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Theorem pythagtrip 13006

Description: Parameterize the Pythagorean triples. If

, and

are naturals, then they obey the Pythagorean triple formula iff they are parameterized by three naturals. This proof follows the Isabelle proof at http://afp.sourceforge.net/entries/Fermat3_4.shtml. This is Metamath 100 proof #23. (Contributed by Scott Fenton, 19-Apr-2014.)

**Assertion**
Ref	Expression
pythagtrip	$/\$ $/\$ $/\$

Distinct variable groups:

**Proof of Theorem pythagtrip**
Step	Hyp	Ref	Expression
1		divgcdodd 12865	. . . . . . 7 $/\$ $\/$
2	1	3adant3 1044	. . . . . 6 $/\$ $/\$ $\/$
3	2	adantr 276	. . . . 5 $/\$ $/\$ $/\$ $\/$
4		pythagtriplem19 13005	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
5	4	3expia 1232	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
6		simp12 1055	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
7		simp11 1054	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
8		simp13 1056	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
9		nnsqcl 10995	. . . . . . . . . . . . . 14
10	9	nncnd 9268	. . . . . . . . . . . . 13
11	10	3ad2ant1 1045	. . . . . . . . . . . 12 $/\$ $/\$
12		nnsqcl 10995	. . . . . . . . . . . . . 14
13	12	nncnd 9268	. . . . . . . . . . . . 13
14	13	3ad2ant2 1046	. . . . . . . . . . . 12 $/\$ $/\$
15	11, 14	addcomd 8440	. . . . . . . . . . 11 $/\$ $/\$
16	15	eqeq1d 2243	. . . . . . . . . 10 $/\$ $/\$
17	16	biimpa 296	. . . . . . . . 9 $/\$ $/\$ $/\$
18	17	3adant3 1044	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
19		nnz 9613	. . . . . . . . . . . . . 14
20	19	3ad2ant1 1045	. . . . . . . . . . . . 13 $/\$ $/\$
21		nnz 9613	. . . . . . . . . . . . . . 15
22	21	3ad2ant2 1046	. . . . . . . . . . . . . 14 $/\$ $/\$
23	22	adantr 276	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
24		gcdcom 12694	. . . . . . . . . . . . 13 $/\$
25	20, 23, 24	syl2an2r 599	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
26	25	oveq2d 6074	. . . . . . . . . . 11 $/\$ $/\$ $/\$
27	26	breq2d 4126	. . . . . . . . . 10 $/\$ $/\$ $/\$
28	27	notbid 673	. . . . . . . . 9 $/\$ $/\$ $/\$
29	28	biimp3a 1382	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
30		pythagtriplem19 13005	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
31	6, 7, 8, 18, 29, 30	syl311anc 1288	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32	31	3expia 1232	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
33	5, 32	orim12d 794	. . . . 5 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $/\$
34	3, 33	mpd 13	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
35		simplll 535	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
36		simpllr 536	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
37		nnz 9613	. . . . . . . . . . . . 13
38	37	adantl 277	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
39		simplrr 538	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
40	39	nnzd 9717	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
41		zsqcl 10996	. . . . . . . . . . . . . 14
42	40, 41	syl 14	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
43		simplrl 537	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
44	43	nnzd 9717	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
45		zsqcl 10996	. . . . . . . . . . . . . 14
46	44, 45	syl 14	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
47	42, 46	zsubcld 9723	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
48	38, 47	zmulcld 9724	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
49		2z 9622	. . . . . . . . . . . . . 14
50	49	a1i 9	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
51	40, 44	zmulcld 9724	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
52	50, 51	zmulcld 9724	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
53	38, 52	zmulcld 9724	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
54		preq12bg 3882	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
55	35, 36, 48, 53, 54	syl22anc 1275	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
56	55	anbi1d 465	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
57	56	rexbidva 2541	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
58	57	2rexbidva 2567	. . . . . . 7 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
59		andir 827	. . . . . . . . . . 11 $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
60		df-3an 1007	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
61		df-3an 1007	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
62	60, 61	orbi12i 772	. . . . . . . . . . 11 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
63		3ancoma 1012	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
64	63	orbi2i 770	. . . . . . . . . . 11 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
65	59, 62, 64	3bitr2i 208	. . . . . . . . . 10 $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
66	65	rexbii 2551	. . . . . . . . 9 $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
67	66	2rexbii 2553	. . . . . . . 8 $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
68		r19.43 2703	. . . . . . . . . 10 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
69	68	2rexbii 2553	. . . . . . . . 9 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
70		r19.43 2703	. . . . . . . . . . 11 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
71	70	rexbii 2551	. . . . . . . . . 10 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
72		r19.43 2703	. . . . . . . . . 10 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
73	71, 72	bitri 184	. . . . . . . . 9 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
74	69, 73	bitri 184	. . . . . . . 8 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
75	67, 74	bitri 184	. . . . . . 7 $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
76	58, 75	bitrdi 196	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
77	76	3adant3 1044	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
78	77	adantr 276	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
79	34, 78	mpbird 167	. . 3 $/\$ $/\$ $/\$ $/\$
80	79	ex 115	. 2 $/\$ $/\$ $/\$
81		pythagtriplem2 12989	. . 3 $/\$ $/\$
82	81	3adant3 1044	. 2 $/\$ $/\$ $/\$
83	80, 82	impbid 129	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 716 $/\$ w3a 1005

wceq 1398

wcel 2205

wrex 2523

cpr 3695 class class class wbr 4114 (class class class)co 6058

cc 8141

caddc 8146

cmul 8148

cmin 8460

cdiv 8963

cn 9254

c2 9305

cz 9594

cexp 10924

cdvds 12498

cgcd 12674

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261 ax-arch 8262 ax-caucvg 8263

This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-xor 1421 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-po 4422 df-iso 4423 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-recs 6549 df-frec 6635 df-1o 6660 df-2o 6661 df-er 6780 df-en 6989 df-sup 7288 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8462 df-neg 8463 df-reap 8866 df-ap 8873 df-div 8964 df-inn 9255 df-2 9313 df-3 9314 df-4 9315 df-n0 9514 df-z 9595 df-uz 9872 df-q 9970 df-rp 10005 df-fz 10362 df-fzo 10499 df-fl 10654 df-mod 10709 df-seqfrec 10834 df-exp 10925 df-cj 11552 df-re 11553 df-im 11554 df-rsqrt 11708 df-abs 11709 df-dvds 12499 df-gcd 12675 df-prm 12830

This theorem is referenced by: (None)

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