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Theorem pythagtriplem13 13038

Description: Lemma for pythagtrip 13045. 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 13035	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
3	2	nnzd 9750	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
4		simp3r 1057	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
5		2z 9655	. . . . . . . . . 10
6		simp3 1030	. . . . . . . . . . . . 13 $/\$ $/\$
7		simp2 1029	. . . . . . . . . . . . 13 $/\$ $/\$
8	6, 7	nnaddcld 9335	. . . . . . . . . . . 12 $/\$ $/\$
9	8	nnzd 9750	. . . . . . . . . . 11 $/\$ $/\$
10	9	3ad2ant1 1049	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
11		nnz 9646	. . . . . . . . . . . 12
12	11	3ad2ant1 1049	. . . . . . . . . . 11 $/\$ $/\$
13	12	3ad2ant1 1049	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
14		dvdsgcdb 12773	. . . . . . . . . 10 $/\$ $/\$ $/\$
15	5, 10, 13, 14	mp3an2i 1383	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
16	15	biimpar 297	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
17	16	simprd 114	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
18	4, 17	mtand 675	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
19		pythagtriplem7 13033	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
20	19	breq2d 4140	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
21	18, 20	mtbird 684	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
22		pythagtriplem8 13034	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
23	22	nnzd 9750	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
24		nnz 9646	. . . . . . . . . . . . 13
25	24	3ad2ant3 1051	. . . . . . . . . . . 12 $/\$ $/\$
26		nnz 9646	. . . . . . . . . . . . 13
27	26	3ad2ant2 1050	. . . . . . . . . . . 12 $/\$ $/\$
28	25, 27	zsubcld 9756	. . . . . . . . . . 11 $/\$ $/\$
29	28	3ad2ant1 1049	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
30		dvdsgcdb 12773	. . . . . . . . . 10 $/\$ $/\$ $/\$
31	5, 29, 13, 30	mp3an2i 1383	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32	31	biimpar 297	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33	32	simprd 114	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	4, 33	mtand 675	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
35		pythagtriplem6 13032	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
36	35	breq2d 4140	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
37	34, 36	mtbird 684	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
38		omoe 12646	. . . . 5 $/\$ $/\$ $/\$
39	3, 21, 23, 37, 38	syl22anc 1279	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
40	28	zred 9751	. . . . . . . . . 10 $/\$ $/\$
41	40	3ad2ant1 1049	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
42		simp13 1060	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
43	42	nnred 9300	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
44	8	nnred 9300	. . . . . . . . . 10 $/\$ $/\$
45	44	3ad2ant1 1049	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
46		nnrp 10047	. . . . . . . . . . . 12
47	46	3ad2ant2 1050	. . . . . . . . . . 11 $/\$ $/\$
48	47	3ad2ant1 1049	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
49	43, 48	ltsubrpd 10113	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
50		nngt0 9312	. . . . . . . . . . . 12
51	50	3ad2ant2 1050	. . . . . . . . . . 11 $/\$ $/\$
52	51	3ad2ant1 1049	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
53		simp12 1059	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
54	53	nnred 9300	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
55	54, 43	ltaddposd 8851	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
56	52, 55	mpbid 147	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
57	41, 43, 45, 49, 56	lttrd 8446	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
58		pythagtriplem10 13031	. . . . . . . . . . 11 $/\$ $/\$ $/\$
59	58	3adant3 1048	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
60		0re 8320	. . . . . . . . . . 11
61		ltle 8407	. . . . . . . . . . 11 $/\$
62	60, 61	mpan 428	. . . . . . . . . 10
63	41, 59, 62	sylc 62	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
64		nngt0 9312	. . . . . . . . . . . . 13
65	64	3ad2ant3 1051	. . . . . . . . . . . 12 $/\$ $/\$
66	65	3ad2ant1 1049	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
67	43, 54, 66, 52	addgt0d 8843	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
68		ltle 8407	. . . . . . . . . . 11 $/\$
69	60, 68	mpan 428	. . . . . . . . . 10
70	45, 67, 69	sylc 62	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
71	41, 63, 45, 70	sqrtltd 11921	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
72	57, 71	mpbid 147	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
73		nnsub 9326	. . . . . . . 8 $/\$
74	22, 2, 73	syl2anc 415	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
75	72, 74	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
76	75	nnzd 9750	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
77		evend2 12639	. . . . 5
78	76, 77	syl 14	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
79	39, 78	mpbid 147	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
80	75	nngt0d 9331	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
81	75	nnred 9300	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
82		halfpos2 9518	. . . . 5
83	81, 82	syl 14	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
84	80, 83	mpbid 147	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
85		elnnz 9637	. . 3 $/\$
86	79, 84, 85	sylanbrc 421	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
87	1, 86	eqeltrid 2325	1 $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 1009

wceq 1402

wcel 2209 class class class wbr 4128

cfv 5375 (class class class)co 6079

cr 8172

cc0 8173

c1 8174

caddc 8176

clt 8354

cle 8355

cmin 8491

cdiv 8996

cn 9287

c2 9338

cz 9627

crp 10037

cexp 10958

csqrt 11745

cdvds 12537

cgcd 12713

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4244 ax-sep 4247 ax-nul 4257 ax-pow 4309 ax-pr 4344 ax-un 4576 ax-setind 4682 ax-iinf 4733 ax-cnex 8264 ax-resscn 8265 ax-1cn 8266 ax-1re 8267 ax-icn 8268 ax-addcl 8269 ax-addrcl 8270 ax-mulcl 8271 ax-mulrcl 8272 ax-addcom 8273 ax-mulcom 8274 ax-addass 8275 ax-mulass 8276 ax-distr 8277 ax-i2m1 8278 ax-0lt1 8279 ax-1rid 8280 ax-0id 8281 ax-rnegex 8282 ax-precex 8283 ax-cnre 8284 ax-pre-ltirr 8285 ax-pre-ltwlin 8286 ax-pre-lttrn 8287 ax-pre-apti 8288 ax-pre-ltadd 8289 ax-pre-mulgt0 8290 ax-pre-mulext 8291 ax-arch 8292 ax-caucvg 8293

This theorem depends on definitions: df-bi 117 df-stab 843 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-xor 1425 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3714 df-pr 3715 df-op 3717 df-uni 3934 df-int 3969 df-iun 4012 df-br 4129 df-opab 4191 df-mpt 4192 df-tr 4228 df-id 4436 df-po 4439 df-iso 4440 df-iord 4509 df-on 4511 df-ilim 4512 df-suc 4514 df-iom 4736 df-xp 4778 df-rel 4779 df-cnv 4780 df-co 4781 df-dm 4782 df-rn 4783 df-res 4784 df-ima 4785 df-iota 5335 df-fun 5377 df-fn 5378 df-f 5379 df-f1 5380 df-fo 5381 df-f1o 5382 df-fv 5383 df-riota 6032 df-ov 6082 df-oprab 6083 df-mpo 6084 df-1st 6368 df-2nd 6369 df-recs 6570 df-frec 6656 df-1o 6681 df-2o 6682 df-er 6801 df-en 7017 df-sup 7318 df-pnf 8356 df-mnf 8357 df-xr 8358 df-ltxr 8359 df-le 8360 df-sub 8493 df-neg 8494 df-reap 8897 df-ap 8904 df-div 8997 df-inn 9288 df-2 9346 df-3 9347 df-4 9348 df-n0 9547 df-z 9628 df-uz 9905 df-q 10003 df-rp 10038 df-fz 10395 df-fzo 10533 df-fl 10688 df-mod 10743 df-seqfrec 10868 df-exp 10959 df-cj 11590 df-re 11591 df-im 11592 df-rsqrt 11747 df-abs 11748 df-dvds 12538 df-gcd 12714 df-prm 12869

This theorem is referenced by: pythagtriplem18 13043

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