Theorem mul0inf 11248

Description: Equality of a product with zero. A bit of a curiosity, in the sense that theorems like abs00ap 11070 and mulap0bd 8613 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 7937	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
2		0cnd 7949	. 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 11189	. . . 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 11189	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( abs ` B ) e. RR )
7		mincl 11238	. . . 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 7985	. 2 |- ( ( A e. CC /\ B e. CC ) -> inf ( { ( abs ` A ) , ( abs ` B ) } , RR , < ) e. CC )
10	3	absge0d 11192	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> 0 <_ ( abs ` A ) )
11	5	absge0d 11192	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> 0 <_ ( abs ` B ) )
12		0red 7957	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> 0 e. RR )
13		lemininf 11241	. . . . . 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 1238	. . . . 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 944	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> 0 <_ inf ( { ( abs ` A ) , ( abs ` B ) } , RR , < ) )
16		ap0gt0 8596	. . . 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 11107	. . . . 5 |- ( A e. CC -> ( A # 0 <-> 0 < ( abs ` A ) ) )
19		absgt0ap 11107	. . . . 5 |- ( B e. CC -> ( B # 0 <-> 0 < ( abs ` B ) ) )
20	18, 19	bi2anan9 606	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A # 0 /\ B # 0 ) <-> ( 0 < ( abs ` A ) /\ 0 < ( abs ` B ) ) ) )
21		ltmininf 11242	. . . . 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 1238	. . . 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 8611	. . 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 8580	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 1353   e. wcel 2148   {cpr 3593   class class class wbr 4003   ` cfv 5216  (class class class)co 5874  infcinf 6981   CCcc 7808   RRcr 7809   0cc0 7810   x. cmul 7815   < clt 7991   <_ cle 7992   # cap 8537   abscabs 11005

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929  ax-caucvg 7930

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-isom 5225  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-frec 6391  df-sup 6982  df-inf 6983  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997  df-sub 8129  df-neg 8130  df-reap 8531  df-ap 8538  df-div 8629  df-inn 8919  df-2 8977  df-3 8978  df-4 8979  df-n0 9176  df-z 9253  df-uz 9528  df-rp 9653  df-seqfrec 10445  df-exp 10519  df-cj 10850  df-re 10851  df-im 10852  df-rsqrt 11006  df-abs 11007

This theorem is referenced by: (None)