Theorem omord 7648

Description: Ordering property of ordinal multiplication. Proposition 8.19 of [TakeutiZaring] p. 63. (Contributed by NM, 14-Dec-2004.)

Assertion

Ref	Expression
omord	|- ( ( A e. On /\ B e. On /\ C e. On ) -> ( ( A e. B /\ (/) e. C ) <-> ( C .o A ) e. ( C .o B ) ) )

Proof of Theorem omord

Step	Hyp	Ref	Expression
1		omord2 7647	. . . 4 |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ (/) e. C ) -> ( A e. B <-> ( C .o A ) e. ( C .o B ) ) )
2	1	ex 450	. . 3 |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( (/) e. C -> ( A e. B <-> ( C .o A ) e. ( C .o B ) ) ) )
3	2	pm5.32rd 672	. 2 |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( ( A e. B /\ (/) e. C ) <-> ( ( C .o A ) e. ( C .o B ) /\ (/) e. C ) ) )
4		simpl 473	. . 3 |- ( ( ( C .o A ) e. ( C .o B ) /\ (/) e. C ) -> ( C .o A ) e. ( C .o B ) )
5		ne0i 3921	. . . . . . . 8 |- ( ( C .o A ) e. ( C .o B ) -> ( C .o B ) =/= (/) )
6		om0r 7619	. . . . . . . . . 10 |- ( B e. On -> ( (/) .o B ) = (/) )
7		oveq1 6657	. . . . . . . . . . 11 |- ( C = (/) -> ( C .o B ) = ( (/) .o B ) )
8	7	eqeq1d 2624	. . . . . . . . . 10 |- ( C = (/) -> ( ( C .o B ) = (/) <-> ( (/) .o B ) = (/) ) )
9	6, 8	syl5ibrcom 237	. . . . . . . . 9 |- ( B e. On -> ( C = (/) -> ( C .o B ) = (/) ) )
10	9	necon3d 2815	. . . . . . . 8 |- ( B e. On -> ( ( C .o B ) =/= (/) -> C =/= (/) ) )
11	5, 10	syl5 34	. . . . . . 7 |- ( B e. On -> ( ( C .o A ) e. ( C .o B ) -> C =/= (/) ) )
12	11	adantr 481	. . . . . 6 |- ( ( B e. On /\ C e. On ) -> ( ( C .o A ) e. ( C .o B ) -> C =/= (/) ) )
13		on0eln0 5780	. . . . . . 7 |- ( C e. On -> ( (/) e. C <-> C =/= (/) ) )
14	13	adantl 482	. . . . . 6 |- ( ( B e. On /\ C e. On ) -> ( (/) e. C <-> C =/= (/) ) )
15	12, 14	sylibrd 249	. . . . 5 |- ( ( B e. On /\ C e. On ) -> ( ( C .o A ) e. ( C .o B ) -> (/) e. C ) )
16	15	3adant1 1079	. . . 4 |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( ( C .o A ) e. ( C .o B ) -> (/) e. C ) )
17	16	ancld 576	. . 3 |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( ( C .o A ) e. ( C .o B ) -> ( ( C .o A ) e. ( C .o B ) /\ (/) e. C ) ) )
18	4, 17	impbid2 216	. 2 |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( ( ( C .o A ) e. ( C .o B ) /\ (/) e. C ) <-> ( C .o A ) e. ( C .o B ) ) )
19	3, 18	bitrd 268	1 |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( ( A e. B /\ (/) e. C ) <-> ( C .o A ) e. ( C .o B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   (/)c0 3915   Oncon0 5723  (class class class)co 6650   .o comu 7558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-omul 7565

This theorem is referenced by:  omlimcl  7658  oneo  7661