Theorem nzprmdif 36738

Description: Subtract one prime's multiples from an unequal prime's. (Contributed by Steve Rodriguez, 20-Jan-2020.)

Hypotheses

Ref	Expression
nzprmdif.m	|- ( ph -> M e. Prime )
nzprmdif.n	|- ( ph -> N e. Prime )
nzprmdif.ne	|- ( ph -> M =/= N )

Assertion

Ref	Expression
nzprmdif	|- ( ph -> ( ( || " { M } ) \ ( || " { N } ) ) = ( ( || " { M } ) \ ( || " { ( M x. N ) } ) ) )

Proof of Theorem nzprmdif

Step	Hyp	Ref	Expression
1		difin 3671	. . 3 |- ( ( || " { M } ) \ ( ( || " { M } ) i^i ( || " { N } ) ) ) = ( ( || " { M } ) \ ( || " { N } ) )
2		nzprmdif.m	. . . . . 6 |- ( ph -> M e. Prime )
3		prmz 14705	. . . . . 6 |- ( M e. Prime -> M e. ZZ )
4	2, 3	syl 17	. . . . 5 |- ( ph -> M e. ZZ )
5		nzprmdif.n	. . . . . 6 |- ( ph -> N e. Prime )
6		prmz 14705	. . . . . 6 |- ( N e. Prime -> N e. ZZ )
7	5, 6	syl 17	. . . . 5 |- ( ph -> N e. ZZ )
8	4, 7	nzin 36737	. . . 4 |- ( ph -> ( ( || " { M } ) i^i ( || " { N } ) ) = ( || " { ( M lcm N ) } ) )
9	8	difeq2d 3540	. . 3 |- ( ph -> ( ( || " { M } ) \ ( ( || " { M } ) i^i ( || " { N } ) ) ) = ( ( || " { M } ) \ ( || " { ( M lcm N ) } ) ) )
10	1, 9	syl5eqr 2519	. 2 |- ( ph -> ( ( || " { M } ) \ ( || " { N } ) ) = ( ( || " { M } ) \ ( || " { ( M lcm N ) } ) ) )
11		lcmgcd 14651	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M lcm N ) x. ( M gcd N ) ) = ( abs ` ( M x. N ) ) )
12	4, 7, 11	syl2anc 673	. . . . . 6 |- ( ph -> ( ( M lcm N ) x. ( M gcd N ) ) = ( abs ` ( M x. N ) ) )
13		nzprmdif.ne	. . . . . . . . 9 |- ( ph -> M =/= N )
14		prmrp 14737	. . . . . . . . . 10 |- ( ( M e. Prime /\ N e. Prime ) -> ( ( M gcd N ) = 1 <-> M =/= N ) )
15	2, 5, 14	syl2anc 673	. . . . . . . . 9 |- ( ph -> ( ( M gcd N ) = 1 <-> M =/= N ) )
16	13, 15	mpbird 240	. . . . . . . 8 |- ( ph -> ( M gcd N ) = 1 )
17	16	oveq2d 6324	. . . . . . 7 |- ( ph -> ( ( M lcm N ) x. ( M gcd N ) ) = ( ( M lcm N ) x. 1 ) )
18		lcmcl 14645	. . . . . . . . . 10 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) e. NN0 )
19	4, 7, 18	syl2anc 673	. . . . . . . . 9 |- ( ph -> ( M lcm N ) e. NN0 )
20	19	nn0cnd 10951	. . . . . . . 8 |- ( ph -> ( M lcm N ) e. CC )
21	20	mulid1d 9678	. . . . . . 7 |- ( ph -> ( ( M lcm N ) x. 1 ) = ( M lcm N ) )
22	17, 21	eqtrd 2505	. . . . . 6 |- ( ph -> ( ( M lcm N ) x. ( M gcd N ) ) = ( M lcm N ) )
23	4	zred 11063	. . . . . . . 8 |- ( ph -> M e. RR )
24	7	zred 11063	. . . . . . . 8 |- ( ph -> N e. RR )
25	23, 24	remulcld 9689	. . . . . . 7 |- ( ph -> ( M x. N ) e. RR )
26		prmnn 14704	. . . . . . . . . . 11 |- ( M e. Prime -> M e. NN )
27	2, 26	syl 17	. . . . . . . . . 10 |- ( ph -> M e. NN )
28	27	nnnn0d 10949	. . . . . . . . 9 |- ( ph -> M e. NN0 )
29	28	nn0ge0d 10952	. . . . . . . 8 |- ( ph -> 0 <_ M )
30		prmnn 14704	. . . . . . . . . . 11 |- ( N e. Prime -> N e. NN )
31	5, 30	syl 17	. . . . . . . . . 10 |- ( ph -> N e. NN )
32	31	nnnn0d 10949	. . . . . . . . 9 |- ( ph -> N e. NN0 )
33	32	nn0ge0d 10952	. . . . . . . 8 |- ( ph -> 0 <_ N )
34	23, 24, 29, 33	mulge0d 10211	. . . . . . 7 |- ( ph -> 0 <_ ( M x. N ) )
35	25, 34	absidd 13561	. . . . . 6 |- ( ph -> ( abs ` ( M x. N ) ) = ( M x. N ) )
36	12, 22, 35	3eqtr3d 2513	. . . . 5 |- ( ph -> ( M lcm N ) = ( M x. N ) )
37	36	sneqd 3971	. . . 4 |- ( ph -> { ( M lcm N ) } = { ( M x. N ) } )
38	37	imaeq2d 5174	. . 3 |- ( ph -> ( || " { ( M lcm N ) } ) = ( || " { ( M x. N ) } ) )
39	38	difeq2d 3540	. 2 |- ( ph -> ( ( || " { M } ) \ ( || " { ( M lcm N ) } ) ) = ( ( || " { M } ) \ ( || " { ( M x. N ) } ) ) )
40	10, 39	eqtrd 2505	1 |- ( ph -> ( ( || " { M } ) \ ( || " { N } ) ) = ( ( || " { M } ) \ ( || " { ( M x. N ) } ) ) )

Syntax hints:   -> wi 4   <-> wb 189   = wceq 1452   e. wcel 1904   =/= wne 2641   \ cdif 3387   i^i cin 3389   {csn 3959   "cima 4842   ` cfv 5589  (class class class)co 6308   1c1 9558   x. cmul 9562   NNcn 10631   NN0cn0 10893   ZZcz 10961   abscabs 13374   || cdvds 14382   gcd cgcd 14547   lcm clcm 14626   Primecprime 14701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-dvds 14383  df-gcd 14548  df-lcm 14630  df-prm 14702

This theorem is referenced by: (None)