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Theorem mul0inf 11040
Description: Equality of a product with zero. A bit of a curiosity, in the sense that theorems like abs00ap 10862 and mulap0bd 8438 may better express the ideas behind it. (Contributed by Jim Kingdon, 31-Jul-2023.)
Assertion
Ref Expression
mul0inf  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  =  0  <-> inf ( { ( abs `  A
) ,  ( abs `  B ) } ,  RR ,  <  )  =  0 ) )

Proof of Theorem mul0inf
StepHypRef Expression
1 mulcl 7767 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
2 0cnd 7779 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  0  e.  CC )
3 simpl 108 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  A  e.  CC )
43abscld 10981 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( abs `  A
)  e.  RR )
5 simpr 109 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  B  e.  CC )
65abscld 10981 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( abs `  B
)  e.  RR )
7 mincl 11030 . . . 4  |-  ( ( ( abs `  A
)  e.  RR  /\  ( abs `  B )  e.  RR )  -> inf ( { ( abs `  A
) ,  ( abs `  B ) } ,  RR ,  <  )  e.  RR )
84, 6, 7syl2anc 409 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  -> inf ( { ( abs `  A ) ,  ( abs `  B ) } ,  RR ,  <  )  e.  RR )
98recnd 7814 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  -> inf ( { ( abs `  A ) ,  ( abs `  B ) } ,  RR ,  <  )  e.  CC )
103absge0d 10984 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  0  <_  ( abs `  A ) )
115absge0d 10984 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  0  <_  ( abs `  B ) )
12 0red 7787 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  0  e.  RR )
13 lemininf 11033 . . . . . 6  |-  ( ( 0  e.  RR  /\  ( abs `  A )  e.  RR  /\  ( abs `  B )  e.  RR )  ->  (
0  <_ inf ( {
( abs `  A
) ,  ( abs `  B ) } ,  RR ,  <  )  <->  ( 0  <_  ( abs `  A
)  /\  0  <_  ( abs `  B ) ) ) )
1412, 4, 6, 13syl3anc 1217 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 0  <_ inf ( { ( abs `  A
) ,  ( abs `  B ) } ,  RR ,  <  )  <->  ( 0  <_  ( abs `  A
)  /\  0  <_  ( abs `  B ) ) ) )
1510, 11, 14mpbir2and 929 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  0  <_ inf ( {
( abs `  A
) ,  ( abs `  B ) } ,  RR ,  <  ) )
16 ap0gt0 8422 . . . 4  |-  ( (inf ( { ( abs `  A ) ,  ( abs `  B ) } ,  RR ,  <  )  e.  RR  /\  0  <_ inf ( { ( abs `  A ) ,  ( abs `  B
) } ,  RR ,  <  ) )  -> 
(inf ( { ( abs `  A ) ,  ( abs `  B
) } ,  RR ,  <  ) #  0  <->  0  < inf ( { ( abs `  A ) ,  ( abs `  B
) } ,  RR ,  <  ) ) )
178, 15, 16syl2anc 409 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  (inf ( { ( abs `  A ) ,  ( abs `  B
) } ,  RR ,  <  ) #  0  <->  0  < inf ( { ( abs `  A ) ,  ( abs `  B
) } ,  RR ,  <  ) ) )
18 absgt0ap 10899 . . . . 5  |-  ( A  e.  CC  ->  ( A #  0  <->  0  <  ( abs `  A ) ) )
19 absgt0ap 10899 . . . . 5  |-  ( B  e.  CC  ->  ( B #  0  <->  0  <  ( abs `  B ) ) )
2018, 19bi2anan9 596 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A #  0  /\  B #  0 )  <-> 
( 0  <  ( abs `  A )  /\  0  <  ( abs `  B
) ) ) )
21 ltmininf 11034 . . . . 5  |-  ( ( 0  e.  RR  /\  ( abs `  A )  e.  RR  /\  ( abs `  B )  e.  RR )  ->  (
0  < inf ( {
( abs `  A
) ,  ( abs `  B ) } ,  RR ,  <  )  <->  ( 0  <  ( abs `  A
)  /\  0  <  ( abs `  B ) ) ) )
2212, 4, 6, 21syl3anc 1217 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 0  < inf ( { ( abs `  A
) ,  ( abs `  B ) } ,  RR ,  <  )  <->  ( 0  <  ( abs `  A
)  /\  0  <  ( abs `  B ) ) ) )
2320, 22bitr4d 190 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A #  0  /\  B #  0 )  <->  0  < inf ( {
( abs `  A
) ,  ( abs `  B ) } ,  RR ,  <  ) ) )
24 mulap0b 8436 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A #  0  /\  B #  0 )  <-> 
( A  x.  B
) #  0 ) )
2517, 23, 243bitr2rd 216 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B ) #  0  <-> inf ( {
( abs `  A
) ,  ( abs `  B ) } ,  RR ,  <  ) #  0 ) )
261, 2, 9, 2, 25apcon4bid 8406 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  =  0  <-> inf ( { ( abs `  A
) ,  ( abs `  B ) } ,  RR ,  <  )  =  0 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1332    e. wcel 1481   {cpr 3529   class class class wbr 3933   ` cfv 5127  (class class class)co 5778  infcinf 6874   CCcc 7638   RRcr 7639   0cc0 7640    x. cmul 7645    < clt 7820    <_ cle 7821   # cap 8363   abscabs 10797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4047  ax-sep 4050  ax-nul 4058  ax-pow 4102  ax-pr 4135  ax-un 4359  ax-setind 4456  ax-iinf 4506  ax-cnex 7731  ax-resscn 7732  ax-1cn 7733  ax-1re 7734  ax-icn 7735  ax-addcl 7736  ax-addrcl 7737  ax-mulcl 7738  ax-mulrcl 7739  ax-addcom 7740  ax-mulcom 7741  ax-addass 7742  ax-mulass 7743  ax-distr 7744  ax-i2m1 7745  ax-0lt1 7746  ax-1rid 7747  ax-0id 7748  ax-rnegex 7749  ax-precex 7750  ax-cnre 7751  ax-pre-ltirr 7752  ax-pre-ltwlin 7753  ax-pre-lttrn 7754  ax-pre-apti 7755  ax-pre-ltadd 7756  ax-pre-mulgt0 7757  ax-pre-mulext 7758  ax-arch 7759  ax-caucvg 7760
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2689  df-sbc 2911  df-csb 3005  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-nul 3365  df-if 3476  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-int 3776  df-iun 3819  df-br 3934  df-opab 3994  df-mpt 3995  df-tr 4031  df-id 4219  df-po 4222  df-iso 4223  df-iord 4292  df-on 4294  df-ilim 4295  df-suc 4297  df-iom 4509  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-res 4555  df-ima 4556  df-iota 5092  df-fun 5129  df-fn 5130  df-f 5131  df-f1 5132  df-fo 5133  df-f1o 5134  df-fv 5135  df-isom 5136  df-riota 5734  df-ov 5781  df-oprab 5782  df-mpo 5783  df-1st 6042  df-2nd 6043  df-recs 6206  df-frec 6292  df-sup 6875  df-inf 6876  df-pnf 7822  df-mnf 7823  df-xr 7824  df-ltxr 7825  df-le 7826  df-sub 7955  df-neg 7956  df-reap 8357  df-ap 8364  df-div 8453  df-inn 8741  df-2 8799  df-3 8800  df-4 8801  df-n0 8998  df-z 9075  df-uz 9347  df-rp 9467  df-seqfrec 10246  df-exp 10320  df-cj 10642  df-re 10643  df-im 10644  df-rsqrt 10798  df-abs 10799
This theorem is referenced by: (None)
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