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Theorem pythagtriplem13 12296

Description: Lemma for pythagtrip 12303. Show that

(which will eventually be closely related to the

in the final statement) is a natural. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

**Hypothesis**
Ref	Expression
pythagtriplem13.1

**Assertion**
Ref	Expression
pythagtriplem13	$/\$ $/\$ $/\$ $/\$ $/\$

**Proof of Theorem pythagtriplem13**
Step	Hyp	Ref	Expression
1		pythagtriplem13.1	. 2
2		pythagtriplem9 12293	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
3	2	nnzd 9394	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
4		simp3r 1028	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
5		2z 9301	. . . . . . . . . 10
6		simp3 1001	. . . . . . . . . . . . 13 $/\$ $/\$
7		simp2 1000	. . . . . . . . . . . . 13 $/\$ $/\$
8	6, 7	nnaddcld 8987	. . . . . . . . . . . 12 $/\$ $/\$
9	8	nnzd 9394	. . . . . . . . . . 11 $/\$ $/\$
10	9	3ad2ant1 1020	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
11		nnz 9292	. . . . . . . . . . . 12
12	11	3ad2ant1 1020	. . . . . . . . . . 11 $/\$ $/\$
13	12	3ad2ant1 1020	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
14		dvdsgcdb 12034	. . . . . . . . . 10 $/\$ $/\$ $/\$
15	5, 10, 13, 14	mp3an2i 1353	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
16	15	biimpar 297	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
17	16	simprd 114	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
18	4, 17	mtand 666	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
19		pythagtriplem7 12291	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
20	19	breq2d 4030	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
21	18, 20	mtbird 674	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
22		pythagtriplem8 12292	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
23	22	nnzd 9394	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
24		nnz 9292	. . . . . . . . . . . . 13
25	24	3ad2ant3 1022	. . . . . . . . . . . 12 $/\$ $/\$
26		nnz 9292	. . . . . . . . . . . . 13
27	26	3ad2ant2 1021	. . . . . . . . . . . 12 $/\$ $/\$
28	25, 27	zsubcld 9400	. . . . . . . . . . 11 $/\$ $/\$
29	28	3ad2ant1 1020	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
30		dvdsgcdb 12034	. . . . . . . . . 10 $/\$ $/\$ $/\$
31	5, 29, 13, 30	mp3an2i 1353	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32	31	biimpar 297	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33	32	simprd 114	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	4, 33	mtand 666	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
35		pythagtriplem6 12290	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
36	35	breq2d 4030	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
37	34, 36	mtbird 674	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
38		omoe 11921	. . . . 5 $/\$ $/\$ $/\$
39	3, 21, 23, 37, 38	syl22anc 1250	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
40	28	zred 9395	. . . . . . . . . 10 $/\$ $/\$
41	40	3ad2ant1 1020	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
42		simp13 1031	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
43	42	nnred 8952	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
44	8	nnred 8952	. . . . . . . . . 10 $/\$ $/\$
45	44	3ad2ant1 1020	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
46		nnrp 9683	. . . . . . . . . . . 12
47	46	3ad2ant2 1021	. . . . . . . . . . 11 $/\$ $/\$
48	47	3ad2ant1 1020	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
49	43, 48	ltsubrpd 9749	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
50		nngt0 8964	. . . . . . . . . . . 12
51	50	3ad2ant2 1021	. . . . . . . . . . 11 $/\$ $/\$
52	51	3ad2ant1 1020	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
53		simp12 1030	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
54	53	nnred 8952	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
55	54, 43	ltaddposd 8506	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
56	52, 55	mpbid 147	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
57	41, 43, 45, 49, 56	lttrd 8103	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
58		pythagtriplem10 12289	. . . . . . . . . . 11 $/\$ $/\$ $/\$
59	58	3adant3 1019	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
60		0re 7977	. . . . . . . . . . 11
61		ltle 8065	. . . . . . . . . . 11 $/\$
62	60, 61	mpan 424	. . . . . . . . . 10
63	41, 59, 62	sylc 62	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
64		nngt0 8964	. . . . . . . . . . . . 13
65	64	3ad2ant3 1022	. . . . . . . . . . . 12 $/\$ $/\$
66	65	3ad2ant1 1020	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
67	43, 54, 66, 52	addgt0d 8498	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
68		ltle 8065	. . . . . . . . . . 11 $/\$
69	60, 68	mpan 424	. . . . . . . . . 10
70	45, 67, 69	sylc 62	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
71	41, 63, 45, 70	sqrtltd 11201	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
72	57, 71	mpbid 147	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
73		nnsub 8978	. . . . . . . 8 $/\$
74	22, 2, 73	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
75	72, 74	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
76	75	nnzd 9394	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
77		evend2 11914	. . . . 5
78	76, 77	syl 14	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
79	39, 78	mpbid 147	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
80	75	nngt0d 8983	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
81	75	nnred 8952	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
82		halfpos2 9169	. . . . 5
83	81, 82	syl 14	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
84	80, 83	mpbid 147	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
85		elnnz 9283	. . 3 $/\$
86	79, 84, 85	sylanbrc 417	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
87	1, 86	eqeltrid 2276	1 $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 980

wceq 1364

wcel 2160 class class class wbr 4018

cfv 5232 (class class class)co 5892

cr 7830

cc0 7831

c1 7832

caddc 7834

clt 8012

cle 8013

cmin 8148

cdiv 8649

cn 8939

c2 8990

cz 9273

crp 9673

cexp 10539

csqrt 11025

cdvds 11814

cgcd 11963

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 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 ax-cnex 7922 ax-resscn 7923 ax-1cn 7924 ax-1re 7925 ax-icn 7926 ax-addcl 7927 ax-addrcl 7928 ax-mulcl 7929 ax-mulrcl 7930 ax-addcom 7931 ax-mulcom 7932 ax-addass 7933 ax-mulass 7934 ax-distr 7935 ax-i2m1 7936 ax-0lt1 7937 ax-1rid 7938 ax-0id 7939 ax-rnegex 7940 ax-precex 7941 ax-cnre 7942 ax-pre-ltirr 7943 ax-pre-ltwlin 7944 ax-pre-lttrn 7945 ax-pre-apti 7946 ax-pre-ltadd 7947 ax-pre-mulgt0 7948 ax-pre-mulext 7949 ax-arch 7950 ax-caucvg 7951

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 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4308 df-po 4311 df-iso 4312 df-iord 4381 df-on 4383 df-ilim 4384 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5234 df-fn 5235 df-f 5236 df-f1 5237 df-fo 5238 df-f1o 5239 df-fv 5240 df-riota 5848 df-ov 5895 df-oprab 5896 df-mpo 5897 df-1st 6160 df-2nd 6161 df-recs 6325 df-frec 6411 df-1o 6436 df-2o 6437 df-er 6554 df-en 6760 df-sup 7003 df-pnf 8014 df-mnf 8015 df-xr 8016 df-ltxr 8017 df-le 8018 df-sub 8150 df-neg 8151 df-reap 8552 df-ap 8559 df-div 8650 df-inn 8940 df-2 8998 df-3 8999 df-4 9000 df-n0 9197 df-z 9274 df-uz 9549 df-q 9640 df-rp 9674 df-fz 10029 df-fzo 10163 df-fl 10290 df-mod 10343 df-seqfrec 10466 df-exp 10540 df-cj 10871 df-re 10872 df-im 10873 df-rsqrt 11027 df-abs 11028 df-dvds 11815 df-gcd 11964 df-prm 12128

This theorem is referenced by: pythagtriplem18 12301

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