Theorem mul0inf 11806

Description: Equality of a product with zero. A bit of a curiosity, in the sense that theorems like abs00ap 11627 and mulap0bd 8837 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 8159	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
2		0cnd 8172	. 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 11746	. . . 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 11746	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( abs ` B ) e. RR )
7		mincl 11796	. . . 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 8208	. 2 |- ( ( A e. CC /\ B e. CC ) -> inf ( { ( abs ` A ) , ( abs ` B ) } , RR , < ) e. CC )
10	3	absge0d 11749	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> 0 <_ ( abs ` A ) )
11	5	absge0d 11749	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> 0 <_ ( abs ` B ) )
12		0red 8180	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> 0 e. RR )
13		lemininf 11799	. . . . . 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 1273	. . . . 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 952	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> 0 <_ inf ( { ( abs ` A ) , ( abs ` B ) } , RR , < ) )
16		ap0gt0 8820	. . . 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 11664	. . . . 5 |- ( A e. CC -> ( A # 0 <-> 0 < ( abs ` A ) ) )
19		absgt0ap 11664	. . . . 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 11800	. . . . 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 1273	. . . 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 8835	. . 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 8804	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 1397   e. wcel 2202   {cpr 3670   class class class wbr 4088   ` cfv 5326  (class class class)co 6018  infcinf 7182   CCcc 8030   RRcr 8031   0cc0 8032   x. cmul 8037   < clt 8214   <_ cle 8215   # cap 8761   abscabs 11562

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-sup 7183  df-inf 7184  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-n0 9403  df-z 9480  df-uz 9756  df-rp 9889  df-seqfrec 10711  df-exp 10802  df-cj 11407  df-re 11408  df-im 11409  df-rsqrt 11563  df-abs 11564

This theorem is referenced by: (None)