Theorem ztprmneprm 42125

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 11391	. . 3 |- ( Z e. ZZ <-> ( Z e. NN0 \/ ( Z e. RR /\ -u Z e. NN ) ) )
2		elnn0 11294	. . . . 5 |- ( Z e. NN0 <-> ( Z e. NN \/ Z = 0 ) )
3		elnn1uz2 11765	. . . . . . 7 |- ( Z e. NN <-> ( Z = 1 \/ Z e. ( ZZ>= ` 2 ) ) )
4		oveq1 6657	. . . . . . . . . . . 12 |- ( Z = 1 -> ( Z x. A ) = ( 1 x. A ) )
5	4	adantr 481	. . . . . . . . . . 11 |- ( ( Z = 1 /\ ( A e. Prime /\ B e. Prime ) ) -> ( Z x. A ) = ( 1 x. A ) )
6	5	eqeq1d 2624	. . . . . . . . . 10 |- ( ( Z = 1 /\ ( A e. Prime /\ B e. Prime ) ) -> ( ( Z x. A ) = B <-> ( 1 x. A ) = B ) )
7		prmz 15389	. . . . . . . . . . . . . . . 16 |- ( A e. Prime -> A e. ZZ )
8	7	zcnd 11483	. . . . . . . . . . . . . . 15 |- ( A e. Prime -> A e. CC )
9	8	mulid2d 10058	. . . . . . . . . . . . . 14 |- ( A e. Prime -> ( 1 x. A ) = A )
10	9	adantr 481	. . . . . . . . . . . . 13 |- ( ( A e. Prime /\ B e. Prime ) -> ( 1 x. A ) = A )
11	10	eqeq1d 2624	. . . . . . . . . . . 12 |- ( ( A e. Prime /\ B e. Prime ) -> ( ( 1 x. A ) = B <-> A = B ) )
12	11	biimpd 219	. . . . . . . . . . 11 |- ( ( A e. Prime /\ B e. Prime ) -> ( ( 1 x. A ) = B -> A = B ) )
13	12	adantl 482	. . . . . . . . . 10 |- ( ( Z = 1 /\ ( A e. Prime /\ B e. Prime ) ) -> ( ( 1 x. A ) = B -> A = B ) )
14	6, 13	sylbid 230	. . . . . . . . 9 |- ( ( Z = 1 /\ ( A e. Prime /\ B e. Prime ) ) -> ( ( Z x. A ) = B -> A = B ) )
15	14	ex 450	. . . . . . . 8 |- ( Z = 1 -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
16		prmuz2 15408	. . . . . . . . . . . 12 |- ( A e. Prime -> A e. ( ZZ>= ` 2 ) )
17	16	adantr 481	. . . . . . . . . . 11 |- ( ( A e. Prime /\ B e. Prime ) -> A e. ( ZZ>= ` 2 ) )
18		nprm 15401	. . . . . . . . . . 11 |- ( ( Z e. ( ZZ>= ` 2 ) /\ A e. ( ZZ>= ` 2 ) ) -> -. ( Z x. A ) e. Prime )
19	17, 18	sylan2 491	. . . . . . . . . 10 |- ( ( Z e. ( ZZ>= ` 2 ) /\ ( A e. Prime /\ B e. Prime ) ) -> -. ( Z x. A ) e. Prime )
20		eleq1 2689	. . . . . . . . . . . . 13 |- ( ( Z x. A ) = B -> ( ( Z x. A ) e. Prime <-> B e. Prime ) )
21	20	notbid 308	. . . . . . . . . . . 12 |- ( ( Z x. A ) = B -> ( -. ( Z x. A ) e. Prime <-> -. B e. Prime ) )
22		pm2.24 121	. . . . . . . . . . . . . . 15 |- ( B e. Prime -> ( -. B e. Prime -> A = B ) )
23	22	adantl 482	. . . . . . . . . . . . . 14 |- ( ( A e. Prime /\ B e. Prime ) -> ( -. B e. Prime -> A = B ) )
24	23	adantl 482	. . . . . . . . . . . . 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 243	. . . . . . . . . . 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 89	. . . . . . . . . 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 38	. . . . . . . . 9 |- ( ( Z e. ( ZZ>= ` 2 ) /\ ( A e. Prime /\ B e. Prime ) ) -> ( ( Z x. A ) = B -> A = B ) )
29	28	ex 450	. . . . . . . 8 |- ( Z e. ( ZZ>= ` 2 ) -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
30	15, 29	jaoi 394	. . . . . . 7 |- ( ( Z = 1 \/ Z e. ( ZZ>= ` 2 ) ) -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
31	3, 30	sylbi 207	. . . . . 6 |- ( Z e. NN -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
32		oveq1 6657	. . . . . . . . 9 |- ( Z = 0 -> ( Z x. A ) = ( 0 x. A ) )
33	32	eqeq1d 2624	. . . . . . . 8 |- ( Z = 0 -> ( ( Z x. A ) = B <-> ( 0 x. A ) = B ) )
34		prmnn 15388	. . . . . . . . . . . . . 14 |- ( A e. Prime -> A e. NN )
35	34	nnred 11035	. . . . . . . . . . . . 13 |- ( A e. Prime -> A e. RR )
36		mul02lem2 10213	. . . . . . . . . . . . 13 |- ( A e. RR -> ( 0 x. A ) = 0 )
37	35, 36	syl 17	. . . . . . . . . . . 12 |- ( A e. Prime -> ( 0 x. A ) = 0 )
38	37	adantr 481	. . . . . . . . . . 11 |- ( ( A e. Prime /\ B e. Prime ) -> ( 0 x. A ) = 0 )
39	38	eqeq1d 2624	. . . . . . . . . 10 |- ( ( A e. Prime /\ B e. Prime ) -> ( ( 0 x. A ) = B <-> 0 = B ) )
40		prmnn 15388	. . . . . . . . . . . 12 |- ( B e. Prime -> B e. NN )
41		elnnne0 11306	. . . . . . . . . . . . 13 |- ( B e. NN <-> ( B e. NN0 /\ B =/= 0 ) )
42		eqneqall 2805	. . . . . . . . . . . . . . . 16 |- ( B = 0 -> ( B =/= 0 -> A = B ) )
43	42	eqcoms 2630	. . . . . . . . . . . . . . 15 |- ( 0 = B -> ( B =/= 0 -> A = B ) )
44	43	com12 32	. . . . . . . . . . . . . 14 |- ( B =/= 0 -> ( 0 = B -> A = B ) )
45	44	adantl 482	. . . . . . . . . . . . 13 |- ( ( B e. NN0 /\ B =/= 0 ) -> ( 0 = B -> A = B ) )
46	41, 45	sylbi 207	. . . . . . . . . . . 12 |- ( B e. NN -> ( 0 = B -> A = B ) )
47	40, 46	syl 17	. . . . . . . . . . 11 |- ( B e. Prime -> ( 0 = B -> A = B ) )
48	47	adantl 482	. . . . . . . . . 10 |- ( ( A e. Prime /\ B e. Prime ) -> ( 0 = B -> A = B ) )
49	39, 48	sylbid 230	. . . . . . . . 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 243	. . . . . . 7 |- ( Z = 0 -> ( ( Z x. A ) = B -> ( ( A e. Prime /\ B e. Prime ) -> A = B ) ) )
52	51	com23 86	. . . . . 6 |- ( Z = 0 -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
53	31, 52	jaoi 394	. . . . 5 |- ( ( Z e. NN \/ Z = 0 ) -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
54	2, 53	sylbi 207	. . . 4 |- ( Z e. NN0 -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
55		elnnz 11387	. . . . . 6 |- ( -u Z e. NN <-> ( -u Z e. ZZ /\ 0 < -u Z ) )
56		lt0neg1 10534	. . . . . . . 8 |- ( Z e. RR -> ( Z < 0 <-> 0 < -u Z ) )
57	34	nngt0d 11064	. . . . . . . . . . . . . . 15 |- ( A e. Prime -> 0 < A )
58	57	adantr 481	. . . . . . . . . . . . . 14 |- ( ( A e. Prime /\ B e. Prime ) -> 0 < A )
59		simpr 477	. . . . . . . . . . . . . 14 |- ( ( Z e. RR /\ Z < 0 ) -> Z < 0 )
60	58, 59	anim12ci 591	. . . . . . . . . . . . 13 |- ( ( ( A e. Prime /\ B e. Prime ) /\ ( Z e. RR /\ Z < 0 ) ) -> ( Z < 0 /\ 0 < A ) )
61	60	orcd 407	. . . . . . . . . . . 12 |- ( ( ( A e. Prime /\ B e. Prime ) /\ ( Z e. RR /\ Z < 0 ) ) -> ( ( Z < 0 /\ 0 < A ) \/ ( 0 < Z /\ A < 0 ) ) )
62		simprl 794	. . . . . . . . . . . . 13 |- ( ( ( A e. Prime /\ B e. Prime ) /\ ( Z e. RR /\ Z < 0 ) ) -> Z e. RR )
63	35	adantr 481	. . . . . . . . . . . . . 14 |- ( ( A e. Prime /\ B e. Prime ) -> A e. RR )
64	63	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( A e. Prime /\ B e. Prime ) /\ ( Z e. RR /\ Z < 0 ) ) -> A e. RR )
65	62, 64	mul2lt0bi 11936	. . . . . . . . . . . 12 |- ( ( ( 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 247	. . . . . . . . . . 11 |- ( ( ( A e. Prime /\ B e. Prime ) /\ ( Z e. RR /\ Z < 0 ) ) -> ( Z x. A ) < 0 )
67	66	ex 450	. . . . . . . . . 10 |- ( ( A e. Prime /\ B e. Prime ) -> ( ( Z e. RR /\ Z < 0 ) -> ( Z x. A ) < 0 ) )
68		breq1 4656	. . . . . . . . . . . . . 14 |- ( ( Z x. A ) = B -> ( ( Z x. A ) < 0 <-> B < 0 ) )
69	68	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( A e. Prime /\ B e. Prime ) /\ ( Z x. A ) = B ) -> ( ( Z x. A ) < 0 <-> B < 0 ) )
70		nnnn0 11299	. . . . . . . . . . . . . . . . 17 |- ( B e. NN -> B e. NN0 )
71		nn0nlt0 11319	. . . . . . . . . . . . . . . . . 18 |- ( B e. NN0 -> -. B < 0 )
72	71	pm2.21d 118	. . . . . . . . . . . . . . . . 17 |- ( B e. NN0 -> ( B < 0 -> A = B ) )
73	70, 72	syl 17	. . . . . . . . . . . . . . . 16 |- ( B e. NN -> ( B < 0 -> A = B ) )
74	40, 73	syl 17	. . . . . . . . . . . . . . 15 |- ( B e. Prime -> ( B < 0 -> A = B ) )
75	74	adantl 482	. . . . . . . . . . . . . 14 |- ( ( A e. Prime /\ B e. Prime ) -> ( B < 0 -> A = B ) )
76	75	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( A e. Prime /\ B e. Prime ) /\ ( Z x. A ) = B ) -> ( B < 0 -> A = B ) )
77	69, 76	sylbid 230	. . . . . . . . . . . 12 |- ( ( ( A e. Prime /\ B e. Prime ) /\ ( Z x. A ) = B ) -> ( ( Z x. A ) < 0 -> A = B ) )
78	77	ex 450	. . . . . . . . . . 11 |- ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> ( ( Z x. A ) < 0 -> A = B ) ) )
79	78	com23 86	. . . . . . . . . 10 |- ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) < 0 -> ( ( Z x. A ) = B -> A = B ) ) )
80	67, 79	syldc 48	. . . . . . . . 9 |- ( ( Z e. RR /\ Z < 0 ) -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
81	80	ex 450	. . . . . . . 8 |- ( Z e. RR -> ( Z < 0 -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) ) )
82	56, 81	sylbird 250	. . . . . . 7 |- ( Z e. RR -> ( 0 < -u Z -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) ) )
83	82	adantld 483	. . . . . 6 |- ( Z e. RR -> ( ( -u Z e. ZZ /\ 0 < -u Z ) -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) ) )
84	55, 83	syl5bi 232	. . . . 5 |- ( Z e. RR -> ( -u Z e. NN -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) ) )
85	84	imp 445	. . . 4 |- ( ( Z e. RR /\ -u Z e. NN ) -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
86	54, 85	jaoi 394	. . 3 |- ( ( Z e. NN0 \/ ( Z e. RR /\ -u Z e. NN ) ) -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
87	1, 86	sylbi 207	. 2 |- ( Z e. ZZ -> ( ( A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) ) )
88	87	3impib 1262	1 |- ( ( Z e. ZZ /\ A e. Prime /\ B e. Prime ) -> ( ( Z x. A ) = B -> A = B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   < clt 10074   -ucneg 10267   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   Primecprime 15385

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-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

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-reu 2919  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-prm 15386

This theorem is referenced by:  zlmodzxznm  42286