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Theorem om00 6569
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 2532 . . . . 5  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  <->  -.  ( A  =  (/)  \/  B  =  (/) ) )
2 eloni 4401 . . . . . . . . . 10  |-  ( A  e.  On  ->  Ord  A )
3 ordge1n0 6493 . . . . . . . . . 10  |-  ( Ord 
A  ->  ( 1o  C_  A  <->  A  =/=  (/) ) )
42, 3syl 15 . . . . . . . . 9  |-  ( A  e.  On  ->  ( 1o  C_  A  <->  A  =/=  (/) ) )
54biimprd 214 . . . . . . . 8  |-  ( A  e.  On  ->  ( A  =/=  (/)  ->  1o  C_  A
) )
65adantr 451 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  =/=  (/)  ->  1o  C_  A ) )
7 on0eln0 4446 . . . . . . . . 9  |-  ( B  e.  On  ->  ( (/) 
e.  B  <->  B  =/=  (/) ) )
87adantl 452 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  e.  B  <->  B  =/=  (/) ) )
9 omword1 6567 . . . . . . . . 9  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  B )  ->  A  C_  ( A  .o  B ) )
109ex 423 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  e.  B  ->  A  C_  ( A  .o  B ) ) )
118, 10sylbird 226 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  =/=  (/)  ->  A  C_  ( A  .o  B
) ) )
126, 11anim12d 546 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  =/=  (/)  /\  B  =/=  (/) )  -> 
( 1o  C_  A  /\  A  C_  ( A  .o  B ) ) ) )
13 sstr 3188 . . . . . 6  |-  ( ( 1o  C_  A  /\  A  C_  ( A  .o  B ) )  ->  1o  C_  ( A  .o  B ) )
1412, 13syl6 29 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  =/=  (/)  /\  B  =/=  (/) )  ->  1o  C_  ( A  .o  B ) ) )
151, 14syl5bir 209 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  ( A  =  (/)  \/  B  =  (/) )  ->  1o  C_  ( A  .o  B
) ) )
16 omcl 6531 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  e.  On )
17 eloni 4401 . . . . 5  |-  ( ( A  .o  B )  e.  On  ->  Ord  ( A  .o  B
) )
18 ordge1n0 6493 . . . . 5  |-  ( Ord  ( A  .o  B
)  ->  ( 1o  C_  ( A  .o  B
)  <->  ( A  .o  B )  =/=  (/) ) )
1916, 17, 183syl 18 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( 1o  C_  ( A  .o  B )  <->  ( A  .o  B )  =/=  (/) ) )
2015, 19sylibd 205 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  ( A  =  (/)  \/  B  =  (/) )  ->  ( A  .o  B )  =/=  (/) ) )
2120necon4bd 2509 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  .o  B )  =  (/)  ->  ( A  =  (/)  \/  B  =  (/) ) ) )
22 oveq1 5827 . . . . . 6  |-  ( A  =  (/)  ->  ( A  .o  B )  =  ( (/)  .o  B
) )
23 om0r 6534 . . . . . 6  |-  ( B  e.  On  ->  ( (/) 
.o  B )  =  (/) )
2422, 23sylan9eqr 2338 . . . . 5  |-  ( ( B  e.  On  /\  A  =  (/) )  -> 
( A  .o  B
)  =  (/) )
2524ex 423 . . . 4  |-  ( B  e.  On  ->  ( A  =  (/)  ->  ( A  .o  B )  =  (/) ) )
2625adantl 452 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  =  (/)  ->  ( A  .o  B
)  =  (/) ) )
27 oveq2 5828 . . . . . 6  |-  ( B  =  (/)  ->  ( A  .o  B )  =  ( A  .o  (/) ) )
28 om0 6512 . . . . . 6  |-  ( A  e.  On  ->  ( A  .o  (/) )  =  (/) )
2927, 28sylan9eqr 2338 . . . . 5  |-  ( ( A  e.  On  /\  B  =  (/) )  -> 
( A  .o  B
)  =  (/) )
3029ex 423 . . . 4  |-  ( A  e.  On  ->  ( B  =  (/)  ->  ( A  .o  B )  =  (/) ) )
3130adantr 451 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  =  (/)  ->  ( A  .o  B
)  =  (/) ) )
3226, 31jaod 369 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  =  (/)  \/  B  =  (/) )  ->  ( A  .o  B )  =  (/) ) )
3321, 32impbid 183 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  .o  B )  =  (/)  <->  ( A  =  (/)  \/  B  =  (/) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1685    =/= wne 2447    C_ wss 3153   (/)c0 3456   Ord word 4390   Oncon0 4391  (class class class)co 5820   1oc1o 6468    .o comu 6473
This theorem is referenced by:  om00el  6570  omlimcl  6572  oeoe  6593
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-recs 6384  df-rdg 6419  df-1o 6475  df-oadd 6479  df-omul 6480
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