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Theorem pythagtriplem4 13049

Description: Lemma for pythagtrip 13064. 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 9665	. . . . . . . . . . . . 13
3		nnz 9665	. . . . . . . . . . . . 13
4		zsubcl 9687	. . . . . . . . . . . . 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 9353	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
11	10	nnzd 9769	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
12		gcddvds 12742	. . . . . . . . . 10 $/\$ $/\$
13	7, 11, 12	syl2anc 415	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
14	13	simprd 114	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
15		breq1 4133	. . . . . . . . 9
16	15	biimpd 144	. . . . . . . 8
17	14, 16	mpan9 281	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
18		2z 9674	. . . . . . . 8
19		simpl13 1105	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
20	19	nnzd 9769	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
21		simpl12 1104	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
22	21	nnzd 9769	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
23	20, 22	zaddcld 9774	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
24	20, 22	zsubcld 9775	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25		dvdsmultr1 12600	. . . . . . . 8 $/\$ $/\$
26	18, 23, 24, 25	mp3an2i 1383	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27	17, 26	mpd 13	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28	19	nncnd 9319	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29	21	nncnd 9319	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30		subsq 11085	. . . . . . 7 $/\$
31	28, 29, 30	syl2anc 415	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32	27, 31	breqtrrd 4158	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33		simpl2 1032	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	33	oveq1d 6100	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		simpl11 1103	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36	35	nnsqcld 11134	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37	36	nncnd 9319	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38	21	nnsqcld 11134	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39	38	nncnd 9319	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	37, 39	pncand 8638	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	34, 40	eqtr3d 2273	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	32, 41	breqtrd 4156	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43		nnz 9665	. . . . . . . 8
44	43	3ad2ant1 1049	. . . . . . 7 $/\$ $/\$
45	44	3ad2ant1 1049	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
46	45	adantr 276	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
47		2prm 12907	. . . . . 6
48		2nn 9468	. . . . . 6
49		prmdvdsexp 12928	. . . . . 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 9678	. . . . . . . 8
55		gcdaddm 12763	. . . . . . . 8 $/\$ $/\$
56	54, 7, 11, 55	mp3an2i 1383	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
57	8	nncnd 9319	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
58	9	nncnd 9319	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
59		pnncan 8567	. . . . . . . . . . 11 $/\$ $/\$
60	59	3anidm23 1338	. . . . . . . . . 10 $/\$
61		subcl 8525	. . . . . . . . . . . . 13 $/\$
62	61	mulm1d 8737	. . . . . . . . . . . 12 $/\$
63	62	oveq2d 6101	. . . . . . . . . . 11 $/\$
64		addcl 8304	. . . . . . . . . . . 12 $/\$
65	64, 61	negsubd 8643	. . . . . . . . . . 11 $/\$
66	63, 65	eqtrd 2271	. . . . . . . . . 10 $/\$
67		2times 9433	. . . . . . . . . . 11
68	67	adantl 277	. . . . . . . . . 10 $/\$
69	60, 66, 68	3eqtr4d 2281	. . . . . . . . 9 $/\$
70	69	oveq2d 6101	. . . . . . . 8 $/\$
71	57, 58, 70	syl2anc 415	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
72	56, 71	eqtrd 2271	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
73	9	nnzd 9769	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
74		zmulcl 9700	. . . . . . . . 9 $/\$
75	18, 73, 74	sylancr 418	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
76		gcddvds 12742	. . . . . . . 8 $/\$ $/\$
77	7, 75, 76	syl2anc 415	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
78	77	simprd 114	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
79	72, 78	eqbrtrd 4152	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
80		1z 9672	. . . . . . . 8
81		gcdaddm 12763	. . . . . . . 8 $/\$ $/\$
82	80, 7, 11, 81	mp3an2i 1383	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
83		ppncan 8568	. . . . . . . . . . 11 $/\$ $/\$
84	83	3anidm13 1337	. . . . . . . . . 10 $/\$
85	61	mullidd 8344	. . . . . . . . . . 11 $/\$
86	85	oveq2d 6101	. . . . . . . . . 10 $/\$
87		2times 9433	. . . . . . . . . . 11
88	87	adantr 276	. . . . . . . . . 10 $/\$
89	84, 86, 88	3eqtr4d 2281	. . . . . . . . 9 $/\$
90	57, 58, 89	syl2anc 415	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
91	90	oveq2d 6101	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
92	82, 91	eqtrd 2271	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
93	8	nnzd 9769	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
94		zmulcl 9700	. . . . . . . . 9 $/\$
95	18, 93, 94	sylancr 418	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
96		gcddvds 12742	. . . . . . . 8 $/\$ $/\$
97	7, 95, 96	syl2anc 415	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
98	97	simprd 114	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
99	92, 98	eqbrtrd 4152	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
100		nnaddcl 9325	. . . . . . . . . . . . . 14 $/\$
101	100	nnne0d 9350	. . . . . . . . . . . . 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 12741	. . . . . . . 8 $/\$ $/\$ $/\$
108	7, 11, 106, 107	syl21anc 1277	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
109	108	nnzd 9769	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
110		dvdsgcd 12791	. . . . . 6 $/\$ $/\$ $/\$
111	109, 75, 95, 110	syl3anc 1278	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
112	79, 99, 111	mp2and 437	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
113		2nn0 9582	. . . . . 6
114		mulgcd 12795	. . . . . 6 $/\$ $/\$
115	113, 73, 93, 114	mp3an2i 1383	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
116		pythagtriplem3 13048	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
117	116	oveq2d 6101	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
118		2t1e2 9459	. . . . . 6
119	117, 118	eqtrdi 2287	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
120	115, 119	eqtrd 2271	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
121	112, 120	breqtrd 4156	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
122		dvdsprime 12902	. . . 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 4130 (class class class)co 6085

cc 8177

cc0 8179

c1 8180

caddc 8182

cmul 8184

cmin 8497

cneg 8498

cn 9305

c2 9356

cn0 9565

cz 9646

cexp 10977

cdvds 12556

cgcd 12732

cprime 12887

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 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 ax-pre-mulext 8297 ax-arch 8298 ax-caucvg 8299

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 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-frec 6662 df-1o 6687 df-2o 6688 df-er 6807 df-en 7023 df-sup 7324 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-reap 8904 df-ap 8911 df-div 9004 df-inn 9306 df-2 9364 df-3 9365 df-4 9366 df-n0 9566 df-z 9647 df-uz 9924 df-q 10022 df-rp 10057 df-fz 10414 df-fzo 10552 df-fl 10707 df-mod 10762 df-seqfrec 10887 df-exp 10978 df-cj 11609 df-re 11610 df-im 11611 df-rsqrt 11766 df-abs 11767 df-dvds 12557 df-gcd 12733 df-prm 12888

This theorem is used by: pythagtriplem6 13051 pythagtriplem7 13052

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