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Theorem pythagtriplem16 12207

Description: Lemma for pythagtrip 12211. Show the relationship between

, and

. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

**Hypotheses**
Ref	Expression
pythagtriplem15.1
pythagtriplem15.2

**Assertion**
Ref	Expression
pythagtriplem16	$/\$ $/\$ $/\$ $/\$ $/\$

**Proof of Theorem pythagtriplem16**
Step	Hyp	Ref	Expression
1		pythagtriplem15.1	. . . . 5
2		pythagtriplem15.2	. . . . 5
3	1, 2	oveq12i 5853	. . . 4
4		simp13 1019	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
5		simp12 1018	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
6	4, 5	nnaddcld 8901	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
7	6	nnrpd 9626	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
8	7	rpsqrtcld 11096	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
9	8	rpcnd 9630	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
10	4	nnzd 9308	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
11	5	nnzd 9308	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
12	10, 11	zsubcld 9314	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
13	12	zred 9309	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
14		pythagtriplem10 12197	. . . . . . . . . . 11 $/\$ $/\$ $/\$
15	14	3adant3 1007	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
16	13, 15	elrpd 9625	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
17	16	rpsqrtcld 11096	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
18	17	rpcnd 9630	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
19	9, 18	addcld 7914	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
20		2cnd 8926	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
21	9, 18	subcld 8205	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
22		2ap0 8946	. . . . . . 7 #
23	22	a1i 9	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ #
24	19, 20, 21, 20, 23, 23	divmuldivapd 8724	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
25	19, 21	mulcld 7915	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
26	25, 20, 20, 23, 23	divdivap1d 8714	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
27		nnre 8860	. . . . . . . . . . . . 13
28		nnre 8860	. . . . . . . . . . . . 13
29		readdcl 7875	. . . . . . . . . . . . 13 $/\$
30	27, 28, 29	syl2anr 288	. . . . . . . . . . . 12 $/\$
31	30	3adant1 1005	. . . . . . . . . . 11 $/\$ $/\$
32	31	3ad2ant1 1008	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
33	27	adantl 275	. . . . . . . . . . . . . 14 $/\$
34	28	adantr 274	. . . . . . . . . . . . . 14 $/\$
35		nngt0 8878	. . . . . . . . . . . . . . 15
36	35	adantl 275	. . . . . . . . . . . . . 14 $/\$
37		nngt0 8878	. . . . . . . . . . . . . . 15
38	37	adantr 274	. . . . . . . . . . . . . 14 $/\$
39	33, 34, 36, 38	addgt0d 8415	. . . . . . . . . . . . 13 $/\$
40		0re 7895	. . . . . . . . . . . . . 14
41		ltle 7982	. . . . . . . . . . . . . 14 $/\$
42	40, 41	mpan 421	. . . . . . . . . . . . 13
43	30, 39, 42	sylc 62	. . . . . . . . . . . 12 $/\$
44	43	3adant1 1005	. . . . . . . . . . 11 $/\$ $/\$
45	44	3ad2ant1 1008	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
46		resqrtth 10969	. . . . . . . . . 10 $/\$
47	32, 45, 46	syl2anc 409	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
48		resubcl 8158	. . . . . . . . . . . . 13 $/\$
49	27, 28, 48	syl2anr 288	. . . . . . . . . . . 12 $/\$
50	49	3adant1 1005	. . . . . . . . . . 11 $/\$ $/\$
51	50	3ad2ant1 1008	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
52		ltle 7982	. . . . . . . . . . . 12 $/\$
53	40, 52	mpan 421	. . . . . . . . . . 11
54	51, 15, 53	sylc 62	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
55		resqrtth 10969	. . . . . . . . . 10 $/\$
56	51, 54, 55	syl2anc 409	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
57	47, 56	oveq12d 5859	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
58	57	oveq1d 5856	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
59		subsq 10557	. . . . . . . . 9 $/\$
60	9, 18, 59	syl2anc 409	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
61	60	oveq1d 5856	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
62		nncn 8861	. . . . . . . . . . . 12
63		nncn 8861	. . . . . . . . . . . 12
64		pnncan 8135	. . . . . . . . . . . . . 14 $/\$ $/\$
65	64	3anidm23 1287	. . . . . . . . . . . . 13 $/\$
66		2times 8981	. . . . . . . . . . . . . 14
67	66	adantl 275	. . . . . . . . . . . . 13 $/\$
68	65, 67	eqtr4d 2201	. . . . . . . . . . . 12 $/\$
69	62, 63, 68	syl2anr 288	. . . . . . . . . . 11 $/\$
70	69	3adant1 1005	. . . . . . . . . 10 $/\$ $/\$
71	70	3ad2ant1 1008	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
72	71	oveq1d 5856	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
73	5	nncnd 8867	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
74	73, 20, 23	divcanap3d 8687	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
75	72, 74	eqtrd 2198	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
76	58, 61, 75	3eqtr3d 2206	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
77	76	oveq1d 5856	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
78	24, 26, 77	3eqtr2d 2204	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
79	3, 78	syl5eq 2210	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
80	79	oveq2d 5857	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
81	73, 20, 23	divcanap2d 8684	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
82	80, 81	eqtr2d 2199	1 $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103 $/\$ w3a 968

wceq 1343

wcel 2136 class class class wbr 3981

cfv 5187 (class class class)co 5841

cc 7747

cr 7748

cc0 7749

c1 7750

caddc 7752

cmul 7754

clt 7929

cle 7930

cmin 8065 # cap 8475

cdiv 8564

cn 8853

c2 8904

cexp 10450

csqrt 10934

cdvds 11723

cgcd 11871

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4096 ax-sep 4099 ax-nul 4107 ax-pow 4152 ax-pr 4186 ax-un 4410 ax-setind 4513 ax-iinf 4564 ax-cnex 7840 ax-resscn 7841 ax-1cn 7842 ax-1re 7843 ax-icn 7844 ax-addcl 7845 ax-addrcl 7846 ax-mulcl 7847 ax-mulrcl 7848 ax-addcom 7849 ax-mulcom 7850 ax-addass 7851 ax-mulass 7852 ax-distr 7853 ax-i2m1 7854 ax-0lt1 7855 ax-1rid 7856 ax-0id 7857 ax-rnegex 7858 ax-precex 7859 ax-cnre 7860 ax-pre-ltirr 7861 ax-pre-ltwlin 7862 ax-pre-lttrn 7863 ax-pre-apti 7864 ax-pre-ltadd 7865 ax-pre-mulgt0 7866 ax-pre-mulext 7867 ax-arch 7868 ax-caucvg 7869

This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2296 df-ne 2336 df-nel 2431 df-ral 2448 df-rex 2449 df-reu 2450 df-rmo 2451 df-rab 2452 df-v 2727 df-sbc 2951 df-csb 3045 df-dif 3117 df-un 3119 df-in 3121 df-ss 3128 df-nul 3409 df-if 3520 df-pw 3560 df-sn 3581 df-pr 3582 df-op 3584 df-uni 3789 df-int 3824 df-iun 3867 df-br 3982 df-opab 4043 df-mpt 4044 df-tr 4080 df-id 4270 df-po 4273 df-iso 4274 df-iord 4343 df-on 4345 df-ilim 4346 df-suc 4348 df-iom 4567 df-xp 4609 df-rel 4610 df-cnv 4611 df-co 4612 df-dm 4613 df-rn 4614 df-res 4615 df-ima 4616 df-iota 5152 df-fun 5189 df-fn 5190 df-f 5191 df-f1 5192 df-fo 5193 df-f1o 5194 df-fv 5195 df-riota 5797 df-ov 5844 df-oprab 5845 df-mpo 5846 df-1st 6105 df-2nd 6106 df-recs 6269 df-frec 6355 df-pnf 7931 df-mnf 7932 df-xr 7933 df-ltxr 7934 df-le 7935 df-sub 8067 df-neg 8068 df-reap 8469 df-ap 8476 df-div 8565 df-inn 8854 df-2 8912 df-3 8913 df-4 8914 df-n0 9111 df-z 9188 df-uz 9463 df-rp 9586 df-seqfrec 10377 df-exp 10451 df-rsqrt 10936

This theorem is referenced by: pythagtriplem18 12209

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