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Theorem pythagtrip 12977

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 12836	. . . . . . 7 $/\$ $\/$
2	1	3adant3 1044	. . . . . 6 $/\$ $/\$ $\/$
3	2	adantr 276	. . . . 5 $/\$ $/\$ $/\$ $\/$
4		pythagtriplem19 12976	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
5	4	3expia 1232	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
6		simp12 1055	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
7		simp11 1054	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
8		simp13 1056	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
9		nnsqcl 10970	. . . . . . . . . . . . . 14
10	9	nncnd 9250	. . . . . . . . . . . . 13
11	10	3ad2ant1 1045	. . . . . . . . . . . 12 $/\$ $/\$
12		nnsqcl 10970	. . . . . . . . . . . . . 14
13	12	nncnd 9250	. . . . . . . . . . . . 13
14	13	3ad2ant2 1046	. . . . . . . . . . . 12 $/\$ $/\$
15	11, 14	addcomd 8423	. . . . . . . . . . 11 $/\$ $/\$
16	15	eqeq1d 2241	. . . . . . . . . 10 $/\$ $/\$
17	16	biimpa 296	. . . . . . . . 9 $/\$ $/\$ $/\$
18	17	3adant3 1044	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
19		nnz 9595	. . . . . . . . . . . . . 14
20	19	3ad2ant1 1045	. . . . . . . . . . . . 13 $/\$ $/\$
21		nnz 9595	. . . . . . . . . . . . . . 15
22	21	3ad2ant2 1046	. . . . . . . . . . . . . 14 $/\$ $/\$
23	22	adantr 276	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
24		gcdcom 12665	. . . . . . . . . . . . 13 $/\$
25	20, 23, 24	syl2an2r 599	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
26	25	oveq2d 6065	. . . . . . . . . . 11 $/\$ $/\$ $/\$
27	26	breq2d 4120	. . . . . . . . . 10 $/\$ $/\$ $/\$
28	27	notbid 673	. . . . . . . . 9 $/\$ $/\$ $/\$
29	28	biimp3a 1382	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
30		pythagtriplem19 12976	. . . . . . . 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 9595	. . . . . . . . . . . . 13
38	37	adantl 277	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
39		simplrr 538	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
40	39	nnzd 9698	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
41		zsqcl 10971	. . . . . . . . . . . . . 14
42	40, 41	syl 14	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
43		simplrl 537	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
44	43	nnzd 9698	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
45		zsqcl 10971	. . . . . . . . . . . . . 14
46	44, 45	syl 14	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
47	42, 46	zsubcld 9704	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
48	38, 47	zmulcld 9705	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
49		2z 9604	. . . . . . . . . . . . . 14
50	49	a1i 9	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
51	40, 44	zmulcld 9705	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
52	50, 51	zmulcld 9705	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
53	38, 52	zmulcld 9705	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
54		preq12bg 3876	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
55	35, 36, 48, 53, 54	syl22anc 1275	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
56	55	anbi1d 465	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
57	56	rexbidva 2539	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
58	57	2rexbidva 2565	. . . . . . 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 2549	. . . . . . . . 9 $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
67	66	2rexbii 2551	. . . . . . . 8 $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
68		r19.43 2701	. . . . . . . . . 10 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
69	68	2rexbii 2551	. . . . . . . . 9 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
70		r19.43 2701	. . . . . . . . . . 11 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
71	70	rexbii 2549	. . . . . . . . . 10 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
72		r19.43 2701	. . . . . . . . . 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 12960	. . 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 2203

wrex 2521

cpr 3689 class class class wbr 4108 (class class class)co 6049

cc 8124

caddc 8129

cmul 8131

cmin 8443

cdiv 8945

cn 9236

c2 9287

cz 9576

cexp 10899

cdvds 12469

cgcd 12645

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 2205 ax-14 2206 ax-ext 2214 ax-coll 4224 ax-sep 4227 ax-nul 4235 ax-pow 4286 ax-pr 4321 ax-un 4553 ax-setind 4658 ax-iinf 4709 ax-cnex 8217 ax-resscn 8218 ax-1cn 8219 ax-1re 8220 ax-icn 8221 ax-addcl 8222 ax-addrcl 8223 ax-mulcl 8224 ax-mulrcl 8225 ax-addcom 8226 ax-mulcom 8227 ax-addass 8228 ax-mulass 8229 ax-distr 8230 ax-i2m1 8231 ax-0lt1 8232 ax-1rid 8233 ax-0id 8234 ax-rnegex 8235 ax-precex 8236 ax-cnre 8237 ax-pre-ltirr 8238 ax-pre-ltwlin 8239 ax-pre-lttrn 8240 ax-pre-apti 8241 ax-pre-ltadd 8242 ax-pre-mulgt0 8243 ax-pre-mulext 8244 ax-arch 8245 ax-caucvg 8246

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 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2814 df-sbc 3042 df-csb 3138 df-dif 3212 df-un 3214 df-in 3216 df-ss 3223 df-nul 3508 df-if 3620 df-pw 3670 df-sn 3694 df-pr 3695 df-op 3697 df-uni 3914 df-int 3949 df-iun 3992 df-br 4109 df-opab 4171 df-mpt 4172 df-tr 4208 df-id 4413 df-po 4416 df-iso 4417 df-iord 4486 df-on 4488 df-ilim 4489 df-suc 4491 df-iom 4712 df-xp 4754 df-rel 4755 df-cnv 4756 df-co 4757 df-dm 4758 df-rn 4759 df-res 4760 df-ima 4761 df-iota 5311 df-fun 5353 df-fn 5354 df-f 5355 df-f1 5356 df-fo 5357 df-f1o 5358 df-fv 5359 df-riota 6002 df-ov 6052 df-oprab 6053 df-mpo 6054 df-1st 6333 df-2nd 6334 df-recs 6535 df-frec 6621 df-1o 6646 df-2o 6647 df-er 6766 df-en 6975 df-sup 7274 df-pnf 8309 df-mnf 8310 df-xr 8311 df-ltxr 8312 df-le 8313 df-sub 8445 df-neg 8446 df-reap 8848 df-ap 8855 df-div 8946 df-inn 9237 df-2 9295 df-3 9296 df-4 9297 df-n0 9496 df-z 9577 df-uz 9853 df-q 9951 df-rp 9986 df-fz 10342 df-fzo 10476 df-fl 10629 df-mod 10684 df-seqfrec 10809 df-exp 10900 df-cj 11523 df-re 11524 df-im 11525 df-rsqrt 11679 df-abs 11680 df-dvds 12470 df-gcd 12646 df-prm 12801

This theorem is referenced by: (None)

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