (𝐴 ·𝑜 𝐵))))
87ancoms 446 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐴 → (𝐶𝐵 → (𝐴 ·𝑜 𝐶) ∈ (𝐴 ·𝑜 𝐵))))
983adant3 1126 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐴 → (𝐶𝐵 → (𝐴 ·𝑜 𝐶) ∈ (𝐴 ·𝑜 𝐵))))
109imp 393 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐶𝐵 → (𝐴 ·𝑜 𝐶) ∈ (𝐴 ·𝑜 𝐵)))
115, 10orim12d 949 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐵𝐶𝐶𝐵) → ((𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 𝐶) ∨ (𝐴 ·𝑜 𝐶) ∈ (𝐴 ·𝑜 𝐵))))
1211con3d 149 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → (¬ ((𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 𝐶) ∨ (𝐴 ·𝑜 𝐶) ∈ (𝐴 ·𝑜 𝐵)) → ¬ (𝐵𝐶𝐶𝐵)))
13 omcl 7774 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On)
14 eloni 5875 . . . . . . 7 ((𝐴 ·𝑜 𝐵) ∈ On → Ord (𝐴 ·𝑜 𝐵))
1513, 14syl 17 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴 ·𝑜 𝐵))
16 omcl 7774 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ·𝑜 𝐶) ∈ On)
17 eloni 5875 . . . . . . 7 ((𝐴 ·𝑜 𝐶) ∈ On → Ord (𝐴 ·𝑜 𝐶))
1816, 17syl 17 . . . . . 6 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → Ord (𝐴 ·𝑜 𝐶))
19 ordtri3 5901 . . . . . 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 583 . . . . 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 1443 . . . 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 466 . . 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 5875 . . . . . 6 (𝐵 ∈ On → Ord 𝐵)
24 eloni 5875 . . . . . 6 (𝐶 ∈ On → Ord 𝐶)
25 ordtri3 5901 . . . . . 6 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝐶𝐶𝐵)))
2623, 24, 25syl2an 583 . . . . 5 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝐶𝐶𝐵)))
27263adant1 1124 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝐶𝐶𝐵)))
2827adantr 466 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝐶𝐶𝐵)))
2912, 22, 283imtr4d 283 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐶) → 𝐵 = 𝐶))
30 oveq2 6804 . 2 (𝐵 = 𝐶 → (𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐶))
3129, 30impbid1 215 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 196  wa 382  wo 836  w3a 1071   = wceq 1631  wcel 2145  c0 4063  Ord word 5864  Oncon0 5865  (class class class)co 6796   ·𝑜 comu 7715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-oadd 7721  df-omul 7722
This theorem is referenced by:  omword  7808  fin1a2lem4  9431
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