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Theorem pythagtriplem19 12236

Description: Lemma for pythagtrip 12237. Introduce

and remove the relative primality requirement. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

**Assertion**
Ref	Expression
pythagtriplem19	$/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Distinct variable groups:

**Proof of Theorem pythagtriplem19**
Step	Hyp	Ref	Expression
1		gcdnncl 11922	. . . . 5 $/\$
2	1	3adant3 1012	. . . 4 $/\$ $/\$
3	2	3ad2ant1 1013	. . 3 $/\$ $/\$ $/\$ $/\$
4		nnz 9231	. . . . . . . . . . 11
5		nnz 9231	. . . . . . . . . . 11
6		gcddvds 11918	. . . . . . . . . . 11 $/\$ $/\$
7	4, 5, 6	syl2an 287	. . . . . . . . . 10 $/\$ $/\$
8	7	3adant3 1012	. . . . . . . . 9 $/\$ $/\$ $/\$
9	8	simpld 111	. . . . . . . 8 $/\$ $/\$
10	2	nnzd 9333	. . . . . . . . 9 $/\$ $/\$
11	2	nnne0d 8923	. . . . . . . . 9 $/\$ $/\$
12	4	3ad2ant1 1013	. . . . . . . . 9 $/\$ $/\$
13		dvdsval2 11752	. . . . . . . . 9 $/\$ $/\$
14	10, 11, 12, 13	syl3anc 1233	. . . . . . . 8 $/\$ $/\$
15	9, 14	mpbid 146	. . . . . . 7 $/\$ $/\$
16		nnre 8885	. . . . . . . . 9
17	16	3ad2ant1 1013	. . . . . . . 8 $/\$ $/\$
18	2	nnred 8891	. . . . . . . 8 $/\$ $/\$
19		nngt0 8903	. . . . . . . . 9
20	19	3ad2ant1 1013	. . . . . . . 8 $/\$ $/\$
21	2	nngt0d 8922	. . . . . . . 8 $/\$ $/\$
22	17, 18, 20, 21	divgt0d 8851	. . . . . . 7 $/\$ $/\$
23		elnnz 9222	. . . . . . 7 $/\$
24	15, 22, 23	sylanbrc 415	. . . . . 6 $/\$ $/\$
25	24	3ad2ant1 1013	. . . . 5 $/\$ $/\$ $/\$ $/\$
26	8	simprd 113	. . . . . . . 8 $/\$ $/\$
27	5	3ad2ant2 1014	. . . . . . . . 9 $/\$ $/\$
28		dvdsval2 11752	. . . . . . . . 9 $/\$ $/\$
29	10, 11, 27, 28	syl3anc 1233	. . . . . . . 8 $/\$ $/\$
30	26, 29	mpbid 146	. . . . . . 7 $/\$ $/\$
31		nnre 8885	. . . . . . . . 9
32	31	3ad2ant2 1014	. . . . . . . 8 $/\$ $/\$
33		nngt0 8903	. . . . . . . . 9
34	33	3ad2ant2 1014	. . . . . . . 8 $/\$ $/\$
35	32, 18, 34, 21	divgt0d 8851	. . . . . . 7 $/\$ $/\$
36		elnnz 9222	. . . . . . 7 $/\$
37	30, 35, 36	sylanbrc 415	. . . . . 6 $/\$ $/\$
38	37	3ad2ant1 1013	. . . . 5 $/\$ $/\$ $/\$ $/\$
39		dvdssq 11986	. . . . . . . . . . . . . . 15 $/\$
40	10, 12, 39	syl2anc 409	. . . . . . . . . . . . . 14 $/\$ $/\$
41		dvdssq 11986	. . . . . . . . . . . . . . 15 $/\$
42	10, 27, 41	syl2anc 409	. . . . . . . . . . . . . 14 $/\$ $/\$
43	40, 42	anbi12d 470	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
44	8, 43	mpbid 146	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
45	2	nnsqcld 10630	. . . . . . . . . . . . . 14 $/\$ $/\$
46	45	nnzd 9333	. . . . . . . . . . . . 13 $/\$ $/\$
47		nnsqcl 10545	. . . . . . . . . . . . . . 15
48	47	3ad2ant1 1013	. . . . . . . . . . . . . 14 $/\$ $/\$
49	48	nnzd 9333	. . . . . . . . . . . . 13 $/\$ $/\$
50		nnsqcl 10545	. . . . . . . . . . . . . . 15
51	50	3ad2ant2 1014	. . . . . . . . . . . . . 14 $/\$ $/\$
52	51	nnzd 9333	. . . . . . . . . . . . 13 $/\$ $/\$
53		dvds2add 11787	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
54	46, 49, 52, 53	syl3anc 1233	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
55	44, 54	mpd 13	. . . . . . . . . . 11 $/\$ $/\$
56	55	adantr 274	. . . . . . . . . 10 $/\$ $/\$ $/\$
57		simpr 109	. . . . . . . . . 10 $/\$ $/\$ $/\$
58	56, 57	breqtrd 4015	. . . . . . . . 9 $/\$ $/\$ $/\$
59		nnz 9231	. . . . . . . . . . . 12
60	59	3ad2ant3 1015	. . . . . . . . . . 11 $/\$ $/\$
61		dvdssq 11986	. . . . . . . . . . 11 $/\$
62	10, 60, 61	syl2anc 409	. . . . . . . . . 10 $/\$ $/\$
63	62	adantr 274	. . . . . . . . 9 $/\$ $/\$ $/\$
64	58, 63	mpbird 166	. . . . . . . 8 $/\$ $/\$ $/\$
65		dvdsval2 11752	. . . . . . . . . 10 $/\$ $/\$
66	10, 11, 60, 65	syl3anc 1233	. . . . . . . . 9 $/\$ $/\$
67	66	adantr 274	. . . . . . . 8 $/\$ $/\$ $/\$
68	64, 67	mpbid 146	. . . . . . 7 $/\$ $/\$ $/\$
69		nnre 8885	. . . . . . . . . 10
70	69	3ad2ant3 1015	. . . . . . . . 9 $/\$ $/\$
71		nngt0 8903	. . . . . . . . . 10
72	71	3ad2ant3 1015	. . . . . . . . 9 $/\$ $/\$
73	70, 18, 72, 21	divgt0d 8851	. . . . . . . 8 $/\$ $/\$
74	73	adantr 274	. . . . . . 7 $/\$ $/\$ $/\$
75		elnnz 9222	. . . . . . 7 $/\$
76	68, 74, 75	sylanbrc 415	. . . . . 6 $/\$ $/\$ $/\$
77	76	3adant3 1012	. . . . 5 $/\$ $/\$ $/\$ $/\$
78	48	nncnd 8892	. . . . . . . 8 $/\$ $/\$
79	51	nncnd 8892	. . . . . . . 8 $/\$ $/\$
80	45	nncnd 8892	. . . . . . . 8 $/\$ $/\$
81	45	nnap0d 8924	. . . . . . . 8 $/\$ $/\$ #
82	78, 79, 80, 81	divdirapd 8746	. . . . . . 7 $/\$ $/\$
83	82	3ad2ant1 1013	. . . . . 6 $/\$ $/\$ $/\$ $/\$
84		nncn 8886	. . . . . . . . . 10
85	84	3ad2ant3 1015	. . . . . . . . 9 $/\$ $/\$
86	2	nncnd 8892	. . . . . . . . 9 $/\$ $/\$
87	2	nnap0d 8924	. . . . . . . . 9 $/\$ $/\$ #
88	85, 86, 87	sqdivapd 10622	. . . . . . . 8 $/\$ $/\$
89	88	3ad2ant1 1013	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
90		oveq1 5860	. . . . . . . 8
91	90	3ad2ant2 1014	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
92	89, 91	eqtr4d 2206	. . . . . 6 $/\$ $/\$ $/\$ $/\$
93		nncn 8886	. . . . . . . . . 10
94	93	3ad2ant1 1013	. . . . . . . . 9 $/\$ $/\$
95	94, 86, 87	sqdivapd 10622	. . . . . . . 8 $/\$ $/\$
96		nncn 8886	. . . . . . . . . 10
97	96	3ad2ant2 1014	. . . . . . . . 9 $/\$ $/\$
98	97, 86, 87	sqdivapd 10622	. . . . . . . 8 $/\$ $/\$
99	95, 98	oveq12d 5871	. . . . . . 7 $/\$ $/\$
100	99	3ad2ant1 1013	. . . . . 6 $/\$ $/\$ $/\$ $/\$
101	83, 92, 100	3eqtr4rd 2214	. . . . 5 $/\$ $/\$ $/\$ $/\$
102		gcddiv 11974	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
103	12, 27, 2, 8, 102	syl31anc 1236	. . . . . . 7 $/\$ $/\$
104	86, 87	dividapd 8703	. . . . . . 7 $/\$ $/\$
105	103, 104	eqtr3d 2205	. . . . . 6 $/\$ $/\$
106	105	3ad2ant1 1013	. . . . 5 $/\$ $/\$ $/\$ $/\$
107		simp3 994	. . . . 5 $/\$ $/\$ $/\$ $/\$
108		pythagtriplem18 12235	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
109	25, 38, 77, 101, 106, 107, 108	syl312anc 1254	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
110	94, 86, 87	divcanap2d 8709	. . . . . . . . . 10 $/\$ $/\$
111	110	eqcomd 2176	. . . . . . . . 9 $/\$ $/\$
112	97, 86, 87	divcanap2d 8709	. . . . . . . . . 10 $/\$ $/\$
113	112	eqcomd 2176	. . . . . . . . 9 $/\$ $/\$
114	85, 86, 87	divcanap2d 8709	. . . . . . . . . 10 $/\$ $/\$
115	114	eqcomd 2176	. . . . . . . . 9 $/\$ $/\$
116	111, 113, 115	3jca 1172	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
117	116	3ad2ant1 1013	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
118		oveq2 5861	. . . . . . . . . 10
119	118	eqeq2d 2182	. . . . . . . . 9
120	119	3ad2ant1 1013	. . . . . . . 8 $/\$ $/\$
121		oveq2 5861	. . . . . . . . . 10
122	121	eqeq2d 2182	. . . . . . . . 9
123	122	3ad2ant2 1014	. . . . . . . 8 $/\$ $/\$
124		oveq2 5861	. . . . . . . . . 10
125	124	eqeq2d 2182	. . . . . . . . 9
126	125	3ad2ant3 1015	. . . . . . . 8 $/\$ $/\$
127	120, 123, 126	3anbi123d 1307	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
128	117, 127	syl5ibcom 154	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
129	128	reximdv 2571	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
130	129	reximdv 2571	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
131	109, 130	mpd 13	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
132		oveq1 5860	. . . . . . 7
133	132	eqeq2d 2182	. . . . . 6
134		oveq1 5860	. . . . . . 7
135	134	eqeq2d 2182	. . . . . 6
136		oveq1 5860	. . . . . . 7
137	136	eqeq2d 2182	. . . . . 6
138	133, 135, 137	3anbi123d 1307	. . . . 5 $/\$ $/\$ $/\$ $/\$
139	138	2rexbidv 2495	. . . 4 $/\$ $/\$ $/\$ $/\$
140	139	rspcev 2834	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
141	3, 131, 140	syl2anc 409	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
142		rexcom 2634	. . 3 $/\$ $/\$ $/\$ $/\$
143		rexcom 2634	. . . 4 $/\$ $/\$ $/\$ $/\$
144	143	rexbii 2477	. . 3 $/\$ $/\$ $/\$ $/\$
145	142, 144	bitri 183	. 2 $/\$ $/\$ $/\$ $/\$
146	141, 145	sylib 121	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $/\$ w3a 973

wceq 1348

wcel 2141

wne 2340

wrex 2449 class class class wbr 3989 (class class class)co 5853

cc 7772

cr 7773

cc0 7774

c1 7775

caddc 7777

cmul 7779

clt 7954

cmin 8090

cdiv 8589

cn 8878

c2 8929

cz 9212

cexp 10475

cdvds 11749

cgcd 11897

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-xor 1371 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: pythagtrip 12237

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