Theorem mul0inf 11738

Description: Equality of a product with zero. A bit of a curiosity, in the sense that theorems like abs00ap 11559 and mulap0bd 8792 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 8114	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
2		0cnd 8127	. 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 11678	. . . 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 11678	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( abs ` B ) e. RR )
7		mincl 11728	. . . 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 8163	. 2 |- ( ( A e. CC /\ B e. CC ) -> inf ( { ( abs ` A ) , ( abs ` B ) } , RR , < ) e. CC )
10	3	absge0d 11681	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> 0 <_ ( abs ` A ) )
11	5	absge0d 11681	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> 0 <_ ( abs ` B ) )
12		0red 8135	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> 0 e. RR )
13		lemininf 11731	. . . . . 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 1271	. . . . 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 950	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> 0 <_ inf ( { ( abs ` A ) , ( abs ` B ) } , RR , < ) )
16		ap0gt0 8775	. . . 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 11596	. . . . 5 |- ( A e. CC -> ( A # 0 <-> 0 < ( abs ` A ) ) )
19		absgt0ap 11596	. . . . 5 |- ( B e. CC -> ( B # 0 <-> 0 < ( abs ` B ) ) )
20	18, 19	bi2anan9 608	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A # 0 /\ B # 0 ) <-> ( 0 < ( abs ` A ) /\ 0 < ( abs ` B ) ) ) )
21		ltmininf 11732	. . . . 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 1271	. . . 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 8790	. . 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 8759	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 1395   e. wcel 2200   {cpr 3667   class class class wbr 4082   ` cfv 5314  (class class class)co 5994  infcinf 7138   CCcc 7985   RRcr 7986   0cc0 7987   x. cmul 7992   < clt 8169   <_ cle 8170   # cap 8716   abscabs 11494

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-iinf 4677  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-mulrcl 8086  ax-addcom 8087  ax-mulcom 8088  ax-addass 8089  ax-mulass 8090  ax-distr 8091  ax-i2m1 8092  ax-0lt1 8093  ax-1rid 8094  ax-0id 8095  ax-rnegex 8096  ax-precex 8097  ax-cnre 8098  ax-pre-ltirr 8099  ax-pre-ltwlin 8100  ax-pre-lttrn 8101  ax-pre-apti 8102  ax-pre-ltadd 8103  ax-pre-mulgt0 8104  ax-pre-mulext 8105  ax-arch 8106  ax-caucvg 8107

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4381  df-po 4384  df-iso 4385  df-iord 4454  df-on 4456  df-ilim 4457  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-isom 5323  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-recs 6441  df-frec 6527  df-sup 7139  df-inf 7140  df-pnf 8171  df-mnf 8172  df-xr 8173  df-ltxr 8174  df-le 8175  df-sub 8307  df-neg 8308  df-reap 8710  df-ap 8717  df-div 8808  df-inn 9099  df-2 9157  df-3 9158  df-4 9159  df-n0 9358  df-z 9435  df-uz 9711  df-rp 9838  df-seqfrec 10657  df-exp 10748  df-cj 11339  df-re 11340  df-im 11341  df-rsqrt 11495  df-abs 11496

This theorem is referenced by: (None)