Theorem nn0opthlem1 13055

Description: A rather pretty lemma for nn0opthi 13057. (Contributed by Raph Levien, 10-Dec-2002.)

Hypotheses

Ref	Expression
nn0opthlem1.1	|- A e. NN0
nn0opthlem1.2	|- C e. NN0

Assertion

Ref	Expression
nn0opthlem1	|- ( A < C <-> ( ( A x. A ) + ( 2 x. A ) ) < ( C x. C ) )

Proof of Theorem nn0opthlem1

Step	Hyp	Ref	Expression
1		nn0opthlem1.1	. . . 4 |- A e. NN0
2		1nn0 11308	. . . 4 |- 1 e. NN0
3	1, 2	nn0addcli 11330	. . 3 |- ( A + 1 ) e. NN0
4		nn0opthlem1.2	. . 3 |- C e. NN0
5	3, 4	nn0le2msqi 13054	. 2 |- ( ( A + 1 ) <_ C <-> ( ( A + 1 ) x. ( A + 1 ) ) <_ ( C x. C ) )
6		nn0ltp1le 11435	. . 3 |- ( ( A e. NN0 /\ C e. NN0 ) -> ( A < C <-> ( A + 1 ) <_ C ) )
7	1, 4, 6	mp2an 708	. 2 |- ( A < C <-> ( A + 1 ) <_ C )
8	1, 1	nn0mulcli 11331	. . . . 5 |- ( A x. A ) e. NN0
9		2nn0 11309	. . . . . 6 |- 2 e. NN0
10	9, 1	nn0mulcli 11331	. . . . 5 |- ( 2 x. A ) e. NN0
11	8, 10	nn0addcli 11330	. . . 4 |- ( ( A x. A ) + ( 2 x. A ) ) e. NN0
12	4, 4	nn0mulcli 11331	. . . 4 |- ( C x. C ) e. NN0
13		nn0ltp1le 11435	. . . 4 |- ( ( ( ( A x. A ) + ( 2 x. A ) ) e. NN0 /\ ( C x. C ) e. NN0 ) -> ( ( ( A x. A ) + ( 2 x. A ) ) < ( C x. C ) <-> ( ( ( A x. A ) + ( 2 x. A ) ) + 1 ) <_ ( C x. C ) ) )
14	11, 12, 13	mp2an 708	. . 3 |- ( ( ( A x. A ) + ( 2 x. A ) ) < ( C x. C ) <-> ( ( ( A x. A ) + ( 2 x. A ) ) + 1 ) <_ ( C x. C ) )
15	1	nn0cni 11304	. . . . . . 7 |- A e. CC
16		ax-1cn 9994	. . . . . . 7 |- 1 e. CC
17	15, 16	binom2i 12974	. . . . . 6 |- ( ( A + 1 ) ^ 2 ) = ( ( ( A ^ 2 ) + ( 2 x. ( A x. 1 ) ) ) + ( 1 ^ 2 ) )
18	15, 16	addcli 10044	. . . . . . 7 |- ( A + 1 ) e. CC
19	18	sqvali 12943	. . . . . 6 |- ( ( A + 1 ) ^ 2 ) = ( ( A + 1 ) x. ( A + 1 ) )
20	15	sqvali 12943	. . . . . . . 8 |- ( A ^ 2 ) = ( A x. A )
21	20	oveq1i 6660	. . . . . . 7 |- ( ( A ^ 2 ) + ( 2 x. ( A x. 1 ) ) ) = ( ( A x. A ) + ( 2 x. ( A x. 1 ) ) )
22	16	sqvali 12943	. . . . . . 7 |- ( 1 ^ 2 ) = ( 1 x. 1 )
23	21, 22	oveq12i 6662	. . . . . 6 |- ( ( ( A ^ 2 ) + ( 2 x. ( A x. 1 ) ) ) + ( 1 ^ 2 ) ) = ( ( ( A x. A ) + ( 2 x. ( A x. 1 ) ) ) + ( 1 x. 1 ) )
24	17, 19, 23	3eqtr3i 2652	. . . . 5 |- ( ( A + 1 ) x. ( A + 1 ) ) = ( ( ( A x. A ) + ( 2 x. ( A x. 1 ) ) ) + ( 1 x. 1 ) )
25	15	mulid1i 10042	. . . . . . . 8 |- ( A x. 1 ) = A
26	25	oveq2i 6661	. . . . . . 7 |- ( 2 x. ( A x. 1 ) ) = ( 2 x. A )
27	26	oveq2i 6661	. . . . . 6 |- ( ( A x. A ) + ( 2 x. ( A x. 1 ) ) ) = ( ( A x. A ) + ( 2 x. A ) )
28	16	mulid1i 10042	. . . . . 6 |- ( 1 x. 1 ) = 1
29	27, 28	oveq12i 6662	. . . . 5 |- ( ( ( A x. A ) + ( 2 x. ( A x. 1 ) ) ) + ( 1 x. 1 ) ) = ( ( ( A x. A ) + ( 2 x. A ) ) + 1 )
30	24, 29	eqtri 2644	. . . 4 |- ( ( A + 1 ) x. ( A + 1 ) ) = ( ( ( A x. A ) + ( 2 x. A ) ) + 1 )
31	30	breq1i 4660	. . 3 |- ( ( ( A + 1 ) x. ( A + 1 ) ) <_ ( C x. C ) <-> ( ( ( A x. A ) + ( 2 x. A ) ) + 1 ) <_ ( C x. C ) )
32	14, 31	bitr4i 267	. 2 |- ( ( ( A x. A ) + ( 2 x. A ) ) < ( C x. C ) <-> ( ( A + 1 ) x. ( A + 1 ) ) <_ ( C x. C ) )
33	5, 7, 32	3bitr4i 292	1 |- ( A < C <-> ( ( A x. A ) + ( 2 x. A ) ) < ( C x. C ) )

Syntax hints:   <-> wb 196   e. wcel 1990   class class class wbr 4653  (class class class)co 6650   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   2c2 11070   NN0cn0 11292   ^cexp 12860

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

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-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-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-seq 12802  df-exp 12861

This theorem is referenced by:  nn0opthlem2  13056