pythagtriplem4 - Metamath Proof Explorer

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Theorem pythagtriplem4 15524

Description: Lemma for pythagtrip 15539. Show that

and

are relatively prime. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

**Assertion**
Ref	Expression
pythagtriplem4	$/\$ $/\$ $/\$ $/\$ $/\$

**Proof of Theorem pythagtriplem4**
Step	Hyp	Ref	Expression
1		simp3r 1090	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
2		nnz 11399	. . . . . . . . . . . . 13
3		nnz 11399	. . . . . . . . . . . . 13
4		zsubcl 11419	. . . . . . . . . . . . 13 $/\$
5	2, 3, 4	syl2anr 495	. . . . . . . . . . . 12 $/\$
6	5	3adant1 1079	. . . . . . . . . . 11 $/\$ $/\$
7	6	3ad2ant1 1082	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
8		simp13 1093	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
9		simp12 1092	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
10	8, 9	nnaddcld 11067	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
11	10	nnzd 11481	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
12		gcddvds 15225	. . . . . . . . . 10 $/\$ $/\$
13	7, 11, 12	syl2anc 693	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
14	13	simprd 479	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
15		breq1 4656	. . . . . . . . 9
16	15	biimpd 219	. . . . . . . 8
17	14, 16	mpan9 486	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
18		simpl13 1138	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
19	18	nnzd 11481	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
20		simpl12 1137	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
21	20	nnzd 11481	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
22	19, 21	zaddcld 11486	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
23	19, 21	zsubcld 11487	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
24		2z 11409	. . . . . . . . 9
25		dvdsmultr1 15019	. . . . . . . . 9 $/\$ $/\$
26	24, 25	mp3an1 1411	. . . . . . . 8 $/\$
27	22, 23, 26	syl2anc 693	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28	17, 27	mpd 15	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29	18	nncnd 11036	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30	20	nncnd 11036	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
31		subsq 12972	. . . . . . 7 $/\$
32	29, 30, 31	syl2anc 693	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33	28, 32	breqtrrd 4681	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34		simpl2 1065	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35	34	oveq1d 6665	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36		simpl11 1136	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37	36	nnsqcld 13029	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38	37	nncnd 11036	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39	20	nnsqcld 13029	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	39	nncnd 11036	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	38, 40	pncand 10393	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	35, 41	eqtr3d 2658	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43	33, 42	breqtrd 4679	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
44		nnz 11399	. . . . . . . 8
45	44	3ad2ant1 1082	. . . . . . 7 $/\$ $/\$
46	45	3ad2ant1 1082	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
47	46	adantr 481	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
48		2prm 15405	. . . . . 6
49		2nn 11185	. . . . . 6
50		prmdvdsexp 15427	. . . . . 6 $/\$ $/\$
51	48, 49, 50	mp3an13 1415	. . . . 5
52	47, 51	syl 17	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
53	43, 52	mpbid 222	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
54	1, 53	mtand 691	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
55		neg1z 11413	. . . . . . . . 9
56		gcdaddm 15246	. . . . . . . . 9 $/\$ $/\$
57	55, 56	mp3an1 1411	. . . . . . . 8 $/\$
58	7, 11, 57	syl2anc 693	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
59	8	nncnd 11036	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
60	9	nncnd 11036	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
61		pnncan 10322	. . . . . . . . . . 11 $/\$ $/\$
62	61	3anidm23 1385	. . . . . . . . . 10 $/\$
63		subcl 10280	. . . . . . . . . . . . 13 $/\$
64	63	mulm1d 10482	. . . . . . . . . . . 12 $/\$
65	64	oveq2d 6666	. . . . . . . . . . 11 $/\$
66		addcl 10018	. . . . . . . . . . . 12 $/\$
67	66, 63	negsubd 10398	. . . . . . . . . . 11 $/\$
68	65, 67	eqtrd 2656	. . . . . . . . . 10 $/\$
69		2times 11145	. . . . . . . . . . 11
70	69	adantl 482	. . . . . . . . . 10 $/\$
71	62, 68, 70	3eqtr4d 2666	. . . . . . . . 9 $/\$
72	71	oveq2d 6666	. . . . . . . 8 $/\$
73	59, 60, 72	syl2anc 693	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
74	58, 73	eqtrd 2656	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
75	9	nnzd 11481	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
76		zmulcl 11426	. . . . . . . . 9 $/\$
77	24, 75, 76	sylancr 695	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
78		gcddvds 15225	. . . . . . . 8 $/\$ $/\$
79	7, 77, 78	syl2anc 693	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
80	79	simprd 479	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
81	74, 80	eqbrtrd 4675	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
82		1z 11407	. . . . . . . . 9
83		gcdaddm 15246	. . . . . . . . 9 $/\$ $/\$
84	82, 83	mp3an1 1411	. . . . . . . 8 $/\$
85	7, 11, 84	syl2anc 693	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
86		ppncan 10323	. . . . . . . . . . 11 $/\$ $/\$
87	86	3anidm13 1384	. . . . . . . . . 10 $/\$
88	63	mulid2d 10058	. . . . . . . . . . 11 $/\$
89	88	oveq2d 6666	. . . . . . . . . 10 $/\$
90		2times 11145	. . . . . . . . . . 11
91	90	adantr 481	. . . . . . . . . 10 $/\$
92	87, 89, 91	3eqtr4d 2666	. . . . . . . . 9 $/\$
93	59, 60, 92	syl2anc 693	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
94	93	oveq2d 6666	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
95	85, 94	eqtrd 2656	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
96	8	nnzd 11481	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
97		zmulcl 11426	. . . . . . . . 9 $/\$
98	24, 96, 97	sylancr 695	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
99		gcddvds 15225	. . . . . . . 8 $/\$ $/\$
100	7, 98, 99	syl2anc 693	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
101	100	simprd 479	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
102	95, 101	eqbrtrd 4675	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
103		nnaddcl 11042	. . . . . . . . . . . . . 14 $/\$
104	103	nnne0d 11065	. . . . . . . . . . . . 13 $/\$
105	104	ancoms 469	. . . . . . . . . . . 12 $/\$
106	105	3adant1 1079	. . . . . . . . . . 11 $/\$ $/\$
107	106	3ad2ant1 1082	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
108	107	neneqd 2799	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
109	108	intnand 962	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
110		gcdn0cl 15224	. . . . . . . 8 $/\$ $/\$ $/\$
111	7, 11, 109, 110	syl21anc 1325	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
112	111	nnzd 11481	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
113		dvdsgcd 15261	. . . . . 6 $/\$ $/\$ $/\$
114	112, 77, 98, 113	syl3anc 1326	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
115	81, 102, 114	mp2and 715	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
116		2nn0 11309	. . . . . . 7
117		mulgcd 15265	. . . . . . 7 $/\$ $/\$
118	116, 117	mp3an1 1411	. . . . . 6 $/\$
119	75, 96, 118	syl2anc 693	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
120		pythagtriplem3 15523	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
121	120	oveq2d 6666	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
122		2t1e2 11176	. . . . . 6
123	121, 122	syl6eq 2672	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
124	119, 123	eqtrd 2656	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
125	115, 124	breqtrd 4679	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
126		dvdsprime 15400	. . . 4 $/\$ $\/$
127	48, 111, 126	sylancr 695	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
128	125, 127	mpbid 222	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
129		orel1 397	. 2 $\/$
130	54, 128, 129	sylc 65	1 $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $\/$ wo 383 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wne 2794 class class class wbr 4653 (class class class)co 6650

cc 9934

cc0 9936

c1 9937

caddc 9939

cmul 9941

cmin 10266

cneg 10267

cn 11020

c2 11070

cn0 11292

cz 11377

cexp 12860

cdvds 14983

cgcd 15216

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-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

This theorem is referenced by: pythagtriplem6 15526 pythagtriplem7 15527

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