Theorem nn0opthlem1d 10622

Description: A rather pretty lemma for nn0opth2 10626. (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 9121	. . . . 5 |- 1 e. NN0
3	2	a1i 9	. . . 4 |- ( ph -> 1 e. NN0 )
4	1, 3	nn0addcld 9162	. . 3 |- ( ph -> ( A + 1 ) e. NN0 )
5		nn0opthlem1d.2	. . 3 |- ( ph -> C e. NN0 )
6	4, 5	nn0le2msqd 10621	. 2 |- ( ph -> ( ( A + 1 ) <_ C <-> ( ( A + 1 ) x. ( A + 1 ) ) <_ ( C x. C ) ) )
7		nn0ltp1le 9244	. . 3 |- ( ( A e. NN0 /\ C e. NN0 ) -> ( A < C <-> ( A + 1 ) <_ C ) )
8	1, 5, 7	syl2anc 409	. 2 |- ( ph -> ( A < C <-> ( A + 1 ) <_ C ) )
9	1, 1	nn0mulcld 9163	. . . . 5 |- ( ph -> ( A x. A ) e. NN0 )
10		2nn0 9122	. . . . . . 7 |- 2 e. NN0
11	10	a1i 9	. . . . . 6 |- ( ph -> 2 e. NN0 )
12	11, 1	nn0mulcld 9163	. . . . 5 |- ( ph -> ( 2 x. A ) e. NN0 )
13	9, 12	nn0addcld 9162	. . . 4 |- ( ph -> ( ( A x. A ) + ( 2 x. A ) ) e. NN0 )
14	5, 5	nn0mulcld 9163	. . . 4 |- ( ph -> ( C x. C ) e. NN0 )
15		nn0ltp1le 9244	. . . 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 409	. . 3 |- ( ph -> ( ( ( A x. A ) + ( 2 x. A ) ) < ( C x. C ) <-> ( ( ( A x. A ) + ( 2 x. A ) ) + 1 ) <_ ( C x. C ) ) )
17	1	nn0cnd 9160	. . . . . . 7 |- ( ph -> A e. CC )
18		1cnd 7906	. . . . . . 7 |- ( ph -> 1 e. CC )
19		binom2 10555	. . . . . . 7 |- ( ( A e. CC /\ 1 e. CC ) -> ( ( A + 1 ) ^ 2 ) = ( ( ( A ^ 2 ) + ( 2 x. ( A x. 1 ) ) ) + ( 1 ^ 2 ) ) )
20	17, 18, 19	syl2anc 409	. . . . . 6 |- ( ph -> ( ( A + 1 ) ^ 2 ) = ( ( ( A ^ 2 ) + ( 2 x. ( A x. 1 ) ) ) + ( 1 ^ 2 ) ) )
21	17, 18	addcld 7909	. . . . . . 7 |- ( ph -> ( A + 1 ) e. CC )
22	21	sqvald 10574	. . . . . 6 |- ( ph -> ( ( A + 1 ) ^ 2 ) = ( ( A + 1 ) x. ( A + 1 ) ) )
23	17	sqvald 10574	. . . . . . . 8 |- ( ph -> ( A ^ 2 ) = ( A x. A ) )
24	23	oveq1d 5851	. . . . . . 7 |- ( ph -> ( ( A ^ 2 ) + ( 2 x. ( A x. 1 ) ) ) = ( ( A x. A ) + ( 2 x. ( A x. 1 ) ) ) )
25	18	sqvald 10574	. . . . . . 7 |- ( ph -> ( 1 ^ 2 ) = ( 1 x. 1 ) )
26	24, 25	oveq12d 5854	. . . . . 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 2205	. . . . 5 |- ( ph -> ( ( A + 1 ) x. ( A + 1 ) ) = ( ( ( A x. A ) + ( 2 x. ( A x. 1 ) ) ) + ( 1 x. 1 ) ) )
28	17	mulid1d 7907	. . . . . . . 8 |- ( ph -> ( A x. 1 ) = A )
29	28	oveq2d 5852	. . . . . . 7 |- ( ph -> ( 2 x. ( A x. 1 ) ) = ( 2 x. A ) )
30	29	oveq2d 5852	. . . . . 6 |- ( ph -> ( ( A x. A ) + ( 2 x. ( A x. 1 ) ) ) = ( ( A x. A ) + ( 2 x. A ) ) )
31	18	mulid1d 7907	. . . . . 6 |- ( ph -> ( 1 x. 1 ) = 1 )
32	30, 31	oveq12d 5854	. . . . 5 |- ( ph -> ( ( ( A x. A ) + ( 2 x. ( A x. 1 ) ) ) + ( 1 x. 1 ) ) = ( ( ( A x. A ) + ( 2 x. A ) ) + 1 ) )
33	27, 32	eqtrd 2197	. . . 4 |- ( ph -> ( ( A + 1 ) x. ( A + 1 ) ) = ( ( ( A x. A ) + ( 2 x. A ) ) + 1 ) )
34	33	breq1d 3986	. . 3 |- ( ph -> ( ( ( A + 1 ) x. ( A + 1 ) ) <_ ( C x. C ) <-> ( ( ( A x. A ) + ( 2 x. A ) ) + 1 ) <_ ( C x. C ) ) )
35	16, 34	bitr4d 190	. 2 |- ( ph -> ( ( ( A x. A ) + ( 2 x. A ) ) < ( C x. C ) <-> ( ( A + 1 ) x. ( A + 1 ) ) <_ ( C x. C ) ) )
36	6, 8, 35	3bitr4d 219	1 |- ( ph -> ( A < C <-> ( ( A x. A ) + ( 2 x. A ) ) < ( C x. C ) ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1342   e. wcel 2135   class class class wbr 3976  (class class class)co 5836   CCcc 7742   1c1 7745   + caddc 7747   x. cmul 7749   < clt 7924   <_ cle 7925   2c2 8899   NN0cn0 9105   ^cexp 10444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4091  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-iinf 4559  ax-cnex 7835  ax-resscn 7836  ax-1cn 7837  ax-1re 7838  ax-icn 7839  ax-addcl 7840  ax-addrcl 7841  ax-mulcl 7842  ax-mulrcl 7843  ax-addcom 7844  ax-mulcom 7845  ax-addass 7846  ax-mulass 7847  ax-distr 7848  ax-i2m1 7849  ax-0lt1 7850  ax-1rid 7851  ax-0id 7852  ax-rnegex 7853  ax-precex 7854  ax-cnre 7855  ax-pre-ltirr 7856  ax-pre-ltwlin 7857  ax-pre-lttrn 7858  ax-pre-apti 7859  ax-pre-ltadd 7860  ax-pre-mulgt0 7861  ax-pre-mulext 7862

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rmo 2450  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-if 3516  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-tr 4075  df-id 4265  df-po 4268  df-iso 4269  df-iord 4338  df-on 4340  df-ilim 4341  df-suc 4343  df-iom 4562  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-riota 5792  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1st 6100  df-2nd 6101  df-recs 6264  df-frec 6350  df-pnf 7926  df-mnf 7927  df-xr 7928  df-ltxr 7929  df-le 7930  df-sub 8062  df-neg 8063  df-reap 8464  df-ap 8471  df-div 8560  df-inn 8849  df-2 8907  df-n0 9106  df-z 9183  df-uz 9458  df-seqfrec 10371  df-exp 10445

This theorem is referenced by:  nn0opthlem2d  10623