Theorem lcmf0val 15335

Description: The value, by convention, of the least common multiple for a set containing 0 is 0. (Contributed by AV, 21-Apr-2020.) (Proof shortened by AV, 16-Sep-2020.)

Assertion

Ref	Expression
lcmf0val	|- ( ( Z C_ ZZ /\ 0 e. Z ) -> (lcm ` Z ) = 0 )

Proof of Theorem lcmf0val

Dummy variables m n z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-lcmf 15304	. . 3 |- lcm = ( z e. ~P ZZ |-> if ( 0 e. z , 0 , inf ( { n e. NN | A. m e. z m || n } , RR , < ) ) )
2	1	a1i 11	. 2 |- ( ( Z C_ ZZ /\ 0 e. Z ) -> lcm = ( z e. ~P ZZ |-> if ( 0 e. z , 0 , inf ( { n e. NN | A. m e. z m || n } , RR , < ) ) ) )
3		eleq2 2690	. . . 4 |- ( z = Z -> ( 0 e. z <-> 0 e. Z ) )
4		raleq 3138	. . . . . 6 |- ( z = Z -> ( A. m e. z m || n <-> A. m e. Z m || n ) )
5	4	rabbidv 3189	. . . . 5 |- ( z = Z -> { n e. NN | A. m e. z m || n } = { n e. NN | A. m e. Z m || n } )
6	5	infeq1d 8383	. . . 4 |- ( z = Z -> inf ( { n e. NN | A. m e. z m || n } , RR , < ) = inf ( { n e. NN | A. m e. Z m || n } , RR , < ) )
7	3, 6	ifbieq2d 4111	. . 3 |- ( z = Z -> if ( 0 e. z , 0 , inf ( { n e. NN | A. m e. z m || n } , RR , < ) ) = if ( 0 e. Z , 0 , inf ( { n e. NN | A. m e. Z m || n } , RR , < ) ) )
8		iftrue 4092	. . . 4 |- ( 0 e. Z -> if ( 0 e. Z , 0 , inf ( { n e. NN | A. m e. Z m || n } , RR , < ) ) = 0 )
9	8	adantl 482	. . 3 |- ( ( Z C_ ZZ /\ 0 e. Z ) -> if ( 0 e. Z , 0 , inf ( { n e. NN | A. m e. Z m || n } , RR , < ) ) = 0 )
10	7, 9	sylan9eqr 2678	. 2 |- ( ( ( Z C_ ZZ /\ 0 e. Z ) /\ z = Z ) -> if ( 0 e. z , 0 , inf ( { n e. NN | A. m e. z m || n } , RR , < ) ) = 0 )
11		zex 11386	. . . . . 6 |- ZZ e. _V
12	11	ssex 4802	. . . . 5 |- ( Z C_ ZZ -> Z e. _V )
13		elpwg 4166	. . . . 5 |- ( Z e. _V -> ( Z e. ~P ZZ <-> Z C_ ZZ ) )
14	12, 13	syl 17	. . . 4 |- ( Z C_ ZZ -> ( Z e. ~P ZZ <-> Z C_ ZZ ) )
15	14	ibir 257	. . 3 |- ( Z C_ ZZ -> Z e. ~P ZZ )
16	15	adantr 481	. 2 |- ( ( Z C_ ZZ /\ 0 e. Z ) -> Z e. ~P ZZ )
17		simpr 477	. 2 |- ( ( Z C_ ZZ /\ 0 e. Z ) -> 0 e. Z )
18	2, 10, 16, 17	fvmptd 6288	1 |- ( ( Z C_ ZZ /\ 0 e. Z ) -> (lcm ` Z ) = 0 )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   C_ wss 3574   ifcif 4086   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  infcinf 8347   RRcr 9935   0cc0 9936   < clt 10074   NNcn 11020   ZZcz 11377   || cdvds 14983  lcmclcmf 15302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-sup 8348  df-inf 8349  df-neg 10269  df-z 11378  df-lcmf 15304

This theorem is referenced by:  lcmfcl  15341  lcmfeq0b  15343  dvdslcmf  15344  lcmftp  15349  lcmfunsnlem2  15353