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Theorem pythagtrip 12452

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 12311	. . . . . . 7 $/\$ $\/$
2	1	3adant3 1019	. . . . . 6 $/\$ $/\$ $\/$
3	2	adantr 276	. . . . 5 $/\$ $/\$ $/\$ $\/$
4		pythagtriplem19 12451	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
5	4	3expia 1207	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
6		simp12 1030	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
7		simp11 1029	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
8		simp13 1031	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
9		nnsqcl 10701	. . . . . . . . . . . . . 14
10	9	nncnd 9004	. . . . . . . . . . . . 13
11	10	3ad2ant1 1020	. . . . . . . . . . . 12 $/\$ $/\$
12		nnsqcl 10701	. . . . . . . . . . . . . 14
13	12	nncnd 9004	. . . . . . . . . . . . 13
14	13	3ad2ant2 1021	. . . . . . . . . . . 12 $/\$ $/\$
15	11, 14	addcomd 8177	. . . . . . . . . . 11 $/\$ $/\$
16	15	eqeq1d 2205	. . . . . . . . . 10 $/\$ $/\$
17	16	biimpa 296	. . . . . . . . 9 $/\$ $/\$ $/\$
18	17	3adant3 1019	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
19		nnz 9345	. . . . . . . . . . . . . 14
20	19	3ad2ant1 1020	. . . . . . . . . . . . 13 $/\$ $/\$
21		nnz 9345	. . . . . . . . . . . . . . 15
22	21	3ad2ant2 1021	. . . . . . . . . . . . . 14 $/\$ $/\$
23	22	adantr 276	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
24		gcdcom 12140	. . . . . . . . . . . . 13 $/\$
25	20, 23, 24	syl2an2r 595	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
26	25	oveq2d 5938	. . . . . . . . . . 11 $/\$ $/\$ $/\$
27	26	breq2d 4045	. . . . . . . . . 10 $/\$ $/\$ $/\$
28	27	notbid 668	. . . . . . . . 9 $/\$ $/\$ $/\$
29	28	biimp3a 1356	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
30		pythagtriplem19 12451	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
31	6, 7, 8, 18, 29, 30	syl311anc 1263	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32	31	3expia 1207	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
33	5, 32	orim12d 787	. . . . 5 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $/\$
34	3, 33	mpd 13	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
35		simplll 533	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
36		simpllr 534	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
37		nnz 9345	. . . . . . . . . . . . 13
38	37	adantl 277	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
39		simplrr 536	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
40	39	nnzd 9447	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
41		zsqcl 10702	. . . . . . . . . . . . . 14
42	40, 41	syl 14	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
43		simplrl 535	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
44	43	nnzd 9447	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
45		zsqcl 10702	. . . . . . . . . . . . . 14
46	44, 45	syl 14	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
47	42, 46	zsubcld 9453	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
48	38, 47	zmulcld 9454	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
49		2z 9354	. . . . . . . . . . . . . 14
50	49	a1i 9	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
51	40, 44	zmulcld 9454	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
52	50, 51	zmulcld 9454	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
53	38, 52	zmulcld 9454	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
54		preq12bg 3803	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
55	35, 36, 48, 53, 54	syl22anc 1250	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
56	55	anbi1d 465	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
57	56	rexbidva 2494	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
58	57	2rexbidva 2520	. . . . . . 7 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
59		andir 820	. . . . . . . . . . 11 $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
60		df-3an 982	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
61		df-3an 982	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
62	60, 61	orbi12i 765	. . . . . . . . . . 11 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
63		3ancoma 987	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
64	63	orbi2i 763	. . . . . . . . . . 11 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
65	59, 62, 64	3bitr2i 208	. . . . . . . . . 10 $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
66	65	rexbii 2504	. . . . . . . . 9 $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
67	66	2rexbii 2506	. . . . . . . 8 $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
68		r19.43 2655	. . . . . . . . . 10 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
69	68	2rexbii 2506	. . . . . . . . 9 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
70		r19.43 2655	. . . . . . . . . . 11 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
71	70	rexbii 2504	. . . . . . . . . 10 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
72		r19.43 2655	. . . . . . . . . 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 1019	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
78	77	adantr 276	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
79	34, 78	mpbird 167	. . 3 $/\$ $/\$ $/\$ $/\$
80	79	ex 115	. 2 $/\$ $/\$ $/\$
81		pythagtriplem2 12435	. . 3 $/\$ $/\$
82	81	3adant3 1019	. 2 $/\$ $/\$ $/\$
83	80, 82	impbid 129	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 709 $/\$ w3a 980

wceq 1364

wcel 2167

wrex 2476

cpr 3623 class class class wbr 4033 (class class class)co 5922

cc 7877

caddc 7882

cmul 7884

cmin 8197

cdiv 8699

cn 8990

c2 9041

cz 9326

cexp 10630

cdvds 11952

cgcd 12120

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-mulrcl 7978 ax-addcom 7979 ax-mulcom 7980 ax-addass 7981 ax-mulass 7982 ax-distr 7983 ax-i2m1 7984 ax-0lt1 7985 ax-1rid 7986 ax-0id 7987 ax-rnegex 7988 ax-precex 7989 ax-cnre 7990 ax-pre-ltirr 7991 ax-pre-ltwlin 7992 ax-pre-lttrn 7993 ax-pre-apti 7994 ax-pre-ltadd 7995 ax-pre-mulgt0 7996 ax-pre-mulext 7997 ax-arch 7998 ax-caucvg 7999

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-xor 1387 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3451 df-if 3562 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-id 4328 df-po 4331 df-iso 4332 df-iord 4401 df-on 4403 df-ilim 4404 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-recs 6363 df-frec 6449 df-1o 6474 df-2o 6475 df-er 6592 df-en 6800 df-sup 7050 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066 df-le 8067 df-sub 8199 df-neg 8200 df-reap 8602 df-ap 8609 df-div 8700 df-inn 8991 df-2 9049 df-3 9050 df-4 9051 df-n0 9250 df-z 9327 df-uz 9602 df-q 9694 df-rp 9729 df-fz 10084 df-fzo 10218 df-fl 10360 df-mod 10415 df-seqfrec 10540 df-exp 10631 df-cj 11007 df-re 11008 df-im 11009 df-rsqrt 11163 df-abs 11164 df-dvds 11953 df-gcd 12121 df-prm 12276

This theorem is referenced by: (None)

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