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Theorem pythagtriplem4 13077

Description: Lemma for pythagtrip 13092. 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 9675	. . . . . . . . . . . . 13
3		nnz 9675	. . . . . . . . . . . . 13
4		zsubcl 9697	. . . . . . . . . . . . 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 9362	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
11	10	nnzd 9779	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
12		gcddvds 12766	. . . . . . . . . 10 $/\$ $/\$
13	7, 11, 12	syl2anc 415	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
14	13	simprd 114	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
15		breq1 4134	. . . . . . . . 9
16	15	biimpd 144	. . . . . . . 8
17	14, 16	mpan9 281	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
18		2z 9684	. . . . . . . 8
19		simpl13 1105	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
20	19	nnzd 9779	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
21		simpl12 1104	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
22	21	nnzd 9779	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
23	20, 22	zaddcld 9784	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
24	20, 22	zsubcld 9785	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25		dvdsmultr1 12624	. . . . . . . 8 $/\$ $/\$
26	18, 23, 24, 25	mp3an2i 1383	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27	17, 26	mpd 13	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28	19	nncnd 9328	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29	21	nncnd 9328	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30		subsq 11105	. . . . . . 7 $/\$
31	28, 29, 30	syl2anc 415	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32	27, 31	breqtrrd 4159	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33		simpl2 1032	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	33	oveq1d 6103	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		simpl11 1103	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36	35	nnsqcld 11154	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37	36	nncnd 9328	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38	21	nnsqcld 11154	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39	38	nncnd 9328	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	37, 39	pncand 8646	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	34, 40	eqtr3d 2273	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	32, 41	breqtrd 4157	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43		nnz 9675	. . . . . . . 8
44	43	3ad2ant1 1049	. . . . . . 7 $/\$ $/\$
45	44	3ad2ant1 1049	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
46	45	adantr 276	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
47		2prm 12931	. . . . . 6
48		2nn 9478	. . . . . 6
49		prmdvdsexp 12953	. . . . . 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 9688	. . . . . . . 8
55		gcdaddm 12787	. . . . . . . 8 $/\$ $/\$
56	54, 7, 11, 55	mp3an2i 1383	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
57	8	nncnd 9328	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
58	9	nncnd 9328	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
59		pnncan 8575	. . . . . . . . . . 11 $/\$ $/\$
60	59	3anidm23 1338	. . . . . . . . . 10 $/\$
61		subcl 8533	. . . . . . . . . . . . 13 $/\$
62	61	mulm1d 8745	. . . . . . . . . . . 12 $/\$
63	62	oveq2d 6104	. . . . . . . . . . 11 $/\$
64		addcl 8311	. . . . . . . . . . . 12 $/\$
65	64, 61	negsubd 8651	. . . . . . . . . . 11 $/\$
66	63, 65	eqtrd 2271	. . . . . . . . . 10 $/\$
67		2times 9442	. . . . . . . . . . 11
68	67	adantl 277	. . . . . . . . . 10 $/\$
69	60, 66, 68	3eqtr4d 2281	. . . . . . . . 9 $/\$
70	69	oveq2d 6104	. . . . . . . 8 $/\$
71	57, 58, 70	syl2anc 415	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
72	56, 71	eqtrd 2271	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
73	9	nnzd 9779	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
74		zmulcl 9710	. . . . . . . . 9 $/\$
75	18, 73, 74	sylancr 418	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
76		gcddvds 12766	. . . . . . . 8 $/\$ $/\$
77	7, 75, 76	syl2anc 415	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
78	77	simprd 114	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
79	72, 78	eqbrtrd 4153	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
80		1z 9682	. . . . . . . 8
81		gcdaddm 12787	. . . . . . . 8 $/\$ $/\$
82	80, 7, 11, 81	mp3an2i 1383	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
83		ppncan 8576	. . . . . . . . . . 11 $/\$ $/\$
84	83	3anidm13 1337	. . . . . . . . . 10 $/\$
85	61	mullidd 8351	. . . . . . . . . . 11 $/\$
86	85	oveq2d 6104	. . . . . . . . . 10 $/\$
87		2times 9442	. . . . . . . . . . 11
88	87	adantr 276	. . . . . . . . . 10 $/\$
89	84, 86, 88	3eqtr4d 2281	. . . . . . . . 9 $/\$
90	57, 58, 89	syl2anc 415	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
91	90	oveq2d 6104	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
92	82, 91	eqtrd 2271	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
93	8	nnzd 9779	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
94		zmulcl 9710	. . . . . . . . 9 $/\$
95	18, 93, 94	sylancr 418	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
96		gcddvds 12766	. . . . . . . 8 $/\$ $/\$
97	7, 95, 96	syl2anc 415	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
98	97	simprd 114	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
99	92, 98	eqbrtrd 4153	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
100		nnaddcl 9334	. . . . . . . . . . . . . 14 $/\$
101	100	nnne0d 9359	. . . . . . . . . . . . 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 12765	. . . . . . . 8 $/\$ $/\$ $/\$
108	7, 11, 106, 107	syl21anc 1277	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
109	108	nnzd 9779	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
110		dvdsgcd 12815	. . . . . 6 $/\$ $/\$ $/\$
111	109, 75, 95, 110	syl3anc 1278	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
112	79, 99, 111	mp2and 437	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
113		2nn0 9592	. . . . . 6
114		mulgcd 12819	. . . . . 6 $/\$ $/\$
115	113, 73, 93, 114	mp3an2i 1383	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
116		pythagtriplem3 13076	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
117	116	oveq2d 6104	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
118		2t1e2 9468	. . . . . 6
119	117, 118	eqtrdi 2287	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
120	115, 119	eqtrd 2271	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
121	112, 120	breqtrd 4157	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
122		dvdsprime 12926	. . . 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

This proof depends on syntax axioms:

wn 3

wi 4 $/\$ wa 104

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

wceq 1402

wcel 2209

wne 2420 class class class wbr 4131 (class class class)co 6088

cc 8184

cc0 8186

c1 8187

caddc 8189

cmul 8191

cmin 8505

cneg 8506

cn 9314

c2 9365

cn0 9575

cz 9656

cexp 10997

cdvds 12580

cgcd 12756

cprime 12911

This proof depends on 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 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8277 ax-resscn 8278 ax-1cn 8279 ax-1re 8280 ax-icn 8281 ax-addcl 8282 ax-addrcl 8283 ax-mulcl 8284 ax-mulrcl 8285 ax-addcom 8286 ax-mulcom 8287 ax-addass 8288 ax-mulass 8289 ax-distr 8290 ax-i2m1 8291 ax-0lt1 8292 ax-1rid 8293 ax-0id 8294 ax-rnegex 8295 ax-precex 8296 ax-cnre 8297 ax-pre-ltirr 8298 ax-pre-ltwlin 8299 ax-pre-lttrn 8300 ax-pre-apti 8301 ax-pre-ltadd 8302 ax-pre-mulgt0 8303 ax-pre-mulext 8304 ax-arch 8305 ax-caucvg 8306

This proof 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 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-po 4443 df-iso 4444 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 df-recs 6580 df-frec 6666 df-1o 6691 df-2o 6692 df-er 6811 df-en 7028 df-sup 7331 df-pnf 8369 df-mnf 8370 df-xr 8371 df-ltxr 8372 df-le 8373 df-sub 8507 df-neg 8508 df-reap 8912 df-ap 8919 df-div 9012 df-inn 9315 df-2 9373 df-3 9374 df-4 9375 df-n0 9576 df-z 9657 df-uz 9939 df-q 10037 df-rp 10073 df-fz 10430 df-fzo 10568 df-fl 10723 df-mod 10782 df-seqfrec 10907 df-exp 10998 df-cj 11630 df-re 11631 df-im 11632 df-rsqrt 11787 df-abs 11788 df-dvds 12581 df-gcd 12757 df-prm 12912

This theorem is used by: pythagtriplem6 13079 pythagtriplem7 13080

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