Theorem rmspecsqrtnqOLD 37471

Description: Obsolete version of rmspecsqrtnq 37470 as of 2-Aug-2021. (Contributed by Stefan O'Rear, 21-Sep-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
rmspecsqrtnqOLD	|- ( A e. ( ZZ>= ` 2 ) -> ( sqr ` ( ( A ^ 2 ) - 1 ) ) e. ( CC \ QQ ) )

Proof of Theorem rmspecsqrtnqOLD

Step	Hyp	Ref	Expression
1		eluzelcn 11699	. . . . 5 |- ( A e. ( ZZ>= ` 2 ) -> A e. CC )
2	1	sqcld 13006	. . . 4 |- ( A e. ( ZZ>= ` 2 ) -> ( A ^ 2 ) e. CC )
3		ax-1cn 9994	. . . 4 |- 1 e. CC
4		subcl 10280	. . . 4 |- ( ( ( A ^ 2 ) e. CC /\ 1 e. CC ) -> ( ( A ^ 2 ) - 1 ) e. CC )
5	2, 3, 4	sylancl 694	. . 3 |- ( A e. ( ZZ>= ` 2 ) -> ( ( A ^ 2 ) - 1 ) e. CC )
6	5	sqrtcld 14176	. 2 |- ( A e. ( ZZ>= ` 2 ) -> ( sqr ` ( ( A ^ 2 ) - 1 ) ) e. CC )
7		eluz2nn 11726	. . . . 5 |- ( A e. ( ZZ>= ` 2 ) -> A e. NN )
8	7	nnsqcld 13029	. . . 4 |- ( A e. ( ZZ>= ` 2 ) -> ( A ^ 2 ) e. NN )
9		nnm1nn0 11334	. . . 4 |- ( ( A ^ 2 ) e. NN -> ( ( A ^ 2 ) - 1 ) e. NN0 )
10	8, 9	syl 17	. . 3 |- ( A e. ( ZZ>= ` 2 ) -> ( ( A ^ 2 ) - 1 ) e. NN0 )
11		nnm1nn0 11334	. . . 4 |- ( A e. NN -> ( A - 1 ) e. NN0 )
12	7, 11	syl 17	. . 3 |- ( A e. ( ZZ>= ` 2 ) -> ( A - 1 ) e. NN0 )
13		binom2sub 12981	. . . . . 6 |- ( ( A e. CC /\ 1 e. CC ) -> ( ( A - 1 ) ^ 2 ) = ( ( ( A ^ 2 ) - ( 2 x. ( A x. 1 ) ) ) + ( 1 ^ 2 ) ) )
14	1, 3, 13	sylancl 694	. . . . 5 |- ( A e. ( ZZ>= ` 2 ) -> ( ( A - 1 ) ^ 2 ) = ( ( ( A ^ 2 ) - ( 2 x. ( A x. 1 ) ) ) + ( 1 ^ 2 ) ) )
15		2re 11090	. . . . . . . 8 |- 2 e. RR
16		eluzelre 11698	. . . . . . . . 9 |- ( A e. ( ZZ>= ` 2 ) -> A e. RR )
17		1re 10039	. . . . . . . . 9 |- 1 e. RR
18		remulcl 10021	. . . . . . . . 9 |- ( ( A e. RR /\ 1 e. RR ) -> ( A x. 1 ) e. RR )
19	16, 17, 18	sylancl 694	. . . . . . . 8 |- ( A e. ( ZZ>= ` 2 ) -> ( A x. 1 ) e. RR )
20		remulcl 10021	. . . . . . . 8 |- ( ( 2 e. RR /\ ( A x. 1 ) e. RR ) -> ( 2 x. ( A x. 1 ) ) e. RR )
21	15, 19, 20	sylancr 695	. . . . . . 7 |- ( A e. ( ZZ>= ` 2 ) -> ( 2 x. ( A x. 1 ) ) e. RR )
22	21	recnd 10068	. . . . . 6 |- ( A e. ( ZZ>= ` 2 ) -> ( 2 x. ( A x. 1 ) ) e. CC )
23	17	resqcli 12949	. . . . . . . 8 |- ( 1 ^ 2 ) e. RR
24	23	recni 10052	. . . . . . 7 |- ( 1 ^ 2 ) e. CC
25	24	a1i 11	. . . . . 6 |- ( A e. ( ZZ>= ` 2 ) -> ( 1 ^ 2 ) e. CC )
26	2, 22, 25	subsubd 10420	. . . . 5 |- ( A e. ( ZZ>= ` 2 ) -> ( ( A ^ 2 ) - ( ( 2 x. ( A x. 1 ) ) - ( 1 ^ 2 ) ) ) = ( ( ( A ^ 2 ) - ( 2 x. ( A x. 1 ) ) ) + ( 1 ^ 2 ) ) )
27	14, 26	eqtr4d 2659	. . . 4 |- ( A e. ( ZZ>= ` 2 ) -> ( ( A - 1 ) ^ 2 ) = ( ( A ^ 2 ) - ( ( 2 x. ( A x. 1 ) ) - ( 1 ^ 2 ) ) ) )
28	17	a1i 11	. . . . 5 |- ( A e. ( ZZ>= ` 2 ) -> 1 e. RR )
29		resubcl 10345	. . . . . 6 |- ( ( ( 2 x. ( A x. 1 ) ) e. RR /\ ( 1 ^ 2 ) e. RR ) -> ( ( 2 x. ( A x. 1 ) ) - ( 1 ^ 2 ) ) e. RR )
30	21, 23, 29	sylancl 694	. . . . 5 |- ( A e. ( ZZ>= ` 2 ) -> ( ( 2 x. ( A x. 1 ) ) - ( 1 ^ 2 ) ) e. RR )
31	8	nnred 11035	. . . . 5 |- ( A e. ( ZZ>= ` 2 ) -> ( A ^ 2 ) e. RR )
32	3	2timesi 11147	. . . . . . . 8 |- ( 2 x. 1 ) = ( 1 + 1 )
33		eluz2b2 11761	. . . . . . . . . 10 |- ( A e. ( ZZ>= ` 2 ) <-> ( A e. NN /\ 1 < A ) )
34	33	simprbi 480	. . . . . . . . 9 |- ( A e. ( ZZ>= ` 2 ) -> 1 < A )
35	15	a1i 11	. . . . . . . . . 10 |- ( A e. ( ZZ>= ` 2 ) -> 2 e. RR )
36		2pos 11112	. . . . . . . . . . 11 |- 0 < 2
37	36	a1i 11	. . . . . . . . . 10 |- ( A e. ( ZZ>= ` 2 ) -> 0 < 2 )
38		ltmul2 10874	. . . . . . . . . 10 |- ( ( 1 e. RR /\ A e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( 1 < A <-> ( 2 x. 1 ) < ( 2 x. A ) ) )
39	28, 16, 35, 37, 38	syl112anc 1330	. . . . . . . . 9 |- ( A e. ( ZZ>= ` 2 ) -> ( 1 < A <-> ( 2 x. 1 ) < ( 2 x. A ) ) )
40	34, 39	mpbid 222	. . . . . . . 8 |- ( A e. ( ZZ>= ` 2 ) -> ( 2 x. 1 ) < ( 2 x. A ) )
41	32, 40	syl5eqbrr 4689	. . . . . . 7 |- ( A e. ( ZZ>= ` 2 ) -> ( 1 + 1 ) < ( 2 x. A ) )
42		remulcl 10021	. . . . . . . . 9 |- ( ( 2 e. RR /\ A e. RR ) -> ( 2 x. A ) e. RR )
43	15, 16, 42	sylancr 695	. . . . . . . 8 |- ( A e. ( ZZ>= ` 2 ) -> ( 2 x. A ) e. RR )
44	28, 28, 43	ltaddsubd 10627	. . . . . . 7 |- ( A e. ( ZZ>= ` 2 ) -> ( ( 1 + 1 ) < ( 2 x. A ) <-> 1 < ( ( 2 x. A ) - 1 ) ) )
45	41, 44	mpbid 222	. . . . . 6 |- ( A e. ( ZZ>= ` 2 ) -> 1 < ( ( 2 x. A ) - 1 ) )
46	1	mulid1d 10057	. . . . . . . 8 |- ( A e. ( ZZ>= ` 2 ) -> ( A x. 1 ) = A )
47	46	oveq2d 6666	. . . . . . 7 |- ( A e. ( ZZ>= ` 2 ) -> ( 2 x. ( A x. 1 ) ) = ( 2 x. A ) )
48		sq1 12958	. . . . . . . 8 |- ( 1 ^ 2 ) = 1
49	48	a1i 11	. . . . . . 7 |- ( A e. ( ZZ>= ` 2 ) -> ( 1 ^ 2 ) = 1 )
50	47, 49	oveq12d 6668	. . . . . 6 |- ( A e. ( ZZ>= ` 2 ) -> ( ( 2 x. ( A x. 1 ) ) - ( 1 ^ 2 ) ) = ( ( 2 x. A ) - 1 ) )
51	45, 50	breqtrrd 4681	. . . . 5 |- ( A e. ( ZZ>= ` 2 ) -> 1 < ( ( 2 x. ( A x. 1 ) ) - ( 1 ^ 2 ) ) )
52	28, 30, 31, 51	ltsub2dd 10640	. . . 4 |- ( A e. ( ZZ>= ` 2 ) -> ( ( A ^ 2 ) - ( ( 2 x. ( A x. 1 ) ) - ( 1 ^ 2 ) ) ) < ( ( A ^ 2 ) - 1 ) )
53	27, 52	eqbrtrd 4675	. . 3 |- ( A e. ( ZZ>= ` 2 ) -> ( ( A - 1 ) ^ 2 ) < ( ( A ^ 2 ) - 1 ) )
54	31	ltm1d 10956	. . . 4 |- ( A e. ( ZZ>= ` 2 ) -> ( ( A ^ 2 ) - 1 ) < ( A ^ 2 ) )
55		npcan 10290	. . . . . 6 |- ( ( A e. CC /\ 1 e. CC ) -> ( ( A - 1 ) + 1 ) = A )
56	1, 3, 55	sylancl 694	. . . . 5 |- ( A e. ( ZZ>= ` 2 ) -> ( ( A - 1 ) + 1 ) = A )
57	56	oveq1d 6665	. . . 4 |- ( A e. ( ZZ>= ` 2 ) -> ( ( ( A - 1 ) + 1 ) ^ 2 ) = ( A ^ 2 ) )
58	54, 57	breqtrrd 4681	. . 3 |- ( A e. ( ZZ>= ` 2 ) -> ( ( A ^ 2 ) - 1 ) < ( ( ( A - 1 ) + 1 ) ^ 2 ) )
59		nonsq 15467	. . 3 |- ( ( ( ( ( A ^ 2 ) - 1 ) e. NN0 /\ ( A - 1 ) e. NN0 ) /\ ( ( ( A - 1 ) ^ 2 ) < ( ( A ^ 2 ) - 1 ) /\ ( ( A ^ 2 ) - 1 ) < ( ( ( A - 1 ) + 1 ) ^ 2 ) ) ) -> -. ( sqr ` ( ( A ^ 2 ) - 1 ) ) e. QQ )
60	10, 12, 53, 58, 59	syl22anc 1327	. 2 |- ( A e. ( ZZ>= ` 2 ) -> -. ( sqr ` ( ( A ^ 2 ) - 1 ) ) e. QQ )
61	6, 60	eldifd 3585	1 |- ( A e. ( ZZ>= ` 2 ) -> ( sqr ` ( ( A ^ 2 ) - 1 ) ) e. ( CC \ QQ ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   \ cdif 3571   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   - cmin 10266   NNcn 11020   2c2 11070   NN0cn0 11292   ZZ>=cuz 11687   QQcq 11788   ^cexp 12860   sqrcsqrt 13973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-numer 15443  df-denom 15444

This theorem is referenced by: (None)