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Theorem pythagtrip 12421

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 12281	. . . . . . 7 $/\$ $\/$
2	1	3adant3 1019	. . . . . 6 $/\$ $/\$ $\/$
3	2	adantr 276	. . . . 5 $/\$ $/\$ $/\$ $\/$
4		pythagtriplem19 12420	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
5	4	3expia 1207	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
6		simp12 1030	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
7		simp11 1029	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
8		simp13 1031	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
9		nnsqcl 10680	. . . . . . . . . . . . . 14
10	9	nncnd 8996	. . . . . . . . . . . . 13
11	10	3ad2ant1 1020	. . . . . . . . . . . 12 $/\$ $/\$
12		nnsqcl 10680	. . . . . . . . . . . . . 14
13	12	nncnd 8996	. . . . . . . . . . . . 13
14	13	3ad2ant2 1021	. . . . . . . . . . . 12 $/\$ $/\$
15	11, 14	addcomd 8170	. . . . . . . . . . 11 $/\$ $/\$
16	15	eqeq1d 2202	. . . . . . . . . 10 $/\$ $/\$
17	16	biimpa 296	. . . . . . . . 9 $/\$ $/\$ $/\$
18	17	3adant3 1019	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
19		nnz 9336	. . . . . . . . . . . . . 14
20	19	3ad2ant1 1020	. . . . . . . . . . . . 13 $/\$ $/\$
21		nnz 9336	. . . . . . . . . . . . . . 15
22	21	3ad2ant2 1021	. . . . . . . . . . . . . 14 $/\$ $/\$
23	22	adantr 276	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
24		gcdcom 12110	. . . . . . . . . . . . 13 $/\$
25	20, 23, 24	syl2an2r 595	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
26	25	oveq2d 5934	. . . . . . . . . . 11 $/\$ $/\$ $/\$
27	26	breq2d 4041	. . . . . . . . . 10 $/\$ $/\$ $/\$
28	27	notbid 668	. . . . . . . . 9 $/\$ $/\$ $/\$
29	28	biimp3a 1356	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
30		pythagtriplem19 12420	. . . . . . . 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 9336	. . . . . . . . . . . . 13
38	37	adantl 277	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
39		simplrr 536	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
40	39	nnzd 9438	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
41		zsqcl 10681	. . . . . . . . . . . . . 14
42	40, 41	syl 14	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
43		simplrl 535	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
44	43	nnzd 9438	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
45		zsqcl 10681	. . . . . . . . . . . . . 14
46	44, 45	syl 14	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
47	42, 46	zsubcld 9444	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
48	38, 47	zmulcld 9445	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
49		2z 9345	. . . . . . . . . . . . . 14
50	49	a1i 9	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
51	40, 44	zmulcld 9445	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
52	50, 51	zmulcld 9445	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
53	38, 52	zmulcld 9445	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
54		preq12bg 3799	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
55	35, 36, 48, 53, 54	syl22anc 1250	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
56	55	anbi1d 465	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
57	56	rexbidva 2491	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
58	57	2rexbidva 2517	. . . . . . 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 2501	. . . . . . . . 9 $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
67	66	2rexbii 2503	. . . . . . . 8 $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
68		r19.43 2652	. . . . . . . . . 10 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
69	68	2rexbii 2503	. . . . . . . . 9 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
70		r19.43 2652	. . . . . . . . . . 11 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
71	70	rexbii 2501	. . . . . . . . . 10 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
72		r19.43 2652	. . . . . . . . . 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 12404	. . 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 2164

wrex 2473

cpr 3619 class class class wbr 4029 (class class class)co 5918

cc 7870

caddc 7875

cmul 7877

cmin 8190

cdiv 8691

cn 8982

c2 9033

cz 9317

cexp 10609

cdvds 11930

cgcd 12079

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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990 ax-arch 7991 ax-caucvg 7992

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 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4324 df-po 4327 df-iso 4328 df-iord 4397 df-on 4399 df-ilim 4400 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-recs 6358 df-frec 6444 df-1o 6469 df-2o 6470 df-er 6587 df-en 6795 df-sup 7043 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 df-inn 8983 df-2 9041 df-3 9042 df-4 9043 df-n0 9241 df-z 9318 df-uz 9593 df-q 9685 df-rp 9720 df-fz 10075 df-fzo 10209 df-fl 10339 df-mod 10394 df-seqfrec 10519 df-exp 10610 df-cj 10986 df-re 10987 df-im 10988 df-rsqrt 11142 df-abs 11143 df-dvds 11931 df-gcd 12080 df-prm 12246

This theorem is referenced by: (None)

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