Theorem ztprmneprm 40181

Description: A prime is not an integer multiple of another prime. (Contributed by AV, 23-May-2019.)

Assertion

Ref	Expression
ztprmneprm	|- ( ( Z e. ZZ /\ A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) )

Proof of Theorem ztprmneprm

Step	Hyp	Ref	Expression
1		elznn0nn 10951	. . 3 |- ( Z e. ZZ <-> ( Z e. NN0 \/ ( Z e. RR /\ -u Z e. NN ) ) )
2		elnn0 10871	. . . . 5 |- ( Z e. NN0 <-> ( Z e. NN \/ Z = 0 ) )
3		elnn1uz2 11235	. . . . . . 7 |- ( Z e. NN <-> ( Z = 1 \/ Z e. ( ZZ>= ` 2 ) ) )
4		oveq1 6297	. . . . . . . . . . . 12 |- ( Z = 1 -> ( Z x. A ) = ( 1 x. A ) )
5	4	adantr 467	. . . . . . . . . . 11 |- ( ( Z = 1 /\ ( A e. Prime /\ B e. Prime ) ) -> ( Z x. A ) = ( 1 x. A ) )
6	5	eqeq1d 2453	. . . . . . . . . 10 |- ( ( Z = 1 /\ ( A e. Prime /\ B e. Prime ) ) -> ( ( Z x. A ) = B <-> ( 1 x. A ) = B ) )
7		prmz 14626	. . . . . . . . . . . . . . . 16 |- ( A e. Prime -> A e. ZZ )
8	7	zcnd 11041	. . . . . . . . . . . . . . 15 |- ( A e. Prime -> A e. CC )
9	8	mulid2d 9661	. . . . . . . . . . . . . 14 |- ( A e. Prime -> ( 1 x. A ) = A )
10	9	adantr 467	. . . . . . . . . . . . 13 |- ( ( A e. Prime /\ B e. Prime ) -> ( 1 x. A ) = A )
11	10	eqeq1d 2453	. . . . . . . . . . . 12 |- ( ( A e. Prime /\ B e. Prime ) -> ( ( 1 x. A ) = B <-> A = B ) )
12	11	biimpd 211	. . . . . . . . . . 11 |- ( ( A e. Prime /\ B e. Prime ) -> ( ( 1 x. A ) = B -> A = B ) )
13	12	adantl 468	. . . . . . . . . 10 |- ( ( Z = 1 /\ ( A e. Prime /\ B e. Prime ) ) -> ( ( 1 x. A ) = B -> A = B ) )
14	6, 13	sylbid 219	. . . . . . . . 9 |- ( ( Z = 1 /\ ( A e. Prime /\ B e. Prime ) ) -> ( ( Z x. A ) = B -> A = B ) )
15	14	ex 436	. . . . . . . 8 |- ( Z = 1 -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
16		prmuz2 14642	. . . . . . . . . . . 12 |- ( A e. Prime -> A e. ( ZZ>= ` 2 ) )
17	16	adantr 467	. . . . . . . . . . 11 |- ( ( A e. Prime /\ B e. Prime ) -> A e. ( ZZ>= ` 2 ) )
18		nprm 14638	. . . . . . . . . . 11 |- ( ( Z e. ( ZZ>= ` 2 ) /\ A e. ( ZZ>= ` 2 ) ) -> -. ( Z x. A ) e. Prime )
19	17, 18	sylan2 477	. . . . . . . . . 10 |- ( ( Z e. ( ZZ>= ` 2 ) /\ ( A e. Prime /\ B e. Prime ) ) -> -. ( Z x. A ) e. Prime )
20		eleq1 2517	. . . . . . . . . . . . 13 |- ( ( Z x. A ) = B -> ( ( Z x. A ) e. Prime <-> B e. Prime ) )
21	20	notbid 296	. . . . . . . . . . . 12 |- ( ( Z x. A ) = B -> ( -. ( Z x. A ) e. Prime <-> -. B e. Prime ) )
22		pm2.24 113	. . . . . . . . . . . . . . 15 |- ( B e. Prime -> ( -. B e. Prime -> A = B ) )
23	22	adantl 468	. . . . . . . . . . . . . 14 |- ( ( A e. Prime /\ B e. Prime ) -> ( -. B e. Prime -> A = B ) )
24	23	adantl 468	. . . . . . . . . . . . 13 |- ( ( Z e. ( ZZ>= ` 2 ) /\ ( A e. Prime /\ B e. Prime ) ) -> ( -. B e. Prime -> A = B ) )
25	24	com12 32	. . . . . . . . . . . 12 |- ( -. B e. Prime -> ( ( Z e. ( ZZ>= ` 2 ) /\ ( A e. Prime /\ B e. Prime ) ) -> A = B ) )
26	21, 25	syl6bi 232	. . . . . . . . . . 11 |- ( ( Z x. A ) = B -> ( -. ( Z x. A ) e. Prime -> ( ( Z e. ( ZZ>= ` 2 ) /\ ( A e. Prime /\ B e. Prime ) ) -> A = B ) ) )
27	26	com3l 84	. . . . . . . . . 10 |- ( -. ( Z x. A ) e. Prime -> ( ( Z e. ( ZZ>= ` 2 ) /\ ( A e. Prime /\ B e. Prime ) ) -> ( ( Z x. A ) = B -> A = B ) ) )
28	19, 27	mpcom 37	. . . . . . . . 9 |- ( ( Z e. ( ZZ>= ` 2 ) /\ ( A e. Prime /\ B e. Prime ) ) -> ( ( Z x. A ) = B -> A = B ) )
29	28	ex 436	. . . . . . . 8 |- ( Z e. ( ZZ>= ` 2 ) -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
30	15, 29	jaoi 381	. . . . . . 7 |- ( ( Z = 1 \/ Z e. ( ZZ>= ` 2 ) ) -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
31	3, 30	sylbi 199	. . . . . 6 |- ( Z e. NN -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
32		oveq1 6297	. . . . . . . . 9 |- ( Z = 0 -> ( Z x. A ) = ( 0 x. A ) )
33	32	eqeq1d 2453	. . . . . . . 8 |- ( Z = 0 -> ( ( Z x. A ) = B <-> ( 0 x. A ) = B ) )
34		prmnn 14625	. . . . . . . . . . . . . 14 |- ( A e. Prime -> A e. NN )
35	34	nnred 10624	. . . . . . . . . . . . 13 |- ( A e. Prime -> A e. RR )
36		mul02lem2 9810	. . . . . . . . . . . . 13 |- ( A e. RR -> ( 0 x. A ) = 0 )
37	35, 36	syl 17	. . . . . . . . . . . 12 |- ( A e. Prime -> ( 0 x. A ) = 0 )
38	37	adantr 467	. . . . . . . . . . 11 |- ( ( A e. Prime /\ B e. Prime ) -> ( 0 x. A ) = 0 )
39	38	eqeq1d 2453	. . . . . . . . . 10 |- ( ( A e. Prime /\ B e. Prime ) -> ( ( 0 x. A ) = B <-> 0 = B ) )
40		prmnn 14625	. . . . . . . . . . . 12 |- ( B e. Prime -> B e. NN )
41		elnnne0 10883	. . . . . . . . . . . . 13 |- ( B e. NN <-> ( B e. NN0 /\ B =/= 0 ) )
42		eqneqall 2634	. . . . . . . . . . . . . . . 16 |- ( B = 0 -> ( B =/= 0 -> A = B ) )
43	42	eqcoms 2459	. . . . . . . . . . . . . . 15 |- ( 0 = B -> ( B =/= 0 -> A = B ) )
44	43	com12 32	. . . . . . . . . . . . . 14 |- ( B =/= 0 -> ( 0 = B -> A = B ) )
45	44	adantl 468	. . . . . . . . . . . . 13 |- ( ( B e. NN0 /\ B =/= 0 ) -> ( 0 = B -> A = B ) )
46	41, 45	sylbi 199	. . . . . . . . . . . 12 |- ( B e. NN -> ( 0 = B -> A = B ) )
47	40, 46	syl 17	. . . . . . . . . . 11 |- ( B e. Prime -> ( 0 = B -> A = B ) )
48	47	adantl 468	. . . . . . . . . 10 |- ( ( A e. Prime /\ B e. Prime ) -> ( 0 = B -> A = B ) )
49	39, 48	sylbid 219	. . . . . . . . 9 |- ( ( A e. Prime /\ B e. Prime ) -> ( ( 0 x. A ) = B -> A = B ) )
50	49	com12 32	. . . . . . . 8 |- ( ( 0 x. A ) = B -> ( ( A e. Prime /\ B e. Prime ) -> A = B ) )
51	33, 50	syl6bi 232	. . . . . . 7 |- ( Z = 0 -> ( ( Z x. A ) = B -> ( ( A e. Prime /\ B e. Prime ) -> A = B ) ) )
52	51	com23 81	. . . . . 6 |- ( Z = 0 -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
53	31, 52	jaoi 381	. . . . 5 |- ( ( Z e. NN \/ Z = 0 ) -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
54	2, 53	sylbi 199	. . . 4 |- ( Z e. NN0 -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
55		elnnz 10947	. . . . . 6 |- ( -u Z e. NN <-> ( -u Z e. ZZ /\ 0 < -u Z ) )
56		lt0neg1 10120	. . . . . . . 8 |- ( Z e. RR -> ( Z < 0 <-> 0 < -u Z ) )
57	34	nngt0d 10653	. . . . . . . . . . . . . . . 16 |- ( A e. Prime -> 0 < A )
58	57	adantr 467	. . . . . . . . . . . . . . 15 |- ( ( A e. Prime /\ B e. Prime ) -> 0 < A )
59		simpr 463	. . . . . . . . . . . . . . 15 |- ( ( Z e. RR /\ Z < 0 ) -> Z < 0 )
60	58, 59	anim12ci 571	. . . . . . . . . . . . . 14 |- ( ( ( A e. Prime /\ B e. Prime ) /\ ( Z e. RR /\ Z < 0 ) ) -> ( Z < 0 /\ 0 < A ) )
61	60	orcd 394	. . . . . . . . . . . . 13 |- ( ( ( A e. Prime /\ B e. Prime ) /\ ( Z e. RR /\ Z < 0 ) ) -> ( ( Z < 0 /\ 0 < A ) \/ ( 0 < Z /\ A < 0 ) ) )
62		simprl 764	. . . . . . . . . . . . . 14 |- ( ( ( A e. Prime /\ B e. Prime ) /\ ( Z e. RR /\ Z < 0 ) ) -> Z e. RR )
63	35	adantr 467	. . . . . . . . . . . . . . 15 |- ( ( A e. Prime /\ B e. Prime ) -> A e. RR )
64	63	adantr 467	. . . . . . . . . . . . . 14 |- ( ( ( A e. Prime /\ B e. Prime ) /\ ( Z e. RR /\ Z < 0 ) ) -> A e. RR )
65	62, 64	mul2lt0bi 11402	. . . . . . . . . . . . 13 |- ( ( ( A e. Prime /\ B e. Prime ) /\ ( Z e. RR /\ Z < 0 ) ) -> ( ( Z x. A ) < 0 <-> ( ( Z < 0 /\ 0 < A ) \/ ( 0 < Z /\ A < 0 ) ) ) )
66	61, 65	mpbird 236	. . . . . . . . . . . 12 |- ( ( ( A e. Prime /\ B e. Prime ) /\ ( Z e. RR /\ Z < 0 ) ) -> ( Z x. A ) < 0 )
67	66	ex 436	. . . . . . . . . . 11 |- ( ( A e. Prime /\ B e. Prime ) -> ( ( Z e. RR /\ Z < 0 ) -> ( Z x. A ) < 0 ) )
68		breq1 4405	. . . . . . . . . . . . . . 15 |- ( ( Z x. A ) = B -> ( ( Z x. A ) < 0 <-> B < 0 ) )
69	68	adantl 468	. . . . . . . . . . . . . 14 |- ( ( ( A e. Prime /\ B e. Prime ) /\ ( Z x. A ) = B ) -> ( ( Z x. A ) < 0 <-> B < 0 ) )
70		nnnn0 10876	. . . . . . . . . . . . . . . . . 18 |- ( B e. NN -> B e. NN0 )
71		nn0nlt0 10896	. . . . . . . . . . . . . . . . . . 19 |- ( B e. NN0 -> -. B < 0 )
72	71	pm2.21d 110	. . . . . . . . . . . . . . . . . 18 |- ( B e. NN0 -> ( B < 0 -> A = B ) )
73	70, 72	syl 17	. . . . . . . . . . . . . . . . 17 |- ( B e. NN -> ( B < 0 -> A = B ) )
74	40, 73	syl 17	. . . . . . . . . . . . . . . 16 |- ( B e. Prime -> ( B < 0 -> A = B ) )
75	74	adantl 468	. . . . . . . . . . . . . . 15 |- ( ( A e. Prime /\ B e. Prime ) -> ( B < 0 -> A = B ) )
76	75	adantr 467	. . . . . . . . . . . . . 14 |- ( ( ( A e. Prime /\ B e. Prime ) /\ ( Z x. A ) = B ) -> ( B < 0 -> A = B ) )
77	69, 76	sylbid 219	. . . . . . . . . . . . 13 |- ( ( ( A e. Prime /\ B e. Prime ) /\ ( Z x. A ) = B ) -> ( ( Z x. A ) < 0 -> A = B ) )
78	77	ex 436	. . . . . . . . . . . 12 |- ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> ( ( Z x. A ) < 0 -> A = B ) ) )
79	78	com23 81	. . . . . . . . . . 11 |- ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) < 0 -> ( ( Z x. A ) = B -> A = B ) ) )
80	67, 79	syld 45	. . . . . . . . . 10 |- ( ( A e. Prime /\ B e. Prime ) -> ( ( Z e. RR /\ Z < 0 ) -> ( ( Z x. A ) = B -> A = B ) ) )
81	80	com12 32	. . . . . . . . 9 |- ( ( Z e. RR /\ Z < 0 ) -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
82	81	ex 436	. . . . . . . 8 |- ( Z e. RR -> ( Z < 0 -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) ) )
83	56, 82	sylbird 239	. . . . . . 7 |- ( Z e. RR -> ( 0 < -u Z -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) ) )
84	83	adantld 469	. . . . . 6 |- ( Z e. RR -> ( ( -u Z e. ZZ /\ 0 < -u Z ) -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) ) )
85	55, 84	syl5bi 221	. . . . 5 |- ( Z e. RR -> ( -u Z e. NN -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) ) )
86	85	imp 431	. . . 4 |- ( ( Z e. RR /\ -u Z e. NN ) -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
87	54, 86	jaoi 381	. . 3 |- ( ( Z e. NN0 \/ ( Z e. RR /\ -u Z e. NN ) ) -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
88	1, 87	sylbi 199	. 2 |- ( Z e. ZZ -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
89	88	3impib 1206	1 |- ( ( Z e. ZZ /\ A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   \/ wo 370   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   =/= wne 2622   class class class wbr 4402   ` cfv 5582  (class class class)co 6290   RRcr 9538   0cc0 9539   1c1 9540   x. cmul 9544   < clt 9675   -ucneg 9861   NNcn 10609   2c2 10659   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159   Primecprime 14622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-dvds 14306  df-prm 14623

This theorem is referenced by:  zlmodzxznm  40343