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Theorem pythagtriplem13 12978

Description: Lemma for pythagtrip 12985. 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 12975	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
3	2	nnzd 9702	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
4		simp3r 1053	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
5		2z 9607	. . . . . . . . . 10
6		simp3 1026	. . . . . . . . . . . . 13 $/\$ $/\$
7		simp2 1025	. . . . . . . . . . . . 13 $/\$ $/\$
8	6, 7	nnaddcld 9287	. . . . . . . . . . . 12 $/\$ $/\$
9	8	nnzd 9702	. . . . . . . . . . 11 $/\$ $/\$
10	9	3ad2ant1 1045	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
11		nnz 9598	. . . . . . . . . . . 12
12	11	3ad2ant1 1045	. . . . . . . . . . 11 $/\$ $/\$
13	12	3ad2ant1 1045	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
14		dvdsgcdb 12713	. . . . . . . . . 10 $/\$ $/\$ $/\$
15	5, 10, 13, 14	mp3an2i 1379	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
16	15	biimpar 297	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
17	16	simprd 114	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
18	4, 17	mtand 671	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
19		pythagtriplem7 12973	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
20	19	breq2d 4123	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
21	18, 20	mtbird 680	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
22		pythagtriplem8 12974	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
23	22	nnzd 9702	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
24		nnz 9598	. . . . . . . . . . . . 13
25	24	3ad2ant3 1047	. . . . . . . . . . . 12 $/\$ $/\$
26		nnz 9598	. . . . . . . . . . . . 13
27	26	3ad2ant2 1046	. . . . . . . . . . . 12 $/\$ $/\$
28	25, 27	zsubcld 9708	. . . . . . . . . . 11 $/\$ $/\$
29	28	3ad2ant1 1045	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
30		dvdsgcdb 12713	. . . . . . . . . 10 $/\$ $/\$ $/\$
31	5, 29, 13, 30	mp3an2i 1379	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32	31	biimpar 297	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33	32	simprd 114	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	4, 33	mtand 671	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
35		pythagtriplem6 12972	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
36	35	breq2d 4123	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
37	34, 36	mtbird 680	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
38		omoe 12586	. . . . 5 $/\$ $/\$ $/\$
39	3, 21, 23, 37, 38	syl22anc 1275	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
40	28	zred 9703	. . . . . . . . . 10 $/\$ $/\$
41	40	3ad2ant1 1045	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
42		simp13 1056	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
43	42	nnred 9252	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
44	8	nnred 9252	. . . . . . . . . 10 $/\$ $/\$
45	44	3ad2ant1 1045	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
46		nnrp 9999	. . . . . . . . . . . 12
47	46	3ad2ant2 1046	. . . . . . . . . . 11 $/\$ $/\$
48	47	3ad2ant1 1045	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
49	43, 48	ltsubrpd 10065	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
50		nngt0 9264	. . . . . . . . . . . 12
51	50	3ad2ant2 1046	. . . . . . . . . . 11 $/\$ $/\$
52	51	3ad2ant1 1045	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
53		simp12 1055	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
54	53	nnred 9252	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
55	54, 43	ltaddposd 8805	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
56	52, 55	mpbid 147	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
57	41, 43, 45, 49, 56	lttrd 8401	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
58		pythagtriplem10 12971	. . . . . . . . . . 11 $/\$ $/\$ $/\$
59	58	3adant3 1044	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
60		0re 8276	. . . . . . . . . . 11
61		ltle 8363	. . . . . . . . . . 11 $/\$
62	60, 61	mpan 424	. . . . . . . . . 10
63	41, 59, 62	sylc 62	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
64		nngt0 9264	. . . . . . . . . . . . 13
65	64	3ad2ant3 1047	. . . . . . . . . . . 12 $/\$ $/\$
66	65	3ad2ant1 1045	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
67	43, 54, 66, 52	addgt0d 8797	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
68		ltle 8363	. . . . . . . . . . 11 $/\$
69	60, 68	mpan 424	. . . . . . . . . 10
70	45, 67, 69	sylc 62	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
71	41, 63, 45, 70	sqrtltd 11861	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
72	57, 71	mpbid 147	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
73		nnsub 9278	. . . . . . . 8 $/\$
74	22, 2, 73	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
75	72, 74	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
76	75	nnzd 9702	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
77		evend2 12579	. . . . 5
78	76, 77	syl 14	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
79	39, 78	mpbid 147	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
80	75	nngt0d 9283	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
81	75	nnred 9252	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
82		halfpos2 9470	. . . . 5
83	81, 82	syl 14	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
84	80, 83	mpbid 147	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
85		elnnz 9589	. . 3 $/\$
86	79, 84, 85	sylanbrc 417	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
87	1, 86	eqeltrid 2321	1 $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 1005

wceq 1398

wcel 2205 class class class wbr 4111

cfv 5354 (class class class)co 6052

cr 8128

cc0 8129

c1 8130

caddc 8132

clt 8310

cle 8311

cmin 8446

cdiv 8948

cn 9239

c2 9290

cz 9579

crp 9989

cexp 10904

csqrt 11685

cdvds 12477

cgcd 12653

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 2207 ax-14 2208 ax-ext 2216 ax-coll 4227 ax-sep 4230 ax-nul 4238 ax-pow 4289 ax-pr 4324 ax-un 4556 ax-setind 4661 ax-iinf 4712 ax-cnex 8220 ax-resscn 8221 ax-1cn 8222 ax-1re 8223 ax-icn 8224 ax-addcl 8225 ax-addrcl 8226 ax-mulcl 8227 ax-mulrcl 8228 ax-addcom 8229 ax-mulcom 8230 ax-addass 8231 ax-mulass 8232 ax-distr 8233 ax-i2m1 8234 ax-0lt1 8235 ax-1rid 8236 ax-0id 8237 ax-rnegex 8238 ax-precex 8239 ax-cnre 8240 ax-pre-ltirr 8241 ax-pre-ltwlin 8242 ax-pre-lttrn 8243 ax-pre-apti 8244 ax-pre-ltadd 8245 ax-pre-mulgt0 8246 ax-pre-mulext 8247 ax-arch 8248 ax-caucvg 8249

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 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3045 df-csb 3141 df-dif 3215 df-un 3217 df-in 3219 df-ss 3226 df-nul 3511 df-if 3623 df-pw 3673 df-sn 3697 df-pr 3698 df-op 3700 df-uni 3917 df-int 3952 df-iun 3995 df-br 4112 df-opab 4174 df-mpt 4175 df-tr 4211 df-id 4416 df-po 4419 df-iso 4420 df-iord 4489 df-on 4491 df-ilim 4492 df-suc 4494 df-iom 4715 df-xp 4757 df-rel 4758 df-cnv 4759 df-co 4760 df-dm 4761 df-rn 4762 df-res 4763 df-ima 4764 df-iota 5314 df-fun 5356 df-fn 5357 df-f 5358 df-f1 5359 df-fo 5360 df-f1o 5361 df-fv 5362 df-riota 6005 df-ov 6055 df-oprab 6056 df-mpo 6057 df-1st 6336 df-2nd 6337 df-recs 6538 df-frec 6624 df-1o 6649 df-2o 6650 df-er 6769 df-en 6978 df-sup 7277 df-pnf 8312 df-mnf 8313 df-xr 8314 df-ltxr 8315 df-le 8316 df-sub 8448 df-neg 8449 df-reap 8851 df-ap 8858 df-div 8949 df-inn 9240 df-2 9298 df-3 9299 df-4 9300 df-n0 9499 df-z 9580 df-uz 9857 df-q 9955 df-rp 9990 df-fz 10346 df-fzo 10481 df-fl 10634 df-mod 10689 df-seqfrec 10814 df-exp 10905 df-cj 11531 df-re 11532 df-im 11533 df-rsqrt 11687 df-abs 11688 df-dvds 12478 df-gcd 12654 df-prm 12809

This theorem is referenced by: pythagtriplem18 12983

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