Theorem lcmcom 15306

Description: The lcm operator is commutative. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.)

Assertion

Ref	Expression
lcmcom	|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) = ( N lcm M ) )

Proof of Theorem lcmcom

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		orcom 402	. . 3 |- ( ( M = 0 \/ N = 0 ) <-> ( N = 0 \/ M = 0 ) )
2		ancom 466	. . . . . 6 |- ( ( M || n /\ N || n ) <-> ( N || n /\ M || n ) )
3	2	a1i 11	. . . . 5 |- ( n e. NN -> ( ( M || n /\ N || n ) <-> ( N || n /\ M || n ) ) )
4	3	rabbiia 3185	. . . 4 |- { n e. NN | ( M || n /\ N || n ) } = { n e. NN | ( N || n /\ M || n ) }
5	4	infeq1i 8384	. . 3 |- inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) = inf ( { n e. NN | ( N || n /\ M || n ) } , RR , < )
6	1, 5	ifbieq2i 4110	. 2 |- if ( ( M = 0 \/ N = 0 ) , 0 , inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) ) = if ( ( N = 0 \/ M = 0 ) , 0 , inf ( { n e. NN | ( N || n /\ M || n ) } , RR , < ) )
7		lcmval 15305	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) = if ( ( M = 0 \/ N = 0 ) , 0 , inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) ) )
8		lcmval 15305	. . 3 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( N lcm M ) = if ( ( N = 0 \/ M = 0 ) , 0 , inf ( { n e. NN | ( N || n /\ M || n ) } , RR , < ) ) )
9	8	ancoms 469	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( N lcm M ) = if ( ( N = 0 \/ M = 0 ) , 0 , inf ( { n e. NN | ( N || n /\ M || n ) } , RR , < ) ) )
10	6, 7, 9	3eqtr4a 2682	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) = ( N lcm M ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   ifcif 4086   class class class wbr 4653  (class class class)co 6650  infcinf 8347   RRcr 9935   0cc0 9936   < clt 10074   NNcn 11020   ZZcz 11377   || cdvds 14983   lcm clcm 15301

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-mulcl 9998  ax-i2m1 10004  ax-pre-lttri 10010  ax-pre-lttrn 10011

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-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rmo 2920  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-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-ltxr 10079  df-lcm 15303

This theorem is referenced by:  dvdslcm  15311  lcmeq0  15313  lcmcl  15314  lcmneg  15316  neglcm  15317  lcmgcd  15320  lcmdvds  15321  lcmftp  15349  lcmfunsnlem2  15353  lcmfunsnlem  15354  lcmf2a3a4e12  15360