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Theorem pythagtriplem4 13030

Description: Lemma for pythagtrip 13045. 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 1057	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
2		nnz 9646	. . . . . . . . . . . . 13
3		nnz 9646	. . . . . . . . . . . . 13
4		zsubcl 9668	. . . . . . . . . . . . 13 $/\$
5	2, 3, 4	syl2anr 290	. . . . . . . . . . . 12 $/\$
6	5	3adant1 1046	. . . . . . . . . . 11 $/\$ $/\$
7	6	3ad2ant1 1049	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
8		simp13 1060	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
9		simp12 1059	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
10	8, 9	nnaddcld 9335	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
11	10	nnzd 9750	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
12		gcddvds 12723	. . . . . . . . . 10 $/\$ $/\$
13	7, 11, 12	syl2anc 415	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
14	13	simprd 114	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
15		breq1 4131	. . . . . . . . 9
16	15	biimpd 144	. . . . . . . 8
17	14, 16	mpan9 281	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
18		2z 9655	. . . . . . . 8
19		simpl13 1105	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
20	19	nnzd 9750	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
21		simpl12 1104	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
22	21	nnzd 9750	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
23	20, 22	zaddcld 9755	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
24	20, 22	zsubcld 9756	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25		dvdsmultr1 12581	. . . . . . . 8 $/\$ $/\$
26	18, 23, 24, 25	mp3an2i 1383	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27	17, 26	mpd 13	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28	19	nncnd 9301	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29	21	nncnd 9301	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30		subsq 11066	. . . . . . 7 $/\$
31	28, 29, 30	syl2anc 415	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32	27, 31	breqtrrd 4156	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33		simpl2 1032	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	33	oveq1d 6094	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		simpl11 1103	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36	35	nnsqcld 11115	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37	36	nncnd 9301	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38	21	nnsqcld 11115	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39	38	nncnd 9301	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	37, 39	pncand 8632	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	34, 40	eqtr3d 2273	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	32, 41	breqtrd 4154	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43		nnz 9646	. . . . . . . 8
44	43	3ad2ant1 1049	. . . . . . 7 $/\$ $/\$
45	44	3ad2ant1 1049	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
46	45	adantr 276	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
47		2prm 12888	. . . . . 6
48		2nn 9449	. . . . . 6
49		prmdvdsexp 12909	. . . . . 6 $/\$ $/\$
50	47, 48, 49	mp3an13 1369	. . . . 5
51	46, 50	syl 14	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52	42, 51	mpbid 147	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
53	1, 52	mtand 675	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
54		neg1z 9659	. . . . . . . 8
55		gcdaddm 12744	. . . . . . . 8 $/\$ $/\$
56	54, 7, 11, 55	mp3an2i 1383	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
57	8	nncnd 9301	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
58	9	nncnd 9301	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
59		pnncan 8561	. . . . . . . . . . 11 $/\$ $/\$
60	59	3anidm23 1338	. . . . . . . . . 10 $/\$
61		subcl 8519	. . . . . . . . . . . . 13 $/\$
62	61	mulm1d 8731	. . . . . . . . . . . 12 $/\$
63	62	oveq2d 6095	. . . . . . . . . . 11 $/\$
64		addcl 8298	. . . . . . . . . . . 12 $/\$
65	64, 61	negsubd 8637	. . . . . . . . . . 11 $/\$
66	63, 65	eqtrd 2271	. . . . . . . . . 10 $/\$
67		2times 9415	. . . . . . . . . . 11
68	67	adantl 277	. . . . . . . . . 10 $/\$
69	60, 66, 68	3eqtr4d 2281	. . . . . . . . 9 $/\$
70	69	oveq2d 6095	. . . . . . . 8 $/\$
71	57, 58, 70	syl2anc 415	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
72	56, 71	eqtrd 2271	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
73	9	nnzd 9750	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
74		zmulcl 9681	. . . . . . . . 9 $/\$
75	18, 73, 74	sylancr 418	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
76		gcddvds 12723	. . . . . . . 8 $/\$ $/\$
77	7, 75, 76	syl2anc 415	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
78	77	simprd 114	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
79	72, 78	eqbrtrd 4150	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
80		1z 9653	. . . . . . . 8
81		gcdaddm 12744	. . . . . . . 8 $/\$ $/\$
82	80, 7, 11, 81	mp3an2i 1383	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
83		ppncan 8562	. . . . . . . . . . 11 $/\$ $/\$
84	83	3anidm13 1337	. . . . . . . . . 10 $/\$
85	61	mullidd 8338	. . . . . . . . . . 11 $/\$
86	85	oveq2d 6095	. . . . . . . . . 10 $/\$
87		2times 9415	. . . . . . . . . . 11
88	87	adantr 276	. . . . . . . . . 10 $/\$
89	84, 86, 88	3eqtr4d 2281	. . . . . . . . 9 $/\$
90	57, 58, 89	syl2anc 415	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
91	90	oveq2d 6095	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
92	82, 91	eqtrd 2271	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
93	8	nnzd 9750	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
94		zmulcl 9681	. . . . . . . . 9 $/\$
95	18, 93, 94	sylancr 418	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
96		gcddvds 12723	. . . . . . . 8 $/\$ $/\$
97	7, 95, 96	syl2anc 415	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
98	97	simprd 114	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
99	92, 98	eqbrtrd 4150	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
100		nnaddcl 9307	. . . . . . . . . . . . . 14 $/\$
101	100	nnne0d 9332	. . . . . . . . . . . . 13 $/\$
102	101	ancoms 268	. . . . . . . . . . . 12 $/\$
103	102	3adant1 1046	. . . . . . . . . . 11 $/\$ $/\$
104	103	3ad2ant1 1049	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
105	104	neneqd 2441	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
106	105	intnand 943	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
107		gcdn0cl 12722	. . . . . . . 8 $/\$ $/\$ $/\$
108	7, 11, 106, 107	syl21anc 1277	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
109	108	nnzd 9750	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
110		dvdsgcd 12772	. . . . . 6 $/\$ $/\$ $/\$
111	109, 75, 95, 110	syl3anc 1278	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
112	79, 99, 111	mp2and 437	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
113		2nn0 9563	. . . . . 6
114		mulgcd 12776	. . . . . 6 $/\$ $/\$
115	113, 73, 93, 114	mp3an2i 1383	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
116		pythagtriplem3 13029	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
117	116	oveq2d 6095	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
118		2t1e2 9441	. . . . . 6
119	117, 118	eqtrdi 2287	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
120	115, 119	eqtrd 2271	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
121	112, 120	breqtrd 4154	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
122		dvdsprime 12883	. . . 4 $/\$ $\/$
123	47, 108, 122	sylancr 418	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
124	121, 123	mpbid 147	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
125		orel1 737	. 2 $\/$
126	53, 124, 125	sylc 62	1 $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 720 $/\$ w3a 1009

wceq 1402

wcel 2209

wne 2420 class class class wbr 4128 (class class class)co 6079

cc 8171

cc0 8173

c1 8174

caddc 8176

cmul 8178

cmin 8491

cneg 8492

cn 9287

c2 9338

cn0 9546

cz 9627

cexp 10958

cdvds 12537

cgcd 12713

cprime 12868

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4244 ax-sep 4247 ax-nul 4257 ax-pow 4309 ax-pr 4344 ax-un 4576 ax-setind 4682 ax-iinf 4733 ax-cnex 8264 ax-resscn 8265 ax-1cn 8266 ax-1re 8267 ax-icn 8268 ax-addcl 8269 ax-addrcl 8270 ax-mulcl 8271 ax-mulrcl 8272 ax-addcom 8273 ax-mulcom 8274 ax-addass 8275 ax-mulass 8276 ax-distr 8277 ax-i2m1 8278 ax-0lt1 8279 ax-1rid 8280 ax-0id 8281 ax-rnegex 8282 ax-precex 8283 ax-cnre 8284 ax-pre-ltirr 8285 ax-pre-ltwlin 8286 ax-pre-lttrn 8287 ax-pre-apti 8288 ax-pre-ltadd 8289 ax-pre-mulgt0 8290 ax-pre-mulext 8291 ax-arch 8292 ax-caucvg 8293

This theorem depends on definitions: df-bi 117 df-stab 843 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3714 df-pr 3715 df-op 3717 df-uni 3934 df-int 3969 df-iun 4012 df-br 4129 df-opab 4191 df-mpt 4192 df-tr 4228 df-id 4436 df-po 4439 df-iso 4440 df-iord 4509 df-on 4511 df-ilim 4512 df-suc 4514 df-iom 4736 df-xp 4778 df-rel 4779 df-cnv 4780 df-co 4781 df-dm 4782 df-rn 4783 df-res 4784 df-ima 4785 df-iota 5335 df-fun 5377 df-fn 5378 df-f 5379 df-f1 5380 df-fo 5381 df-f1o 5382 df-fv 5383 df-riota 6032 df-ov 6082 df-oprab 6083 df-mpo 6084 df-1st 6368 df-2nd 6369 df-recs 6570 df-frec 6656 df-1o 6681 df-2o 6682 df-er 6801 df-en 7017 df-sup 7318 df-pnf 8356 df-mnf 8357 df-xr 8358 df-ltxr 8359 df-le 8360 df-sub 8493 df-neg 8494 df-reap 8897 df-ap 8904 df-div 8997 df-inn 9288 df-2 9346 df-3 9347 df-4 9348 df-n0 9547 df-z 9628 df-uz 9905 df-q 10003 df-rp 10038 df-fz 10395 df-fzo 10533 df-fl 10688 df-mod 10743 df-seqfrec 10868 df-exp 10959 df-cj 11590 df-re 11591 df-im 11592 df-rsqrt 11747 df-abs 11748 df-dvds 12538 df-gcd 12714 df-prm 12869

This theorem is referenced by: pythagtriplem6 13032 pythagtriplem7 13033

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