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Theorem omcan 7536
 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 7533 . . . . . . . . 9 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐵𝐶 → (𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 𝐶)))
21ex 449 . . . . . . . 8 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (∅ ∈ 𝐴 → (𝐵𝐶 → (𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 𝐶))))
32ancoms 468 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐴 → (𝐵𝐶 → (𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 𝐶))))
433adant2 1073 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐴 → (𝐵𝐶 → (𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 𝐶))))
54imp 444 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐵𝐶 → (𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 𝐶)))
6 omordi 7533 . . . . . . . . 9 (((𝐵 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐶𝐵 → (𝐴 ·𝑜 𝐶) ∈ (𝐴 ·𝑜 𝐵)))
76ex 449 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (∅ ∈ 𝐴 → (𝐶𝐵 → (𝐴 ·𝑜 𝐶) ∈ (𝐴 ·𝑜 𝐵))))
87ancoms 468 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐴 → (𝐶𝐵 → (𝐴 ·𝑜 𝐶) ∈ (𝐴 ·𝑜 𝐵))))
983adant3 1074 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐴 → (𝐶𝐵 → (𝐴 ·𝑜 𝐶) ∈ (𝐴 ·𝑜 𝐵))))
109imp 444 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐶𝐵 → (𝐴 ·𝑜 𝐶) ∈ (𝐴 ·𝑜 𝐵)))
115, 10orim12d 879 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐵𝐶𝐶𝐵) → ((𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 𝐶) ∨ (𝐴 ·𝑜 𝐶) ∈ (𝐴 ·𝑜 𝐵))))
1211con3d 147 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → (¬ ((𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 𝐶) ∨ (𝐴 ·𝑜 𝐶) ∈ (𝐴 ·𝑜 𝐵)) → ¬ (𝐵𝐶𝐶𝐵)))
13 omcl 7503 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On)
14 eloni 5650 . . . . . . 7 ((𝐴 ·𝑜 𝐵) ∈ On → Ord (𝐴 ·𝑜 𝐵))
1513, 14syl 17 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴 ·𝑜 𝐵))
16 omcl 7503 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ·𝑜 𝐶) ∈ On)
17 eloni 5650 . . . . . . 7 ((𝐴 ·𝑜 𝐶) ∈ On → Ord (𝐴 ·𝑜 𝐶))
1816, 17syl 17 . . . . . 6 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → Ord (𝐴 ·𝑜 𝐶))
19 ordtri3 5676 . . . . . 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 493 . . . . 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 1373 . . . 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 5650 . . . . . 6 (𝐵 ∈ On → Ord 𝐵)
24 eloni 5650 . . . . . 6 (𝐶 ∈ On → Ord 𝐶)
25 ordtri3 5676 . . . . . 6 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝐶𝐶𝐵)))
2623, 24, 25syl2an 493 . . . . 5 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝐶𝐶𝐵)))
27263adant1 1072 . . . 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 282 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐶) → 𝐵 = 𝐶))
30 oveq2 6557 . 2 (𝐵 = 𝐶 → (𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐶))
3129, 30impbid1 214 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 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∅c0 3874  Ord word 5639  Oncon0 5640  (class class class)co 6549   ·𝑜 comu 7445 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-omul 7452 This theorem is referenced by:  omword  7537  fin1a2lem4  9108
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