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Theorem pythagtriplem4 12222

Description: Lemma for pythagtrip 12237. 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 1021	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
2		nnz 9231	. . . . . . . . . . . . 13
3		nnz 9231	. . . . . . . . . . . . 13
4		zsubcl 9253	. . . . . . . . . . . . 13 $/\$
5	2, 3, 4	syl2anr 288	. . . . . . . . . . . 12 $/\$
6	5	3adant1 1010	. . . . . . . . . . 11 $/\$ $/\$
7	6	3ad2ant1 1013	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
8		simp13 1024	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
9		simp12 1023	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
10	8, 9	nnaddcld 8926	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
11	10	nnzd 9333	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
12		gcddvds 11918	. . . . . . . . . 10 $/\$ $/\$
13	7, 11, 12	syl2anc 409	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
14	13	simprd 113	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
15		breq1 3992	. . . . . . . . 9
16	15	biimpd 143	. . . . . . . 8
17	14, 16	mpan9 279	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
18		2z 9240	. . . . . . . 8
19		simpl13 1069	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
20	19	nnzd 9333	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
21		simpl12 1068	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
22	21	nnzd 9333	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
23	20, 22	zaddcld 9338	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
24	20, 22	zsubcld 9339	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25		dvdsmultr1 11793	. . . . . . . 8 $/\$ $/\$
26	18, 23, 24, 25	mp3an2i 1337	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27	17, 26	mpd 13	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28	19	nncnd 8892	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29	21	nncnd 8892	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30		subsq 10582	. . . . . . 7 $/\$
31	28, 29, 30	syl2anc 409	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32	27, 31	breqtrrd 4017	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33		simpl2 996	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	33	oveq1d 5868	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		simpl11 1067	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36	35	nnsqcld 10630	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37	36	nncnd 8892	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38	21	nnsqcld 10630	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39	38	nncnd 8892	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	37, 39	pncand 8231	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	34, 40	eqtr3d 2205	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	32, 41	breqtrd 4015	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43		nnz 9231	. . . . . . . 8
44	43	3ad2ant1 1013	. . . . . . 7 $/\$ $/\$
45	44	3ad2ant1 1013	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
46	45	adantr 274	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
47		2prm 12081	. . . . . 6
48		2nn 9039	. . . . . 6
49		prmdvdsexp 12102	. . . . . 6 $/\$ $/\$
50	47, 48, 49	mp3an13 1323	. . . . 5
51	46, 50	syl 14	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52	42, 51	mpbid 146	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
53	1, 52	mtand 660	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
54		neg1z 9244	. . . . . . . 8
55		gcdaddm 11939	. . . . . . . 8 $/\$ $/\$
56	54, 7, 11, 55	mp3an2i 1337	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
57	8	nncnd 8892	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
58	9	nncnd 8892	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
59		pnncan 8160	. . . . . . . . . . 11 $/\$ $/\$
60	59	3anidm23 1292	. . . . . . . . . 10 $/\$
61		subcl 8118	. . . . . . . . . . . . 13 $/\$
62	61	mulm1d 8329	. . . . . . . . . . . 12 $/\$
63	62	oveq2d 5869	. . . . . . . . . . 11 $/\$
64		addcl 7899	. . . . . . . . . . . 12 $/\$
65	64, 61	negsubd 8236	. . . . . . . . . . 11 $/\$
66	63, 65	eqtrd 2203	. . . . . . . . . 10 $/\$
67		2times 9006	. . . . . . . . . . 11
68	67	adantl 275	. . . . . . . . . 10 $/\$
69	60, 66, 68	3eqtr4d 2213	. . . . . . . . 9 $/\$
70	69	oveq2d 5869	. . . . . . . 8 $/\$
71	57, 58, 70	syl2anc 409	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
72	56, 71	eqtrd 2203	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
73	9	nnzd 9333	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
74		zmulcl 9265	. . . . . . . . 9 $/\$
75	18, 73, 74	sylancr 412	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
76		gcddvds 11918	. . . . . . . 8 $/\$ $/\$
77	7, 75, 76	syl2anc 409	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
78	77	simprd 113	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
79	72, 78	eqbrtrd 4011	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
80		1z 9238	. . . . . . . 8
81		gcdaddm 11939	. . . . . . . 8 $/\$ $/\$
82	80, 7, 11, 81	mp3an2i 1337	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
83		ppncan 8161	. . . . . . . . . . 11 $/\$ $/\$
84	83	3anidm13 1291	. . . . . . . . . 10 $/\$
85	61	mulid2d 7938	. . . . . . . . . . 11 $/\$
86	85	oveq2d 5869	. . . . . . . . . 10 $/\$
87		2times 9006	. . . . . . . . . . 11
88	87	adantr 274	. . . . . . . . . 10 $/\$
89	84, 86, 88	3eqtr4d 2213	. . . . . . . . 9 $/\$
90	57, 58, 89	syl2anc 409	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
91	90	oveq2d 5869	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
92	82, 91	eqtrd 2203	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
93	8	nnzd 9333	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
94		zmulcl 9265	. . . . . . . . 9 $/\$
95	18, 93, 94	sylancr 412	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
96		gcddvds 11918	. . . . . . . 8 $/\$ $/\$
97	7, 95, 96	syl2anc 409	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
98	97	simprd 113	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
99	92, 98	eqbrtrd 4011	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
100		nnaddcl 8898	. . . . . . . . . . . . . 14 $/\$
101	100	nnne0d 8923	. . . . . . . . . . . . 13 $/\$
102	101	ancoms 266	. . . . . . . . . . . 12 $/\$
103	102	3adant1 1010	. . . . . . . . . . 11 $/\$ $/\$
104	103	3ad2ant1 1013	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
105	104	neneqd 2361	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
106	105	intnand 926	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
107		gcdn0cl 11917	. . . . . . . 8 $/\$ $/\$ $/\$
108	7, 11, 106, 107	syl21anc 1232	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
109	108	nnzd 9333	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
110		dvdsgcd 11967	. . . . . 6 $/\$ $/\$ $/\$
111	109, 75, 95, 110	syl3anc 1233	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
112	79, 99, 111	mp2and 431	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
113		2nn0 9152	. . . . . 6
114		mulgcd 11971	. . . . . 6 $/\$ $/\$
115	113, 73, 93, 114	mp3an2i 1337	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
116		pythagtriplem3 12221	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
117	116	oveq2d 5869	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
118		2t1e2 9031	. . . . . 6
119	117, 118	eqtrdi 2219	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
120	115, 119	eqtrd 2203	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
121	112, 120	breqtrd 4015	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
122		dvdsprime 12076	. . . 4 $/\$ $\/$
123	47, 108, 122	sylancr 412	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
124	121, 123	mpbid 146	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
125		orel1 720	. 2 $\/$
126	53, 124, 125	sylc 62	1 $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $\/$ wo 703 $/\$ w3a 973

wceq 1348

wcel 2141

wne 2340 class class class wbr 3989 (class class class)co 5853

cc 7772

cc0 7774

c1 7775

caddc 7777

cmul 7779

cmin 8090

cneg 8091

cn 8878

c2 8929

cn0 9135

cz 9212

cexp 10475

cdvds 11749

cgcd 11897

cprime 12061

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894

This theorem depends on definitions: df-bi 116 df-stab 826 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-1o 6395 df-2o 6396 df-er 6513 df-en 6719 df-sup 6961 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-fz 9966 df-fzo 10099 df-fl 10226 df-mod 10279 df-seqfrec 10402 df-exp 10476 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-dvds 11750 df-gcd 11898 df-prm 12062

This theorem is referenced by: pythagtriplem6 12224 pythagtriplem7 12225

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