Theorem ztprmneprm 40636

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 10975	. . 3 |- ( Z e. ZZ <-> ( Z e. NN0 \/ ( Z e. RR /\ -u Z e. NN ) ) )
2		elnn0 10895	. . . . 5 |- ( Z e. NN0 <-> ( Z e. NN \/ Z = 0 ) )
3		elnn1uz2 11258	. . . . . . 7 |- ( Z e. NN <-> ( Z = 1 \/ Z e. ( ZZ>= ` 2 ) ) )
4		oveq1 6315	. . . . . . . . . . . 12 |- ( Z = 1 -> ( Z x. A ) = ( 1 x. A ) )
5	4	adantr 472	. . . . . . . . . . 11 |- ( ( Z = 1 /\ ( A e. Prime /\ B e. Prime ) ) -> ( Z x. A ) = ( 1 x. A ) )
6	5	eqeq1d 2473	. . . . . . . . . 10 |- ( ( Z = 1 /\ ( A e. Prime /\ B e. Prime ) ) -> ( ( Z x. A ) = B <-> ( 1 x. A ) = B ) )
7		prmz 14705	. . . . . . . . . . . . . . . 16 |- ( A e. Prime -> A e. ZZ )
8	7	zcnd 11064	. . . . . . . . . . . . . . 15 |- ( A e. Prime -> A e. CC )
9	8	mulid2d 9679	. . . . . . . . . . . . . 14 |- ( A e. Prime -> ( 1 x. A ) = A )
10	9	adantr 472	. . . . . . . . . . . . 13 |- ( ( A e. Prime /\ B e. Prime ) -> ( 1 x. A ) = A )
11	10	eqeq1d 2473	. . . . . . . . . . . 12 |- ( ( A e. Prime /\ B e. Prime ) -> ( ( 1 x. A ) = B <-> A = B ) )
12	11	biimpd 212	. . . . . . . . . . 11 |- ( ( A e. Prime /\ B e. Prime ) -> ( ( 1 x. A ) = B -> A = B ) )
13	12	adantl 473	. . . . . . . . . 10 |- ( ( Z = 1 /\ ( A e. Prime /\ B e. Prime ) ) -> ( ( 1 x. A ) = B -> A = B ) )
14	6, 13	sylbid 223	. . . . . . . . 9 |- ( ( Z = 1 /\ ( A e. Prime /\ B e. Prime ) ) -> ( ( Z x. A ) = B -> A = B ) )
15	14	ex 441	. . . . . . . 8 |- ( Z = 1 -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
16		prmuz2 14721	. . . . . . . . . . . 12 |- ( A e. Prime -> A e. ( ZZ>= ` 2 ) )
17	16	adantr 472	. . . . . . . . . . 11 |- ( ( A e. Prime /\ B e. Prime ) -> A e. ( ZZ>= ` 2 ) )
18		nprm 14717	. . . . . . . . . . 11 |- ( ( Z e. ( ZZ>= ` 2 ) /\ A e. ( ZZ>= ` 2 ) ) -> -. ( Z x. A ) e. Prime )
19	17, 18	sylan2 482	. . . . . . . . . 10 |- ( ( Z e. ( ZZ>= ` 2 ) /\ ( A e. Prime /\ B e. Prime ) ) -> -. ( Z x. A ) e. Prime )
20		eleq1 2537	. . . . . . . . . . . . 13 |- ( ( Z x. A ) = B -> ( ( Z x. A ) e. Prime <-> B e. Prime ) )
21	20	notbid 301	. . . . . . . . . . . 12 |- ( ( Z x. A ) = B -> ( -. ( Z x. A ) e. Prime <-> -. B e. Prime ) )
22		pm2.24 112	. . . . . . . . . . . . . . 15 |- ( B e. Prime -> ( -. B e. Prime -> A = B ) )
23	22	adantl 473	. . . . . . . . . . . . . 14 |- ( ( A e. Prime /\ B e. Prime ) -> ( -. B e. Prime -> A = B ) )
24	23	adantl 473	. . . . . . . . . . . . 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 236	. . . . . . . . . . 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 83	. . . . . . . . . 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 441	. . . . . . . 8 |- ( Z e. ( ZZ>= ` 2 ) -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
30	15, 29	jaoi 386	. . . . . . 7 |- ( ( Z = 1 \/ Z e. ( ZZ>= ` 2 ) ) -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
31	3, 30	sylbi 200	. . . . . 6 |- ( Z e. NN -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
32		oveq1 6315	. . . . . . . . 9 |- ( Z = 0 -> ( Z x. A ) = ( 0 x. A ) )
33	32	eqeq1d 2473	. . . . . . . 8 |- ( Z = 0 -> ( ( Z x. A ) = B <-> ( 0 x. A ) = B ) )
34		prmnn 14704	. . . . . . . . . . . . . 14 |- ( A e. Prime -> A e. NN )
35	34	nnred 10646	. . . . . . . . . . . . 13 |- ( A e. Prime -> A e. RR )
36		mul02lem2 9828	. . . . . . . . . . . . 13 |- ( A e. RR -> ( 0 x. A ) = 0 )
37	35, 36	syl 17	. . . . . . . . . . . 12 |- ( A e. Prime -> ( 0 x. A ) = 0 )
38	37	adantr 472	. . . . . . . . . . 11 |- ( ( A e. Prime /\ B e. Prime ) -> ( 0 x. A ) = 0 )
39	38	eqeq1d 2473	. . . . . . . . . 10 |- ( ( A e. Prime /\ B e. Prime ) -> ( ( 0 x. A ) = B <-> 0 = B ) )
40		prmnn 14704	. . . . . . . . . . . 12 |- ( B e. Prime -> B e. NN )
41		elnnne0 10907	. . . . . . . . . . . . 13 |- ( B e. NN <-> ( B e. NN0 /\ B =/= 0 ) )
42		eqneqall 2654	. . . . . . . . . . . . . . . 16 |- ( B = 0 -> ( B =/= 0 -> A = B ) )
43	42	eqcoms 2479	. . . . . . . . . . . . . . 15 |- ( 0 = B -> ( B =/= 0 -> A = B ) )
44	43	com12 31	. . . . . . . . . . . . . 14 |- ( B =/= 0 -> ( 0 = B -> A = B ) )
45	44	adantl 473	. . . . . . . . . . . . 13 |- ( ( B e. NN0 /\ B =/= 0 ) -> ( 0 = B -> A = B ) )
46	41, 45	sylbi 200	. . . . . . . . . . . 12 |- ( B e. NN -> ( 0 = B -> A = B ) )
47	40, 46	syl 17	. . . . . . . . . . 11 |- ( B e. Prime -> ( 0 = B -> A = B ) )
48	47	adantl 473	. . . . . . . . . 10 |- ( ( A e. Prime /\ B e. Prime ) -> ( 0 = B -> A = B ) )
49	39, 48	sylbid 223	. . . . . . . . 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 236	. . . . . . 7 |- ( Z = 0 -> ( ( Z x. A ) = B -> ( ( A e. Prime /\ B e. Prime ) -> A = B ) ) )
52	51	com23 80	. . . . . 6 |- ( Z = 0 -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
53	31, 52	jaoi 386	. . . . 5 |- ( ( Z e. NN \/ Z = 0 ) -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
54	2, 53	sylbi 200	. . . 4 |- ( Z e. NN0 -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
55		elnnz 10971	. . . . . 6 |- ( -u Z e. NN <-> ( -u Z e. ZZ /\ 0 < -u Z ) )
56		lt0neg1 10141	. . . . . . . 8 |- ( Z e. RR -> ( Z < 0 <-> 0 < -u Z ) )
57	34	nngt0d 10675	. . . . . . . . . . . . . . . 16 |- ( A e. Prime -> 0 < A )
58	57	adantr 472	. . . . . . . . . . . . . . 15 |- ( ( A e. Prime /\ B e. Prime ) -> 0 < A )
59		simpr 468	. . . . . . . . . . . . . . 15 |- ( ( Z e. RR /\ Z < 0 ) -> Z < 0 )
60	58, 59	anim12ci 577	. . . . . . . . . . . . . 14 |- ( ( ( A e. Prime /\ B e. Prime ) /\ ( Z e. RR /\ Z < 0 ) ) -> ( Z < 0 /\ 0 < A ) )
61	60	orcd 399	. . . . . . . . . . . . 13 |- ( ( ( A e. Prime /\ B e. Prime ) /\ ( Z e. RR /\ Z < 0 ) ) -> ( ( Z < 0 /\ 0 < A ) \/ ( 0 < Z /\ A < 0 ) ) )
62		simprl 772	. . . . . . . . . . . . . 14 |- ( ( ( A e. Prime /\ B e. Prime ) /\ ( Z e. RR /\ Z < 0 ) ) -> Z e. RR )
63	35	adantr 472	. . . . . . . . . . . . . . 15 |- ( ( A e. Prime /\ B e. Prime ) -> A e. RR )
64	63	adantr 472	. . . . . . . . . . . . . 14 |- ( ( ( A e. Prime /\ B e. Prime ) /\ ( Z e. RR /\ Z < 0 ) ) -> A e. RR )
65	62, 64	mul2lt0bi 11425	. . . . . . . . . . . . 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 240	. . . . . . . . . . . 12 |- ( ( ( A e. Prime /\ B e. Prime ) /\ ( Z e. RR /\ Z < 0 ) ) -> ( Z x. A ) < 0 )
67	66	ex 441	. . . . . . . . . . 11 |- ( ( A e. Prime /\ B e. Prime ) -> ( ( Z e. RR /\ Z < 0 ) -> ( Z x. A ) < 0 ) )
68		breq1 4398	. . . . . . . . . . . . . . 15 |- ( ( Z x. A ) = B -> ( ( Z x. A ) < 0 <-> B < 0 ) )
69	68	adantl 473	. . . . . . . . . . . . . 14 |- ( ( ( A e. Prime /\ B e. Prime ) /\ ( Z x. A ) = B ) -> ( ( Z x. A ) < 0 <-> B < 0 ) )
70		nnnn0 10900	. . . . . . . . . . . . . . . . . 18 |- ( B e. NN -> B e. NN0 )
71		nn0nlt0 10920	. . . . . . . . . . . . . . . . . . 19 |- ( B e. NN0 -> -. B < 0 )
72	71	pm2.21d 109	. . . . . . . . . . . . . . . . . 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 473	. . . . . . . . . . . . . . 15 |- ( ( A e. Prime /\ B e. Prime ) -> ( B < 0 -> A = B ) )
76	75	adantr 472	. . . . . . . . . . . . . 14 |- ( ( ( A e. Prime /\ B e. Prime ) /\ ( Z x. A ) = B ) -> ( B < 0 -> A = B ) )
77	69, 76	sylbid 223	. . . . . . . . . . . . 13 |- ( ( ( A e. Prime /\ B e. Prime ) /\ ( Z x. A ) = B ) -> ( ( Z x. A ) < 0 -> A = B ) )
78	77	ex 441	. . . . . . . . . . . 12 |- ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> ( ( Z x. A ) < 0 -> A = B ) ) )
79	78	com23 80	. . . . . . . . . . 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 441	. . . . . . . 8 |- ( Z e. RR -> ( Z < 0 -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) ) )
83	56, 82	sylbird 243	. . . . . . 7 |- ( Z e. RR -> ( 0 < -u Z -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) ) )
84	83	adantld 474	. . . . . 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 225	. . . . 5 |- ( Z e. RR -> ( -u Z e. NN -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) ) )
86	85	imp 436	. . . 4 |- ( ( Z e. RR /\ -u Z e. NN ) -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
87	54, 86	jaoi 386	. . 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 200	. 2 |- ( Z e. ZZ -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
89	88	3impib 1229	1 |- ( ( Z e. ZZ /\ A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   \/ wo 375   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   =/= wne 2641   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   RRcr 9556   0cc0 9557   1c1 9558   x. cmul 9562   < clt 9693   -ucneg 9881   NNcn 10631   2c2 10681   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   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

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-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-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-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-dvds 14383  df-prm 14702

This theorem is referenced by:  zlmodzxznm  40798