Theorem sqrt2irrlem 12649

Description: Lemma for sqrt2irr 12650. This is the core of the proof: - if A / B = sqr ( 2 ), then A and B are even, so A / 2 and B / 2 are smaller representatives, which is absurd by the method of infinite descent (here implemented by strong induction). (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 12-Sep-2015.)

Hypotheses

Ref	Expression
sqrt2irrlem.1	|- ( ph -> A e. ZZ )
sqrt2irrlem.2	|- ( ph -> B e. NN )
sqrt2irrlem.3	|- ( ph -> ( sqr ` 2 ) = ( A / B ) )

Assertion

Ref	Expression
sqrt2irrlem	|- ( ph -> ( ( A / 2 ) e. ZZ /\ ( B / 2 ) e. NN ) )

Proof of Theorem sqrt2irrlem

Step	Hyp	Ref	Expression
1		2re 9148	. . . . . . . . . . . 12 |- 2 e. RR
2		0le2 9168	. . . . . . . . . . . 12 |- 0 <_ 2
3		resqrtth 11508	. . . . . . . . . . . 12 |- ( ( 2 e. RR /\ 0 <_ 2 ) -> ( ( sqr ` 2 ) ^ 2 ) = 2 )
4	1, 2, 3	mp2an 426	. . . . . . . . . . 11 |- ( ( sqr ` 2 ) ^ 2 ) = 2
5		sqrt2irrlem.3	. . . . . . . . . . . 12 |- ( ph -> ( sqr ` 2 ) = ( A / B ) )
6	5	oveq1d 5989	. . . . . . . . . . 11 |- ( ph -> ( ( sqr ` 2 ) ^ 2 ) = ( ( A / B ) ^ 2 ) )
7	4, 6	eqtr3id 2256	. . . . . . . . . 10 |- ( ph -> 2 = ( ( A / B ) ^ 2 ) )
8		sqrt2irrlem.1	. . . . . . . . . . . 12 |- ( ph -> A e. ZZ )
9	8	zcnd 9538	. . . . . . . . . . 11 |- ( ph -> A e. CC )
10		sqrt2irrlem.2	. . . . . . . . . . . 12 |- ( ph -> B e. NN )
11	10	nncnd 9092	. . . . . . . . . . 11 |- ( ph -> B e. CC )
12	10	nnap0d 9124	. . . . . . . . . . 11 |- ( ph -> B # 0 )
13	9, 11, 12	sqdivapd 10875	. . . . . . . . . 10 |- ( ph -> ( ( A / B ) ^ 2 ) = ( ( A ^ 2 ) / ( B ^ 2 ) ) )
14	7, 13	eqtrd 2242	. . . . . . . . 9 |- ( ph -> 2 = ( ( A ^ 2 ) / ( B ^ 2 ) ) )
15	14	oveq1d 5989	. . . . . . . 8 |- ( ph -> ( 2 x. ( B ^ 2 ) ) = ( ( ( A ^ 2 ) / ( B ^ 2 ) ) x. ( B ^ 2 ) ) )
16	9	sqcld 10860	. . . . . . . . 9 |- ( ph -> ( A ^ 2 ) e. CC )
17	10	nnsqcld 10883	. . . . . . . . . 10 |- ( ph -> ( B ^ 2 ) e. NN )
18	17	nncnd 9092	. . . . . . . . 9 |- ( ph -> ( B ^ 2 ) e. CC )
19	17	nnap0d 9124	. . . . . . . . 9 |- ( ph -> ( B ^ 2 ) # 0 )
20	16, 18, 19	divcanap1d 8906	. . . . . . . 8 |- ( ph -> ( ( ( A ^ 2 ) / ( B ^ 2 ) ) x. ( B ^ 2 ) ) = ( A ^ 2 ) )
21	15, 20	eqtrd 2242	. . . . . . 7 |- ( ph -> ( 2 x. ( B ^ 2 ) ) = ( A ^ 2 ) )
22	21	oveq1d 5989	. . . . . 6 |- ( ph -> ( ( 2 x. ( B ^ 2 ) ) / 2 ) = ( ( A ^ 2 ) / 2 ) )
23		2cnd 9151	. . . . . . 7 |- ( ph -> 2 e. CC )
24		2ap0 9171	. . . . . . . 8 |- 2 # 0
25	24	a1i 9	. . . . . . 7 |- ( ph -> 2 # 0 )
26	18, 23, 25	divcanap3d 8910	. . . . . 6 |- ( ph -> ( ( 2 x. ( B ^ 2 ) ) / 2 ) = ( B ^ 2 ) )
27	22, 26	eqtr3d 2244	. . . . 5 |- ( ph -> ( ( A ^ 2 ) / 2 ) = ( B ^ 2 ) )
28	27, 17	eqeltrd 2286	. . . 4 |- ( ph -> ( ( A ^ 2 ) / 2 ) e. NN )
29	28	nnzd 9536	. . 3 |- ( ph -> ( ( A ^ 2 ) / 2 ) e. ZZ )
30		zesq 10847	. . . 4 |- ( A e. ZZ -> ( ( A / 2 ) e. ZZ <-> ( ( A ^ 2 ) / 2 ) e. ZZ ) )
31	8, 30	syl 14	. . 3 |- ( ph -> ( ( A / 2 ) e. ZZ <-> ( ( A ^ 2 ) / 2 ) e. ZZ ) )
32	29, 31	mpbird 167	. 2 |- ( ph -> ( A / 2 ) e. ZZ )
33		2cn 9149	. . . . . . . . 9 |- 2 e. CC
34	33	sqvali 10808	. . . . . . . 8 |- ( 2 ^ 2 ) = ( 2 x. 2 )
35	34	oveq2i 5985	. . . . . . 7 |- ( ( A ^ 2 ) / ( 2 ^ 2 ) ) = ( ( A ^ 2 ) / ( 2 x. 2 ) )
36	9, 23, 25	sqdivapd 10875	. . . . . . 7 |- ( ph -> ( ( A / 2 ) ^ 2 ) = ( ( A ^ 2 ) / ( 2 ^ 2 ) ) )
37	16, 23, 23, 25, 25	divdivap1d 8937	. . . . . . 7 |- ( ph -> ( ( ( A ^ 2 ) / 2 ) / 2 ) = ( ( A ^ 2 ) / ( 2 x. 2 ) ) )
38	35, 36, 37	3eqtr4a 2268	. . . . . 6 |- ( ph -> ( ( A / 2 ) ^ 2 ) = ( ( ( A ^ 2 ) / 2 ) / 2 ) )
39	27	oveq1d 5989	. . . . . 6 |- ( ph -> ( ( ( A ^ 2 ) / 2 ) / 2 ) = ( ( B ^ 2 ) / 2 ) )
40	38, 39	eqtrd 2242	. . . . 5 |- ( ph -> ( ( A / 2 ) ^ 2 ) = ( ( B ^ 2 ) / 2 ) )
41		zsqcl 10799	. . . . . 6 |- ( ( A / 2 ) e. ZZ -> ( ( A / 2 ) ^ 2 ) e. ZZ )
42	32, 41	syl 14	. . . . 5 |- ( ph -> ( ( A / 2 ) ^ 2 ) e. ZZ )
43	40, 42	eqeltrrd 2287	. . . 4 |- ( ph -> ( ( B ^ 2 ) / 2 ) e. ZZ )
44	17	nnrpd 9858	. . . . . 6 |- ( ph -> ( B ^ 2 ) e. RR+ )
45	44	rphalfcld 9873	. . . . 5 |- ( ph -> ( ( B ^ 2 ) / 2 ) e. RR+ )
46	45	rpgt0d 9863	. . . 4 |- ( ph -> 0 < ( ( B ^ 2 ) / 2 ) )
47		elnnz 9424	. . . 4 |- ( ( ( B ^ 2 ) / 2 ) e. NN <-> ( ( ( B ^ 2 ) / 2 ) e. ZZ /\ 0 < ( ( B ^ 2 ) / 2 ) ) )
48	43, 46, 47	sylanbrc 417	. . 3 |- ( ph -> ( ( B ^ 2 ) / 2 ) e. NN )
49		nnesq 10848	. . . 4 |- ( B e. NN -> ( ( B / 2 ) e. NN <-> ( ( B ^ 2 ) / 2 ) e. NN ) )
50	10, 49	syl 14	. . 3 |- ( ph -> ( ( B / 2 ) e. NN <-> ( ( B ^ 2 ) / 2 ) e. NN ) )
51	48, 50	mpbird 167	. 2 |- ( ph -> ( B / 2 ) e. NN )
52	32, 51	jca 306	1 |- ( ph -> ( ( A / 2 ) e. ZZ /\ ( B / 2 ) e. NN ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1375   e. wcel 2180   class class class wbr 4062   ` cfv 5294  (class class class)co 5974   RRcr 7966   0cc0 7967   x. cmul 7972   < clt 8149   <_ cle 8150   # cap 8696   / cdiv 8787   NNcn 9078   2c2 9129   ZZcz 9414   ^cexp 10727   sqrcsqrt 11473

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-iinf 4657  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-mulrcl 8066  ax-addcom 8067  ax-mulcom 8068  ax-addass 8069  ax-mulass 8070  ax-distr 8071  ax-i2m1 8072  ax-0lt1 8073  ax-1rid 8074  ax-0id 8075  ax-rnegex 8076  ax-precex 8077  ax-cnre 8078  ax-pre-ltirr 8079  ax-pre-ltwlin 8080  ax-pre-lttrn 8081  ax-pre-apti 8082  ax-pre-ltadd 8083  ax-pre-mulgt0 8084  ax-pre-mulext 8085  ax-arch 8086  ax-caucvg 8087

This theorem depends on definitions:  df-bi 117  df-dc 839  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-nel 2476  df-ral 2493  df-rex 2494  df-reu 2495  df-rmo 2496  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-if 3583  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-tr 4162  df-id 4361  df-po 4364  df-iso 4365  df-iord 4434  df-on 4436  df-ilim 4437  df-suc 4439  df-iom 4660  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-recs 6421  df-frec 6507  df-pnf 8151  df-mnf 8152  df-xr 8153  df-ltxr 8154  df-le 8155  df-sub 8287  df-neg 8288  df-reap 8690  df-ap 8697  df-div 8788  df-inn 9079  df-2 9137  df-3 9138  df-4 9139  df-n0 9338  df-z 9415  df-uz 9691  df-rp 9818  df-seqfrec 10637  df-exp 10728  df-rsqrt 11475

This theorem is referenced by:  sqrt2irr  12650