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

Theorem omword 7602
Description: Weak ordering property of ordinal multiplication. (Contributed by NM, 21-Dec-2004.)
Assertion
Ref Expression
omword (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 ↔ (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))

Proof of Theorem omword
StepHypRef Expression
1 omord2 7599 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))
2 3anrot 1041 . . . . 5 ((𝐶 ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) ↔ (𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On))
3 omcan 7601 . . . . 5 (((𝐶 ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐶) → ((𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵) ↔ 𝐴 = 𝐵))
42, 3sylanbr 490 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → ((𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵) ↔ 𝐴 = 𝐵))
54bicomd 213 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 = 𝐵 ↔ (𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵)))
61, 5orbi12d 745 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → ((𝐴𝐵𝐴 = 𝐵) ↔ ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) ∨ (𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵))))
7 onsseleq 5729 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
873adant3 1079 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
98adantr 481 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
10 omcl 7568 . . . . . . 7 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ·𝑜 𝐴) ∈ On)
11 omcl 7568 . . . . . . 7 ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ·𝑜 𝐵) ∈ On)
1210, 11anim12dan 881 . . . . . 6 ((𝐶 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → ((𝐶 ·𝑜 𝐴) ∈ On ∧ (𝐶 ·𝑜 𝐵) ∈ On))
1312ancoms 469 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ((𝐶 ·𝑜 𝐴) ∈ On ∧ (𝐶 ·𝑜 𝐵) ∈ On))
14133impa 1256 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 ·𝑜 𝐴) ∈ On ∧ (𝐶 ·𝑜 𝐵) ∈ On))
15 onsseleq 5729 . . . 4 (((𝐶 ·𝑜 𝐴) ∈ On ∧ (𝐶 ·𝑜 𝐵) ∈ On) → ((𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵) ↔ ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) ∨ (𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵))))
1614, 15syl 17 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵) ↔ ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) ∨ (𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵))))
1716adantr 481 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → ((𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵) ↔ ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) ∨ (𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵))))
186, 9, 173bitr4d 300 1 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 ↔ (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wss 3559  c0 3896  Oncon0 5687  (class class class)co 6610   ·𝑜 comu 7510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-oadd 7516  df-omul 7517
This theorem is referenced by:  omwordi  7603  omeulem2  7615  oeeui  7634
  Copyright terms: Public domain W3C validator