Theorem sqrt2irrlem 12691

Description: Lemma for sqrt2irr 12692. 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 9188	. . . . . . . . . . . 12 |- 2 e. RR
2		0le2 9208	. . . . . . . . . . . 12 |- 0 <_ 2
3		resqrtth 11550	. . . . . . . . . . . 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 6022	. . . . . . . . . . 11 |- ( ph -> ( ( sqr ` 2 ) ^ 2 ) = ( ( A / B ) ^ 2 ) )
7	4, 6	eqtr3id 2276	. . . . . . . . . 10 |- ( ph -> 2 = ( ( A / B ) ^ 2 ) )
8		sqrt2irrlem.1	. . . . . . . . . . . 12 |- ( ph -> A e. ZZ )
9	8	zcnd 9578	. . . . . . . . . . 11 |- ( ph -> A e. CC )
10		sqrt2irrlem.2	. . . . . . . . . . . 12 |- ( ph -> B e. NN )
11	10	nncnd 9132	. . . . . . . . . . 11 |- ( ph -> B e. CC )
12	10	nnap0d 9164	. . . . . . . . . . 11 |- ( ph -> B # 0 )
13	9, 11, 12	sqdivapd 10916	. . . . . . . . . 10 |- ( ph -> ( ( A / B ) ^ 2 ) = ( ( A ^ 2 ) / ( B ^ 2 ) ) )
14	7, 13	eqtrd 2262	. . . . . . . . 9 |- ( ph -> 2 = ( ( A ^ 2 ) / ( B ^ 2 ) ) )
15	14	oveq1d 6022	. . . . . . . 8 |- ( ph -> ( 2 x. ( B ^ 2 ) ) = ( ( ( A ^ 2 ) / ( B ^ 2 ) ) x. ( B ^ 2 ) ) )
16	9	sqcld 10901	. . . . . . . . 9 |- ( ph -> ( A ^ 2 ) e. CC )
17	10	nnsqcld 10924	. . . . . . . . . 10 |- ( ph -> ( B ^ 2 ) e. NN )
18	17	nncnd 9132	. . . . . . . . 9 |- ( ph -> ( B ^ 2 ) e. CC )
19	17	nnap0d 9164	. . . . . . . . 9 |- ( ph -> ( B ^ 2 ) # 0 )
20	16, 18, 19	divcanap1d 8946	. . . . . . . 8 |- ( ph -> ( ( ( A ^ 2 ) / ( B ^ 2 ) ) x. ( B ^ 2 ) ) = ( A ^ 2 ) )
21	15, 20	eqtrd 2262	. . . . . . 7 |- ( ph -> ( 2 x. ( B ^ 2 ) ) = ( A ^ 2 ) )
22	21	oveq1d 6022	. . . . . 6 |- ( ph -> ( ( 2 x. ( B ^ 2 ) ) / 2 ) = ( ( A ^ 2 ) / 2 ) )
23		2cnd 9191	. . . . . . 7 |- ( ph -> 2 e. CC )
24		2ap0 9211	. . . . . . . 8 |- 2 # 0
25	24	a1i 9	. . . . . . 7 |- ( ph -> 2 # 0 )
26	18, 23, 25	divcanap3d 8950	. . . . . 6 |- ( ph -> ( ( 2 x. ( B ^ 2 ) ) / 2 ) = ( B ^ 2 ) )
27	22, 26	eqtr3d 2264	. . . . 5 |- ( ph -> ( ( A ^ 2 ) / 2 ) = ( B ^ 2 ) )
28	27, 17	eqeltrd 2306	. . . 4 |- ( ph -> ( ( A ^ 2 ) / 2 ) e. NN )
29	28	nnzd 9576	. . 3 |- ( ph -> ( ( A ^ 2 ) / 2 ) e. ZZ )
30		zesq 10888	. . . 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 9189	. . . . . . . . 9 |- 2 e. CC
34	33	sqvali 10849	. . . . . . . 8 |- ( 2 ^ 2 ) = ( 2 x. 2 )
35	34	oveq2i 6018	. . . . . . 7 |- ( ( A ^ 2 ) / ( 2 ^ 2 ) ) = ( ( A ^ 2 ) / ( 2 x. 2 ) )
36	9, 23, 25	sqdivapd 10916	. . . . . . 7 |- ( ph -> ( ( A / 2 ) ^ 2 ) = ( ( A ^ 2 ) / ( 2 ^ 2 ) ) )
37	16, 23, 23, 25, 25	divdivap1d 8977	. . . . . . 7 |- ( ph -> ( ( ( A ^ 2 ) / 2 ) / 2 ) = ( ( A ^ 2 ) / ( 2 x. 2 ) ) )
38	35, 36, 37	3eqtr4a 2288	. . . . . 6 |- ( ph -> ( ( A / 2 ) ^ 2 ) = ( ( ( A ^ 2 ) / 2 ) / 2 ) )
39	27	oveq1d 6022	. . . . . 6 |- ( ph -> ( ( ( A ^ 2 ) / 2 ) / 2 ) = ( ( B ^ 2 ) / 2 ) )
40	38, 39	eqtrd 2262	. . . . 5 |- ( ph -> ( ( A / 2 ) ^ 2 ) = ( ( B ^ 2 ) / 2 ) )
41		zsqcl 10840	. . . . . 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 2307	. . . 4 |- ( ph -> ( ( B ^ 2 ) / 2 ) e. ZZ )
44	17	nnrpd 9898	. . . . . 6 |- ( ph -> ( B ^ 2 ) e. RR+ )
45	44	rphalfcld 9913	. . . . 5 |- ( ph -> ( ( B ^ 2 ) / 2 ) e. RR+ )
46	45	rpgt0d 9903	. . . 4 |- ( ph -> 0 < ( ( B ^ 2 ) / 2 ) )
47		elnnz 9464	. . . 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 10889	. . . 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 1395   e. wcel 2200   class class class wbr 4083   ` cfv 5318  (class class class)co 6007   RRcr 8006   0cc0 8007   x. cmul 8012   < clt 8189   <_ cle 8190   # cap 8736   / cdiv 8827   NNcn 9118   2c2 9169   ZZcz 9454   ^cexp 10768   sqrcsqrt 11515

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 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-mulrcl 8106  ax-addcom 8107  ax-mulcom 8108  ax-addass 8109  ax-mulass 8110  ax-distr 8111  ax-i2m1 8112  ax-0lt1 8113  ax-1rid 8114  ax-0id 8115  ax-rnegex 8116  ax-precex 8117  ax-cnre 8118  ax-pre-ltirr 8119  ax-pre-ltwlin 8120  ax-pre-lttrn 8121  ax-pre-apti 8122  ax-pre-ltadd 8123  ax-pre-mulgt0 8124  ax-pre-mulext 8125  ax-arch 8126  ax-caucvg 8127

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  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 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-pnf 8191  df-mnf 8192  df-xr 8193  df-ltxr 8194  df-le 8195  df-sub 8327  df-neg 8328  df-reap 8730  df-ap 8737  df-div 8828  df-inn 9119  df-2 9177  df-3 9178  df-4 9179  df-n0 9378  df-z 9455  df-uz 9731  df-rp 9858  df-seqfrec 10678  df-exp 10769  df-rsqrt 11517

This theorem is referenced by:  sqrt2irr  12692