Theorem nn0opthlem1d 10985

Description: A rather pretty lemma for nn0opth2 10989. (Contributed by Jim Kingdon, 31-Oct-2021.)

Hypotheses

Ref	Expression
nn0opthlem1d.1	|- ( ph -> A e. NN0 )
nn0opthlem1d.2	|- ( ph -> C e. NN0 )

Assertion

Ref	Expression
nn0opthlem1d	|- ( ph -> ( A < C <-> ( ( A x. A ) + ( 2 x. A ) ) < ( C x. C ) ) )

Proof of Theorem nn0opthlem1d

Step	Hyp	Ref	Expression
1		nn0opthlem1d.1	. . . 4 |- ( ph -> A e. NN0 )
2		1nn0 9420	. . . . 5 |- 1 e. NN0
3	2	a1i 9	. . . 4 |- ( ph -> 1 e. NN0 )
4	1, 3	nn0addcld 9461	. . 3 |- ( ph -> ( A + 1 ) e. NN0 )
5		nn0opthlem1d.2	. . 3 |- ( ph -> C e. NN0 )
6	4, 5	nn0le2msqd 10984	. 2 |- ( ph -> ( ( A + 1 ) <_ C <-> ( ( A + 1 ) x. ( A + 1 ) ) <_ ( C x. C ) ) )
7		nn0ltp1le 9544	. . 3 |- ( ( A e. NN0 /\ C e. NN0 ) -> ( A < C <-> ( A + 1 ) <_ C ) )
8	1, 5, 7	syl2anc 411	. 2 |- ( ph -> ( A < C <-> ( A + 1 ) <_ C ) )
9	1, 1	nn0mulcld 9462	. . . . 5 |- ( ph -> ( A x. A ) e. NN0 )
10		2nn0 9421	. . . . . . 7 |- 2 e. NN0
11	10	a1i 9	. . . . . 6 |- ( ph -> 2 e. NN0 )
12	11, 1	nn0mulcld 9462	. . . . 5 |- ( ph -> ( 2 x. A ) e. NN0 )
13	9, 12	nn0addcld 9461	. . . 4 |- ( ph -> ( ( A x. A ) + ( 2 x. A ) ) e. NN0 )
14	5, 5	nn0mulcld 9462	. . . 4 |- ( ph -> ( C x. C ) e. NN0 )
15		nn0ltp1le 9544	. . . 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 ) ) )
16	13, 14, 15	syl2anc 411	. . 3 |- ( ph -> ( ( ( A x. A ) + ( 2 x. A ) ) < ( C x. C ) <-> ( ( ( A x. A ) + ( 2 x. A ) ) + 1 ) <_ ( C x. C ) ) )
17	1	nn0cnd 9459	. . . . . . 7 |- ( ph -> A e. CC )
18		1cnd 8197	. . . . . . 7 |- ( ph -> 1 e. CC )
19		binom2 10916	. . . . . . 7 |- ( ( A e. CC /\ 1 e. CC ) -> ( ( A + 1 ) ^ 2 ) = ( ( ( A ^ 2 ) + ( 2 x. ( A x. 1 ) ) ) + ( 1 ^ 2 ) ) )
20	17, 18, 19	syl2anc 411	. . . . . 6 |- ( ph -> ( ( A + 1 ) ^ 2 ) = ( ( ( A ^ 2 ) + ( 2 x. ( A x. 1 ) ) ) + ( 1 ^ 2 ) ) )
21	17, 18	addcld 8201	. . . . . . 7 |- ( ph -> ( A + 1 ) e. CC )
22	21	sqvald 10935	. . . . . 6 |- ( ph -> ( ( A + 1 ) ^ 2 ) = ( ( A + 1 ) x. ( A + 1 ) ) )
23	17	sqvald 10935	. . . . . . . 8 |- ( ph -> ( A ^ 2 ) = ( A x. A ) )
24	23	oveq1d 6035	. . . . . . 7 |- ( ph -> ( ( A ^ 2 ) + ( 2 x. ( A x. 1 ) ) ) = ( ( A x. A ) + ( 2 x. ( A x. 1 ) ) ) )
25	18	sqvald 10935	. . . . . . 7 |- ( ph -> ( 1 ^ 2 ) = ( 1 x. 1 ) )
26	24, 25	oveq12d 6038	. . . . . 6 |- ( ph -> ( ( ( A ^ 2 ) + ( 2 x. ( A x. 1 ) ) ) + ( 1 ^ 2 ) ) = ( ( ( A x. A ) + ( 2 x. ( A x. 1 ) ) ) + ( 1 x. 1 ) ) )
27	20, 22, 26	3eqtr3d 2271	. . . . 5 |- ( ph -> ( ( A + 1 ) x. ( A + 1 ) ) = ( ( ( A x. A ) + ( 2 x. ( A x. 1 ) ) ) + ( 1 x. 1 ) ) )
28	17	mulridd 8198	. . . . . . . 8 |- ( ph -> ( A x. 1 ) = A )
29	28	oveq2d 6036	. . . . . . 7 |- ( ph -> ( 2 x. ( A x. 1 ) ) = ( 2 x. A ) )
30	29	oveq2d 6036	. . . . . 6 |- ( ph -> ( ( A x. A ) + ( 2 x. ( A x. 1 ) ) ) = ( ( A x. A ) + ( 2 x. A ) ) )
31	18	mulridd 8198	. . . . . 6 |- ( ph -> ( 1 x. 1 ) = 1 )
32	30, 31	oveq12d 6038	. . . . 5 |- ( ph -> ( ( ( A x. A ) + ( 2 x. ( A x. 1 ) ) ) + ( 1 x. 1 ) ) = ( ( ( A x. A ) + ( 2 x. A ) ) + 1 ) )
33	27, 32	eqtrd 2263	. . . 4 |- ( ph -> ( ( A + 1 ) x. ( A + 1 ) ) = ( ( ( A x. A ) + ( 2 x. A ) ) + 1 ) )
34	33	breq1d 4097	. . 3 |- ( ph -> ( ( ( A + 1 ) x. ( A + 1 ) ) <_ ( C x. C ) <-> ( ( ( A x. A ) + ( 2 x. A ) ) + 1 ) <_ ( C x. C ) ) )
35	16, 34	bitr4d 191	. 2 |- ( ph -> ( ( ( A x. A ) + ( 2 x. A ) ) < ( C x. C ) <-> ( ( A + 1 ) x. ( A + 1 ) ) <_ ( C x. C ) ) )
36	6, 8, 35	3bitr4d 220	1 |- ( ph -> ( A < C <-> ( ( A x. A ) + ( 2 x. A ) ) < ( C x. C ) ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1397   e. wcel 2201   class class class wbr 4087  (class class class)co 6020   CCcc 8032   1c1 8035   + caddc 8037   x. cmul 8039   < clt 8216   <_ cle 8217   2c2 9196   NN0cn0 9404   ^cexp 10803

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4203  ax-sep 4206  ax-nul 4214  ax-pow 4263  ax-pr 4298  ax-un 4529  ax-setind 4634  ax-iinf 4685  ax-cnex 8125  ax-resscn 8126  ax-1cn 8127  ax-1re 8128  ax-icn 8129  ax-addcl 8130  ax-addrcl 8131  ax-mulcl 8132  ax-mulrcl 8133  ax-addcom 8134  ax-mulcom 8135  ax-addass 8136  ax-mulass 8137  ax-distr 8138  ax-i2m1 8139  ax-0lt1 8140  ax-1rid 8141  ax-0id 8142  ax-rnegex 8143  ax-precex 8144  ax-cnre 8145  ax-pre-ltirr 8146  ax-pre-ltwlin 8147  ax-pre-lttrn 8148  ax-pre-apti 8149  ax-pre-ltadd 8150  ax-pre-mulgt0 8151  ax-pre-mulext 8152

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-reu 2516  df-rmo 2517  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-if 3605  df-pw 3653  df-sn 3674  df-pr 3675  df-op 3677  df-uni 3893  df-int 3928  df-iun 3971  df-br 4088  df-opab 4150  df-mpt 4151  df-tr 4187  df-id 4389  df-po 4392  df-iso 4393  df-iord 4462  df-on 4464  df-ilim 4465  df-suc 4467  df-iom 4688  df-xp 4730  df-rel 4731  df-cnv 4732  df-co 4733  df-dm 4734  df-rn 4735  df-res 4736  df-ima 4737  df-iota 5285  df-fun 5327  df-fn 5328  df-f 5329  df-f1 5330  df-fo 5331  df-f1o 5332  df-fv 5333  df-riota 5973  df-ov 6023  df-oprab 6024  df-mpo 6025  df-1st 6305  df-2nd 6306  df-recs 6473  df-frec 6559  df-pnf 8218  df-mnf 8219  df-xr 8220  df-ltxr 8221  df-le 8222  df-sub 8354  df-neg 8355  df-reap 8757  df-ap 8764  df-div 8855  df-inn 9146  df-2 9204  df-n0 9405  df-z 9482  df-uz 9758  df-seqfrec 10713  df-exp 10804

This theorem is referenced by:  nn0opthlem2d  10986