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Theorem pythagtriplem2 12900

Description: Lemma for pythagtrip 12917. Prove the full version of one direction of the theorem. (Contributed by Scott Fenton, 28-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

**Assertion**
Ref	Expression
pythagtriplem2	$/\$ $/\$

Distinct variable groups:

**Proof of Theorem pythagtriplem2**
Step	Hyp	Ref	Expression
1		simplll 535	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
2		simpllr 536	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
3		nnz 9541	. . . . . . . . . 10
4	3	adantl 277	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
5		simplrr 538	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
6	5	nnzd 9644	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
7		zsqcl 10916	. . . . . . . . . . 11
8	6, 7	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
9		simplrl 537	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
10	9	nnzd 9644	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
11		zsqcl 10916	. . . . . . . . . . 11
12	10, 11	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
13	8, 12	zsubcld 9650	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
14	4, 13	zmulcld 9651	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
15		2z 9550	. . . . . . . . . . 11
16	15	a1i 9	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
17	6, 10	zmulcld 9651	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
18	16, 17	zmulcld 9651	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
19	4, 18	zmulcld 9651	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
20		preq12bg 3861	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
21	1, 2, 14, 19, 20	syl22anc 1275	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
22	21	anbi1d 465	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
23		andir 827	. . . . . . 7 $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
24		df-3an 1007	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
25		df-3an 1007	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
26	24, 25	orbi12i 772	. . . . . . 7 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
27	23, 26	bitr4i 187	. . . . . 6 $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
28	22, 27	bitrdi 196	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
29	28	rexbidva 2530	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
30	29	2rexbidva 2556	. . 3 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
31		r19.43 2692	. . . . 5 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
32	31	2rexbii 2542	. . . 4 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
33		r19.43 2692	. . . . 5 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
34	33	rexbii 2540	. . . 4 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
35		r19.43 2692	. . . 4 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
36	32, 34, 35	3bitri 206	. . 3 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
37	30, 36	bitrdi 196	. 2 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
38		pythagtriplem1 12899	. . . 4 $/\$ $/\$
39	38	a1i 9	. . 3 $/\$ $/\$ $/\$
40		3ancoma 1012	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
41	40	rexbii 2540	. . . . . 6 $/\$ $/\$ $/\$ $/\$
42	41	2rexbii 2542	. . . . 5 $/\$ $/\$ $/\$ $/\$
43		pythagtriplem1 12899	. . . . 5 $/\$ $/\$
44	42, 43	sylbi 121	. . . 4 $/\$ $/\$
45		nncn 9194	. . . . . . 7
46	45	sqcld 10977	. . . . . 6
47		nncn 9194	. . . . . . 7
48	47	sqcld 10977	. . . . . 6
49		addcom 8359	. . . . . 6 $/\$
50	46, 48, 49	syl2an 289	. . . . 5 $/\$
51	50	eqeq1d 2240	. . . 4 $/\$
52	44, 51	imbitrrid 156	. . 3 $/\$ $/\$ $/\$
53	39, 52	jaod 725	. 2 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
54	37, 53	sylbid 150	1 $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $\/$ wo 716 $/\$ w3a 1005

wceq 1398

wcel 2202

wrex 2512

cpr 3674 (class class class)co 6028

cc 8073

caddc 8078

cmul 8080

cmin 8393

cn 9186

c2 9237

cz 9522

cexp 10844

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 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692 ax-cnex 8166 ax-resscn 8167 ax-1cn 8168 ax-1re 8169 ax-icn 8170 ax-addcl 8171 ax-addrcl 8172 ax-mulcl 8173 ax-mulrcl 8174 ax-addcom 8175 ax-mulcom 8176 ax-addass 8177 ax-mulass 8178 ax-distr 8179 ax-i2m1 8180 ax-0lt1 8181 ax-1rid 8182 ax-0id 8183 ax-rnegex 8184 ax-precex 8185 ax-cnre 8186 ax-pre-ltirr 8187 ax-pre-ltwlin 8188 ax-pre-lttrn 8189 ax-pre-apti 8190 ax-pre-ltadd 8191 ax-pre-mulgt0 8192 ax-pre-mulext 8193

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-reu 2518 df-rmo 2519 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-if 3608 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-id 4396 df-po 4399 df-iso 4400 df-iord 4469 df-on 4471 df-ilim 4472 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-recs 6514 df-frec 6600 df-pnf 8259 df-mnf 8260 df-xr 8261 df-ltxr 8262 df-le 8263 df-sub 8395 df-neg 8396 df-reap 8798 df-ap 8805 df-div 8896 df-inn 9187 df-2 9245 df-3 9246 df-4 9247 df-n0 9446 df-z 9523 df-uz 9799 df-seqfrec 10754 df-exp 10845

This theorem is referenced by: pythagtrip 12917

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