Theorem mul0inf 11951

Description: Equality of a product with zero. A bit of a curiosity, in the sense that theorems like abs00ap 11772 and mulap0bd 8948 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

Step	Hyp	Ref	Expression
1		mulcl 8270	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
2		0cnd 8283	. 2 |- ( ( A e. CC /\ B e. CC ) -> 0 e. CC )
3		simpl 109	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> A e. CC )
4	3	abscld 11891	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( abs ` A ) e. RR )
5		simpr 110	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> B e. CC )
6	5	abscld 11891	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( abs ` B ) e. RR )
7		mincl 11941	. . . 4 |- ( ( ( abs ` A ) e. RR /\ ( abs ` B ) e. RR ) -> inf ( { ( abs ` A ) , ( abs ` B ) } , RR , < ) e. RR )
8	4, 6, 7	syl2anc 411	. . 3 |- ( ( A e. CC /\ B e. CC ) -> inf ( { ( abs ` A ) , ( abs ` B ) } , RR , < ) e. RR )
9	8	recnd 8318	. 2 |- ( ( A e. CC /\ B e. CC ) -> inf ( { ( abs ` A ) , ( abs ` B ) } , RR , < ) e. CC )
10	3	absge0d 11894	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> 0 <_ ( abs ` A ) )
11	5	absge0d 11894	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> 0 <_ ( abs ` B ) )
12		0red 8291	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> 0 e. RR )
13		lemininf 11944	. . . . . 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 ) ) ) )
14	12, 4, 6, 13	syl3anc 1274	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( 0 <_ inf ( { ( abs ` A ) , ( abs ` B ) } , RR , < ) <-> ( 0 <_ ( abs ` A ) /\ 0 <_ ( abs ` B ) ) ) )
15	10, 11, 14	mpbir2and 953	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> 0 <_ inf ( { ( abs ` A ) , ( abs ` B ) } , RR , < ) )
16		ap0gt0 8931	. . . 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 , < ) ) )
17	8, 15, 16	syl2anc 411	. . 3 |- ( ( A e. CC /\ B e. CC ) -> (inf ( { ( abs ` A ) , ( abs ` B ) } , RR , < ) # 0 <-> 0 < inf ( { ( abs ` A ) , ( abs ` B ) } , RR , < ) ) )
18		absgt0ap 11809	. . . . 5 |- ( A e. CC -> ( A # 0 <-> 0 < ( abs ` A ) ) )
19		absgt0ap 11809	. . . . 5 |- ( B e. CC -> ( B # 0 <-> 0 < ( abs ` B ) ) )
20	18, 19	bi2anan9 610	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A # 0 /\ B # 0 ) <-> ( 0 < ( abs ` A ) /\ 0 < ( abs ` B ) ) ) )
21		ltmininf 11945	. . . . 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 ) ) ) )
22	12, 4, 6, 21	syl3anc 1274	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( 0 < inf ( { ( abs ` A ) , ( abs ` B ) } , RR , < ) <-> ( 0 < ( abs ` A ) /\ 0 < ( abs ` B ) ) ) )
23	20, 22	bitr4d 191	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A # 0 /\ B # 0 ) <-> 0 < inf ( { ( abs ` A ) , ( abs ` B ) } , RR , < ) ) )
24		mulap0b 8946	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A # 0 /\ B # 0 ) <-> ( A x. B ) # 0 ) )
25	17, 23, 24	3bitr2rd 217	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) # 0 <-> inf ( { ( abs ` A ) , ( abs ` B ) } , RR , < ) # 0 ) )
26	1, 2, 9, 2, 25	apcon4bid 8915	1 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) = 0 <-> inf ( { ( abs ` A ) , ( abs ` B ) } , RR , < ) = 0 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   {cpr 3695   class class class wbr 4114   ` cfv 5357  (class class class)co 6058  infcinf 7287   CCcc 8141   RRcr 8142   0cc0 8143   x. cmul 8148   < clt 8324   <_ cle 8325   # cap 8872   abscabs 11707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-sup 7288  df-inf 7289  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-rp 10005  df-seqfrec 10834  df-exp 10925  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709

This theorem is referenced by: (None)