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Theorem om00 7236
Description: The product of two ordinal numbers is zero iff at least one of them is zero. Proposition 8.22 of [TakeutiZaring] p. 64. (Contributed by NM, 21-Dec-2004.)
Assertion
Ref Expression
om00  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  .o  B )  =  (/)  <->  ( A  =  (/)  \/  B  =  (/) ) ) )

Proof of Theorem om00
StepHypRef Expression
1 neanior 2792 . . . . 5  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  <->  -.  ( A  =  (/)  \/  B  =  (/) ) )
2 eloni 4894 . . . . . . . . . 10  |-  ( A  e.  On  ->  Ord  A )
3 ordge1n0 7160 . . . . . . . . . 10  |-  ( Ord 
A  ->  ( 1o  C_  A  <->  A  =/=  (/) ) )
42, 3syl 16 . . . . . . . . 9  |-  ( A  e.  On  ->  ( 1o  C_  A  <->  A  =/=  (/) ) )
54biimprd 223 . . . . . . . 8  |-  ( A  e.  On  ->  ( A  =/=  (/)  ->  1o  C_  A
) )
65adantr 465 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  =/=  (/)  ->  1o  C_  A ) )
7 on0eln0 4939 . . . . . . . . 9  |-  ( B  e.  On  ->  ( (/) 
e.  B  <->  B  =/=  (/) ) )
87adantl 466 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  e.  B  <->  B  =/=  (/) ) )
9 omword1 7234 . . . . . . . . 9  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  B )  ->  A  C_  ( A  .o  B ) )
109ex 434 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  e.  B  ->  A  C_  ( A  .o  B ) ) )
118, 10sylbird 235 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  =/=  (/)  ->  A  C_  ( A  .o  B
) ) )
126, 11anim12d 563 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  =/=  (/)  /\  B  =/=  (/) )  -> 
( 1o  C_  A  /\  A  C_  ( A  .o  B ) ) ) )
13 sstr 3517 . . . . . 6  |-  ( ( 1o  C_  A  /\  A  C_  ( A  .o  B ) )  ->  1o  C_  ( A  .o  B ) )
1412, 13syl6 33 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  =/=  (/)  /\  B  =/=  (/) )  ->  1o  C_  ( A  .o  B ) ) )
151, 14syl5bir 218 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  ( A  =  (/)  \/  B  =  (/) )  ->  1o  C_  ( A  .o  B
) ) )
16 omcl 7198 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  e.  On )
17 eloni 4894 . . . . 5  |-  ( ( A  .o  B )  e.  On  ->  Ord  ( A  .o  B
) )
18 ordge1n0 7160 . . . . 5  |-  ( Ord  ( A  .o  B
)  ->  ( 1o  C_  ( A  .o  B
)  <->  ( A  .o  B )  =/=  (/) ) )
1916, 17, 183syl 20 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( 1o  C_  ( A  .o  B )  <->  ( A  .o  B )  =/=  (/) ) )
2015, 19sylibd 214 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  ( A  =  (/)  \/  B  =  (/) )  ->  ( A  .o  B )  =/=  (/) ) )
2120necon4bd 2689 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  .o  B )  =  (/)  ->  ( A  =  (/)  \/  B  =  (/) ) ) )
22 oveq1 6302 . . . . . 6  |-  ( A  =  (/)  ->  ( A  .o  B )  =  ( (/)  .o  B
) )
23 om0r 7201 . . . . . 6  |-  ( B  e.  On  ->  ( (/) 
.o  B )  =  (/) )
2422, 23sylan9eqr 2530 . . . . 5  |-  ( ( B  e.  On  /\  A  =  (/) )  -> 
( A  .o  B
)  =  (/) )
2524ex 434 . . . 4  |-  ( B  e.  On  ->  ( A  =  (/)  ->  ( A  .o  B )  =  (/) ) )
2625adantl 466 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  =  (/)  ->  ( A  .o  B
)  =  (/) ) )
27 oveq2 6303 . . . . . 6  |-  ( B  =  (/)  ->  ( A  .o  B )  =  ( A  .o  (/) ) )
28 om0 7179 . . . . . 6  |-  ( A  e.  On  ->  ( A  .o  (/) )  =  (/) )
2927, 28sylan9eqr 2530 . . . . 5  |-  ( ( A  e.  On  /\  B  =  (/) )  -> 
( A  .o  B
)  =  (/) )
3029ex 434 . . . 4  |-  ( A  e.  On  ->  ( B  =  (/)  ->  ( A  .o  B )  =  (/) ) )
3130adantr 465 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  =  (/)  ->  ( A  .o  B
)  =  (/) ) )
3226, 31jaod 380 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  =  (/)  \/  B  =  (/) )  ->  ( A  .o  B )  =  (/) ) )
3321, 32impbid 191 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  .o  B )  =  (/)  <->  ( A  =  (/)  \/  B  =  (/) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662    C_ wss 3481   (/)c0 3790   Ord word 4883   Oncon0 4884  (class class class)co 6295   1oc1o 7135    .o comu 7140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-omul 7147
This theorem is referenced by:  om00el  7237  omlimcl  7239  oeoe  7260
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