difsqpwdvds - Metamath Proof Explorer

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Theorem difsqpwdvds 15591

Description: If the difference of two squares is a power of a prime, the prime divides twice the second squared number. (Contributed by AV, 13-Aug-2021.)

**Assertion**
Ref	Expression
difsqpwdvds	$/\$ $/\$ $/\$ $/\$

Proof of Theorem difsqpwdvds Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0cn 11302	. . . . . . 7
2		nn0cn 11302	. . . . . . 7
3	1, 2	anim12i 590	. . . . . 6 $/\$ $/\$
4	3	3adant3 1081	. . . . 5 $/\$ $/\$ $/\$
5		subsq 12972	. . . . 5 $/\$
6	4, 5	syl 17	. . . 4 $/\$ $/\$
7	6	adantr 481	. . 3 $/\$ $/\$ $/\$ $/\$
8	7	eqeq2d 2632	. 2 $/\$ $/\$ $/\$ $/\$
9		simprl 794	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
10		nn0z 11400	. . . . . . . . . . . 12
11		nn0z 11400	. . . . . . . . . . . 12
12	10, 11	anim12i 590	. . . . . . . . . . 11 $/\$ $/\$
13		zaddcl 11417	. . . . . . . . . . 11 $/\$
14	12, 13	syl 17	. . . . . . . . . 10 $/\$
15	14	3adant3 1081	. . . . . . . . 9 $/\$ $/\$
16		nn0re 11301	. . . . . . . . . . . . 13
17	16	adantl 482	. . . . . . . . . . . 12 $/\$
18		1red 10055	. . . . . . . . . . . 12 $/\$
19		nn0re 11301	. . . . . . . . . . . . 13
20	19	adantr 481	. . . . . . . . . . . 12 $/\$
21	17, 18, 20	ltaddsub2d 10628	. . . . . . . . . . 11 $/\$
22		simpr 477	. . . . . . . . . . . . 13 $/\$
23	20, 22, 18	3jca 1242	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
24		difgtsumgt 11346	. . . . . . . . . . . 12 $/\$ $/\$
25	23, 24	syl 17	. . . . . . . . . . 11 $/\$
26	21, 25	sylbid 230	. . . . . . . . . 10 $/\$
27	26	3impia 1261	. . . . . . . . 9 $/\$ $/\$
28		eluz2b1 11759	. . . . . . . . 9 $/\$
29	15, 27, 28	sylanbrc 698	. . . . . . . 8 $/\$ $/\$
30	29	adantr 481	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
31		simprr 796	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
32	9, 30, 31	3jca 1242	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33	32	adantr 481	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34		zsubcl 11419	. . . . . . . . . . 11 $/\$
35	13, 34	jca 554	. . . . . . . . . 10 $/\$ $/\$
36	12, 35	syl 17	. . . . . . . . 9 $/\$ $/\$
37	36	3adant3 1081	. . . . . . . 8 $/\$ $/\$ $/\$
38		dvdsmul1 15003	. . . . . . . 8 $/\$
39	37, 38	syl 17	. . . . . . 7 $/\$ $/\$
40	39	ad2antrr 762	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
41		breq2 4657	. . . . . . 7
42	41	adantl 482	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
43	40, 42	mpbird 247	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
44		dvdsprmpweqnn 15589	. . . . 5 $/\$ $/\$
45	33, 43, 44	sylc 65	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
46		prmz 15389	. . . . . . . . . . 11
47		iddvdsexp 15005	. . . . . . . . . . 11 $/\$
48	46, 47	sylan 488	. . . . . . . . . 10 $/\$
49		breq2 4657	. . . . . . . . . 10
50	48, 49	syl5ibrcom 237	. . . . . . . . 9 $/\$
51	50	rexlimdva 3031	. . . . . . . 8
52	51	adantr 481	. . . . . . 7 $/\$
53	52	adantl 482	. . . . . 6 $/\$ $/\$ $/\$ $/\$
54	53	adantr 481	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
55	12, 34	syl 17	. . . . . . . . . . . 12 $/\$
56	55	3adant3 1081	. . . . . . . . . . 11 $/\$ $/\$
57	21	biimp3a 1432	. . . . . . . . . . 11 $/\$ $/\$
58		eluz2b1 11759	. . . . . . . . . . 11 $/\$
59	56, 57, 58	sylanbrc 698	. . . . . . . . . 10 $/\$ $/\$
60	59	adantr 481	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
61	9, 60, 31	3jca 1242	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
62	61	adantr 481	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
63		dvdsmul2 15004	. . . . . . . . . 10 $/\$
64	37, 63	syl 17	. . . . . . . . 9 $/\$ $/\$
65	64	ad2antrr 762	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
66		breq2 4657	. . . . . . . . 9
67	66	adantl 482	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
68	65, 67	mpbird 247	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
69		dvdsprmpweqnn 15589	. . . . . . 7 $/\$ $/\$
70	62, 68, 69	sylc 65	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
71		iddvdsexp 15005	. . . . . . . . . . . . 13 $/\$
72	46, 71	sylan 488	. . . . . . . . . . . 12 $/\$
73		breq2 4657	. . . . . . . . . . . 12
74	72, 73	syl5ibrcom 237	. . . . . . . . . . 11 $/\$
75	74	rexlimdva 3031	. . . . . . . . . 10
76	75	adantr 481	. . . . . . . . 9 $/\$
77	76	adantl 482	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
78	77	adantr 481	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
79	46	adantr 481	. . . . . . . . . . . . 13 $/\$
80	37, 79	anim12ci 591	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
81		3anass 1042	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
82	80, 81	sylibr 224	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
83		dvds2sub 15016	. . . . . . . . . . 11 $/\$ $/\$ $/\$
84	82, 83	syl 17	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
85	1	3ad2ant1 1082	. . . . . . . . . . . . . . 15 $/\$ $/\$
86	2	3ad2ant2 1083	. . . . . . . . . . . . . . 15 $/\$ $/\$
87	85, 86, 86	pnncand 10431	. . . . . . . . . . . . . 14 $/\$ $/\$
88	2	2timesd 11275	. . . . . . . . . . . . . . . 16
89	88	eqcomd 2628	. . . . . . . . . . . . . . 15
90	89	3ad2ant2 1083	. . . . . . . . . . . . . 14 $/\$ $/\$
91	87, 90	eqtrd 2656	. . . . . . . . . . . . 13 $/\$ $/\$
92	91	breq2d 4665	. . . . . . . . . . . 12 $/\$ $/\$
93	92	biimpd 219	. . . . . . . . . . 11 $/\$ $/\$
94	93	adantr 481	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
95	84, 94	syld 47	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
96	95	expcomd 454	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
97	96	adantr 481	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
98	78, 97	syld 47	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
99	70, 98	mpd 15	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
100	54, 99	syld 47	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
101	45, 100	mpd 15	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
102	101	ex 450	. 2 $/\$ $/\$ $/\$ $/\$
103	8, 102	sylbid 230	1 $/\$ $/\$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wrex 2913 class class class wbr 4653

cfv 5888 (class class class)co 6650

cc 9934

cr 9935

c1 9937

caddc 9939

cmul 9941

clt 10074

cmin 10266

cn 11020

c2 11070

cn0 11292

cz 11377

cuz 11687

cexp 12860

cdvds 14983

cprime 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-1st 7168 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-inf 8349 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-q 11789 df-rp 11833 df-fz 12327 df-fl 12593 df-mod 12669 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-gcd 15217 df-prm 15386 df-pc 15542

This theorem is referenced by: lighneallem2 41523

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