pythagtriplem13 - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > pythagtriplem13		Unicode version

Theorem pythagtriplem13 12820

Description: Lemma for pythagtrip 12827. 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 12817	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
3	2	nnzd 9584	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
4		simp3r 1050	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
5		2z 9490	. . . . . . . . . 10
6		simp3 1023	. . . . . . . . . . . . 13 $/\$ $/\$
7		simp2 1022	. . . . . . . . . . . . 13 $/\$ $/\$
8	6, 7	nnaddcld 9174	. . . . . . . . . . . 12 $/\$ $/\$
9	8	nnzd 9584	. . . . . . . . . . 11 $/\$ $/\$
10	9	3ad2ant1 1042	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
11		nnz 9481	. . . . . . . . . . . 12
12	11	3ad2ant1 1042	. . . . . . . . . . 11 $/\$ $/\$
13	12	3ad2ant1 1042	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
14		dvdsgcdb 12555	. . . . . . . . . 10 $/\$ $/\$ $/\$
15	5, 10, 13, 14	mp3an2i 1376	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
16	15	biimpar 297	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
17	16	simprd 114	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
18	4, 17	mtand 669	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
19		pythagtriplem7 12815	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
20	19	breq2d 4095	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
21	18, 20	mtbird 677	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
22		pythagtriplem8 12816	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
23	22	nnzd 9584	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
24		nnz 9481	. . . . . . . . . . . . 13
25	24	3ad2ant3 1044	. . . . . . . . . . . 12 $/\$ $/\$
26		nnz 9481	. . . . . . . . . . . . 13
27	26	3ad2ant2 1043	. . . . . . . . . . . 12 $/\$ $/\$
28	25, 27	zsubcld 9590	. . . . . . . . . . 11 $/\$ $/\$
29	28	3ad2ant1 1042	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
30		dvdsgcdb 12555	. . . . . . . . . 10 $/\$ $/\$ $/\$
31	5, 29, 13, 30	mp3an2i 1376	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32	31	biimpar 297	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33	32	simprd 114	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	4, 33	mtand 669	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
35		pythagtriplem6 12814	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
36	35	breq2d 4095	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
37	34, 36	mtbird 677	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
38		omoe 12428	. . . . 5 $/\$ $/\$ $/\$
39	3, 21, 23, 37, 38	syl22anc 1272	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
40	28	zred 9585	. . . . . . . . . 10 $/\$ $/\$
41	40	3ad2ant1 1042	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
42		simp13 1053	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
43	42	nnred 9139	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
44	8	nnred 9139	. . . . . . . . . 10 $/\$ $/\$
45	44	3ad2ant1 1042	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
46		nnrp 9876	. . . . . . . . . . . 12
47	46	3ad2ant2 1043	. . . . . . . . . . 11 $/\$ $/\$
48	47	3ad2ant1 1042	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
49	43, 48	ltsubrpd 9942	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
50		nngt0 9151	. . . . . . . . . . . 12
51	50	3ad2ant2 1043	. . . . . . . . . . 11 $/\$ $/\$
52	51	3ad2ant1 1042	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
53		simp12 1052	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
54	53	nnred 9139	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
55	54, 43	ltaddposd 8692	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
56	52, 55	mpbid 147	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
57	41, 43, 45, 49, 56	lttrd 8288	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
58		pythagtriplem10 12813	. . . . . . . . . . 11 $/\$ $/\$ $/\$
59	58	3adant3 1041	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
60		0re 8162	. . . . . . . . . . 11
61		ltle 8250	. . . . . . . . . . 11 $/\$
62	60, 61	mpan 424	. . . . . . . . . 10
63	41, 59, 62	sylc 62	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
64		nngt0 9151	. . . . . . . . . . . . 13
65	64	3ad2ant3 1044	. . . . . . . . . . . 12 $/\$ $/\$
66	65	3ad2ant1 1042	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
67	43, 54, 66, 52	addgt0d 8684	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
68		ltle 8250	. . . . . . . . . . 11 $/\$
69	60, 68	mpan 424	. . . . . . . . . 10
70	45, 67, 69	sylc 62	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
71	41, 63, 45, 70	sqrtltd 11704	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
72	57, 71	mpbid 147	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
73		nnsub 9165	. . . . . . . 8 $/\$
74	22, 2, 73	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
75	72, 74	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
76	75	nnzd 9584	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
77		evend2 12421	. . . . 5
78	76, 77	syl 14	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
79	39, 78	mpbid 147	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
80	75	nngt0d 9170	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
81	75	nnred 9139	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
82		halfpos2 9357	. . . . 5
83	81, 82	syl 14	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
84	80, 83	mpbid 147	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
85		elnnz 9472	. . 3 $/\$
86	79, 84, 85	sylanbrc 417	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
87	1, 86	eqeltrid 2316	1 $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 1002

wceq 1395

wcel 2200 class class class wbr 4083

cfv 5321 (class class class)co 6010

cr 8014

cc0 8015

c1 8016

caddc 8018

clt 8197

cle 8198

cmin 8333

cdiv 8835

cn 9126

c2 9177

cz 9462

crp 9866

cexp 10777

csqrt 11528

cdvds 12319

cgcd 12495

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4259 ax-pr 4294 ax-un 4525 ax-setind 4630 ax-iinf 4681 ax-cnex 8106 ax-resscn 8107 ax-1cn 8108 ax-1re 8109 ax-icn 8110 ax-addcl 8111 ax-addrcl 8112 ax-mulcl 8113 ax-mulrcl 8114 ax-addcom 8115 ax-mulcom 8116 ax-addass 8117 ax-mulass 8118 ax-distr 8119 ax-i2m1 8120 ax-0lt1 8121 ax-1rid 8122 ax-0id 8123 ax-rnegex 8124 ax-precex 8125 ax-cnre 8126 ax-pre-ltirr 8127 ax-pre-ltwlin 8128 ax-pre-lttrn 8129 ax-pre-apti 8130 ax-pre-ltadd 8131 ax-pre-mulgt0 8132 ax-pre-mulext 8133 ax-arch 8134 ax-caucvg 8135

This theorem depends on definitions: df-bi 117 df-stab 836 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-xor 1418 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4385 df-po 4388 df-iso 4389 df-iord 4458 df-on 4460 df-ilim 4461 df-suc 4463 df-iom 4684 df-xp 4726 df-rel 4727 df-cnv 4728 df-co 4729 df-dm 4730 df-rn 4731 df-res 4732 df-ima 4733 df-iota 5281 df-fun 5323 df-fn 5324 df-f 5325 df-f1 5326 df-fo 5327 df-f1o 5328 df-fv 5329 df-riota 5963 df-ov 6013 df-oprab 6014 df-mpo 6015 df-1st 6295 df-2nd 6296 df-recs 6462 df-frec 6548 df-1o 6573 df-2o 6574 df-er 6693 df-en 6901 df-sup 7167 df-pnf 8199 df-mnf 8200 df-xr 8201 df-ltxr 8202 df-le 8203 df-sub 8335 df-neg 8336 df-reap 8738 df-ap 8745 df-div 8836 df-inn 9127 df-2 9185 df-3 9186 df-4 9187 df-n0 9386 df-z 9463 df-uz 9739 df-q 9832 df-rp 9867 df-fz 10222 df-fzo 10356 df-fl 10507 df-mod 10562 df-seqfrec 10687 df-exp 10778 df-cj 11374 df-re 11375 df-im 11376 df-rsqrt 11530 df-abs 11531 df-dvds 12320 df-gcd 12496 df-prm 12651

This theorem is referenced by: pythagtriplem18 12825

W3C validator