Theorem ztprmneprm 33123

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 10899	. . 3 |- ( Z e. ZZ <-> ( Z e. NN0 \/ ( Z e. RR /\ -u Z e. NN ) ) )
2		elnn0 10818	. . . . 5 |- ( Z e. NN0 <-> ( Z e. NN \/ Z = 0 ) )
3		elnn1uz2 11183	. . . . . . 7 |- ( Z e. NN <-> ( Z = 1 \/ Z e. ( ZZ>= ` 2 ) ) )
4		oveq1 6303	. . . . . . . . . . . 12 |- ( Z = 1 -> ( Z x. A ) = ( 1 x. A ) )
5	4	adantr 465	. . . . . . . . . . 11 |- ( ( Z = 1 /\ ( A e. Prime /\ B e. Prime ) ) -> ( Z x. A ) = ( 1 x. A ) )
6	5	eqeq1d 2459	. . . . . . . . . 10 |- ( ( Z = 1 /\ ( A e. Prime /\ B e. Prime ) ) -> ( ( Z x. A ) = B <-> ( 1 x. A ) = B ) )
7		prmz 14324	. . . . . . . . . . . . . . . 16 |- ( A e. Prime -> A e. ZZ )
8	7	zcnd 10991	. . . . . . . . . . . . . . 15 |- ( A e. Prime -> A e. CC )
9	8	mulid2d 9631	. . . . . . . . . . . . . 14 |- ( A e. Prime -> ( 1 x. A ) = A )
10	9	adantr 465	. . . . . . . . . . . . 13 |- ( ( A e. Prime /\ B e. Prime ) -> ( 1 x. A ) = A )
11	10	eqeq1d 2459	. . . . . . . . . . . 12 |- ( ( A e. Prime /\ B e. Prime ) -> ( ( 1 x. A ) = B <-> A = B ) )
12	11	biimpd 207	. . . . . . . . . . 11 |- ( ( A e. Prime /\ B e. Prime ) -> ( ( 1 x. A ) = B -> A = B ) )
13	12	adantl 466	. . . . . . . . . 10 |- ( ( Z = 1 /\ ( A e. Prime /\ B e. Prime ) ) -> ( ( 1 x. A ) = B -> A = B ) )
14	6, 13	sylbid 215	. . . . . . . . 9 |- ( ( Z = 1 /\ ( A e. Prime /\ B e. Prime ) ) -> ( ( Z x. A ) = B -> A = B ) )
15	14	ex 434	. . . . . . . 8 |- ( Z = 1 -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
16		prmuz2 14338	. . . . . . . . . . . 12 |- ( A e. Prime -> A e. ( ZZ>= ` 2 ) )
17	16	adantr 465	. . . . . . . . . . 11 |- ( ( A e. Prime /\ B e. Prime ) -> A e. ( ZZ>= ` 2 ) )
18		nprm 14334	. . . . . . . . . . 11 |- ( ( Z e. ( ZZ>= ` 2 ) /\ A e. ( ZZ>= ` 2 ) ) -> -. ( Z x. A ) e. Prime )
19	17, 18	sylan2 474	. . . . . . . . . 10 |- ( ( Z e. ( ZZ>= ` 2 ) /\ ( A e. Prime /\ B e. Prime ) ) -> -. ( Z x. A ) e. Prime )
20		eleq1 2529	. . . . . . . . . . . . 13 |- ( ( Z x. A ) = B -> ( ( Z x. A ) e. Prime <-> B e. Prime ) )
21	20	notbid 294	. . . . . . . . . . . 12 |- ( ( Z x. A ) = B -> ( -. ( Z x. A ) e. Prime <-> -. B e. Prime ) )
22		pm2.24 109	. . . . . . . . . . . . . . 15 |- ( B e. Prime -> ( -. B e. Prime -> A = B ) )
23	22	adantl 466	. . . . . . . . . . . . . 14 |- ( ( A e. Prime /\ B e. Prime ) -> ( -. B e. Prime -> A = B ) )
24	23	adantl 466	. . . . . . . . . . . . 13 |- ( ( Z e. ( ZZ>= ` 2 ) /\ ( A e. Prime /\ B e. Prime ) ) -> ( -. B e. Prime -> A = B ) )
25	24	com12 31	. . . . . . . . . . . 12 |- ( -. B e. Prime -> ( ( Z e. ( ZZ>= ` 2 ) /\ ( A e. Prime /\ B e. Prime ) ) -> A = B ) )
26	21, 25	syl6bi 228	. . . . . . . . . . 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 81	. . . . . . . . . 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 36	. . . . . . . . 9 |- ( ( Z e. ( ZZ>= ` 2 ) /\ ( A e. Prime /\ B e. Prime ) ) -> ( ( Z x. A ) = B -> A = B ) )
29	28	ex 434	. . . . . . . 8 |- ( Z e. ( ZZ>= ` 2 ) -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
30	15, 29	jaoi 379	. . . . . . 7 |- ( ( Z = 1 \/ Z e. ( ZZ>= ` 2 ) ) -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
31	3, 30	sylbi 195	. . . . . 6 |- ( Z e. NN -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
32		oveq1 6303	. . . . . . . . 9 |- ( Z = 0 -> ( Z x. A ) = ( 0 x. A ) )
33	32	eqeq1d 2459	. . . . . . . 8 |- ( Z = 0 -> ( ( Z x. A ) = B <-> ( 0 x. A ) = B ) )
34		prmnn 14323	. . . . . . . . . . . . . 14 |- ( A e. Prime -> A e. NN )
35	34	nnred 10571	. . . . . . . . . . . . 13 |- ( A e. Prime -> A e. RR )
36		mul02lem2 9774	. . . . . . . . . . . . 13 |- ( A e. RR -> ( 0 x. A ) = 0 )
37	35, 36	syl 16	. . . . . . . . . . . 12 |- ( A e. Prime -> ( 0 x. A ) = 0 )
38	37	adantr 465	. . . . . . . . . . 11 |- ( ( A e. Prime /\ B e. Prime ) -> ( 0 x. A ) = 0 )
39	38	eqeq1d 2459	. . . . . . . . . 10 |- ( ( A e. Prime /\ B e. Prime ) -> ( ( 0 x. A ) = B <-> 0 = B ) )
40		prmnn 14323	. . . . . . . . . . . 12 |- ( B e. Prime -> B e. NN )
41		elnnne0 10830	. . . . . . . . . . . . 13 |- ( B e. NN <-> ( B e. NN0 /\ B =/= 0 ) )
42		eqneqall 2664	. . . . . . . . . . . . . . . 16 |- ( B = 0 -> ( B =/= 0 -> A = B ) )
43	42	eqcoms 2469	. . . . . . . . . . . . . . 15 |- ( 0 = B -> ( B =/= 0 -> A = B ) )
44	43	com12 31	. . . . . . . . . . . . . 14 |- ( B =/= 0 -> ( 0 = B -> A = B ) )
45	44	adantl 466	. . . . . . . . . . . . 13 |- ( ( B e. NN0 /\ B =/= 0 ) -> ( 0 = B -> A = B ) )
46	41, 45	sylbi 195	. . . . . . . . . . . 12 |- ( B e. NN -> ( 0 = B -> A = B ) )
47	40, 46	syl 16	. . . . . . . . . . 11 |- ( B e. Prime -> ( 0 = B -> A = B ) )
48	47	adantl 466	. . . . . . . . . 10 |- ( ( A e. Prime /\ B e. Prime ) -> ( 0 = B -> A = B ) )
49	39, 48	sylbid 215	. . . . . . . . 9 |- ( ( A e. Prime /\ B e. Prime ) -> ( ( 0 x. A ) = B -> A = B ) )
50	49	com12 31	. . . . . . . 8 |- ( ( 0 x. A ) = B -> ( ( A e. Prime /\ B e. Prime ) -> A = B ) )
51	33, 50	syl6bi 228	. . . . . . 7 |- ( Z = 0 -> ( ( Z x. A ) = B -> ( ( A e. Prime /\ B e. Prime ) -> A = B ) ) )
52	51	com23 78	. . . . . 6 |- ( Z = 0 -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
53	31, 52	jaoi 379	. . . . 5 |- ( ( Z e. NN \/ Z = 0 ) -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
54	2, 53	sylbi 195	. . . 4 |- ( Z e. NN0 -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
55		elnnz 10895	. . . . . 6 |- ( -u Z e. NN <-> ( -u Z e. ZZ /\ 0 < -u Z ) )
56		lt0neg1 10079	. . . . . . . 8 |- ( Z e. RR -> ( Z < 0 <-> 0 < -u Z ) )
57	34	nngt0d 10600	. . . . . . . . . . . . . . . 16 |- ( A e. Prime -> 0 < A )
58	57	adantr 465	. . . . . . . . . . . . . . 15 |- ( ( A e. Prime /\ B e. Prime ) -> 0 < A )
59		simpr 461	. . . . . . . . . . . . . . 15 |- ( ( Z e. RR /\ Z < 0 ) -> Z < 0 )
60	58, 59	anim12ci 567	. . . . . . . . . . . . . 14 |- ( ( ( A e. Prime /\ B e. Prime ) /\ ( Z e. RR /\ Z < 0 ) ) -> ( Z < 0 /\ 0 < A ) )
61	60	orcd 392	. . . . . . . . . . . . 13 |- ( ( ( A e. Prime /\ B e. Prime ) /\ ( Z e. RR /\ Z < 0 ) ) -> ( ( Z < 0 /\ 0 < A ) \/ ( 0 < Z /\ A < 0 ) ) )
62		simprl 756	. . . . . . . . . . . . . 14 |- ( ( ( A e. Prime /\ B e. Prime ) /\ ( Z e. RR /\ Z < 0 ) ) -> Z e. RR )
63	35	adantr 465	. . . . . . . . . . . . . . 15 |- ( ( A e. Prime /\ B e. Prime ) -> A e. RR )
64	63	adantr 465	. . . . . . . . . . . . . 14 |- ( ( ( A e. Prime /\ B e. Prime ) /\ ( Z e. RR /\ Z < 0 ) ) -> A e. RR )
65	62, 64	mul2lt0bi 27813	. . . . . . . . . . . . 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 232	. . . . . . . . . . . 12 |- ( ( ( A e. Prime /\ B e. Prime ) /\ ( Z e. RR /\ Z < 0 ) ) -> ( Z x. A ) < 0 )
67	66	ex 434	. . . . . . . . . . 11 |- ( ( A e. Prime /\ B e. Prime ) -> ( ( Z e. RR /\ Z < 0 ) -> ( Z x. A ) < 0 ) )
68		breq1 4459	. . . . . . . . . . . . . . 15 |- ( ( Z x. A ) = B -> ( ( Z x. A ) < 0 <-> B < 0 ) )
69	68	adantl 466	. . . . . . . . . . . . . 14 |- ( ( ( A e. Prime /\ B e. Prime ) /\ ( Z x. A ) = B ) -> ( ( Z x. A ) < 0 <-> B < 0 ) )
70		nnnn0 10823	. . . . . . . . . . . . . . . . . 18 |- ( B e. NN -> B e. NN0 )
71		nn0nlt0 10843	. . . . . . . . . . . . . . . . . . 19 |- ( B e. NN0 -> -. B < 0 )
72	71	pm2.21d 106	. . . . . . . . . . . . . . . . . 18 |- ( B e. NN0 -> ( B < 0 -> A = B ) )
73	70, 72	syl 16	. . . . . . . . . . . . . . . . 17 |- ( B e. NN -> ( B < 0 -> A = B ) )
74	40, 73	syl 16	. . . . . . . . . . . . . . . 16 |- ( B e. Prime -> ( B < 0 -> A = B ) )
75	74	adantl 466	. . . . . . . . . . . . . . 15 |- ( ( A e. Prime /\ B e. Prime ) -> ( B < 0 -> A = B ) )
76	75	adantr 465	. . . . . . . . . . . . . 14 |- ( ( ( A e. Prime /\ B e. Prime ) /\ ( Z x. A ) = B ) -> ( B < 0 -> A = B ) )
77	69, 76	sylbid 215	. . . . . . . . . . . . 13 |- ( ( ( A e. Prime /\ B e. Prime ) /\ ( Z x. A ) = B ) -> ( ( Z x. A ) < 0 -> A = B ) )
78	77	ex 434	. . . . . . . . . . . 12 |- ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> ( ( Z x. A ) < 0 -> A = B ) ) )
79	78	com23 78	. . . . . . . . . . 11 |- ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) < 0 -> ( ( Z x. A ) = B -> A = B ) ) )
80	67, 79	syld 44	. . . . . . . . . 10 |- ( ( A e. Prime /\ B e. Prime ) -> ( ( Z e. RR /\ Z < 0 ) -> ( ( Z x. A ) = B -> A = B ) ) )
81	80	com12 31	. . . . . . . . 9 |- ( ( Z e. RR /\ Z < 0 ) -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
82	81	ex 434	. . . . . . . 8 |- ( Z e. RR -> ( Z < 0 -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) ) )
83	56, 82	sylbird 235	. . . . . . 7 |- ( Z e. RR -> ( 0 < -u Z -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) ) )
84	83	adantld 467	. . . . . 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 217	. . . . 5 |- ( Z e. RR -> ( -u Z e. NN -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) ) )
86	85	imp 429	. . . 4 |- ( ( Z e. RR /\ -u Z e. NN ) -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
87	54, 86	jaoi 379	. . 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 195	. 2 |- ( Z e. ZZ -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
89	88	3impib 1194	1 |- ( ( Z e. ZZ /\ A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   /\ w3a 973   = wceq 1395   e. wcel 1819   =/= wne 2652   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   RRcr 9508   0cc0 9509   1c1 9510   x. cmul 9514   < clt 9645   -ucneg 9825   NNcn 10556   2c2 10606   NN0cn0 10816   ZZcz 10885   ZZ>=cuz 11106   Primecprime 14320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-dvds 14090  df-prm 14321

This theorem is referenced by:  zlmodzxznm  33285