Theorem nn0opthlem1d 10865

Description: A rather pretty lemma for nn0opth2 10869. (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 9311	. . . . 5 |- 1 e. NN0
3	2	a1i 9	. . . 4 |- ( ph -> 1 e. NN0 )
4	1, 3	nn0addcld 9352	. . 3 |- ( ph -> ( A + 1 ) e. NN0 )
5		nn0opthlem1d.2	. . 3 |- ( ph -> C e. NN0 )
6	4, 5	nn0le2msqd 10864	. 2 |- ( ph -> ( ( A + 1 ) <_ C <-> ( ( A + 1 ) x. ( A + 1 ) ) <_ ( C x. C ) ) )
7		nn0ltp1le 9435	. . 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 9353	. . . . 5 |- ( ph -> ( A x. A ) e. NN0 )
10		2nn0 9312	. . . . . . 7 |- 2 e. NN0
11	10	a1i 9	. . . . . 6 |- ( ph -> 2 e. NN0 )
12	11, 1	nn0mulcld 9353	. . . . 5 |- ( ph -> ( 2 x. A ) e. NN0 )
13	9, 12	nn0addcld 9352	. . . 4 |- ( ph -> ( ( A x. A ) + ( 2 x. A ) ) e. NN0 )
14	5, 5	nn0mulcld 9353	. . . 4 |- ( ph -> ( C x. C ) e. NN0 )
15		nn0ltp1le 9435	. . . 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 9350	. . . . . . 7 |- ( ph -> A e. CC )
18		1cnd 8088	. . . . . . 7 |- ( ph -> 1 e. CC )
19		binom2 10796	. . . . . . 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 8092	. . . . . . 7 |- ( ph -> ( A + 1 ) e. CC )
22	21	sqvald 10815	. . . . . 6 |- ( ph -> ( ( A + 1 ) ^ 2 ) = ( ( A + 1 ) x. ( A + 1 ) ) )
23	17	sqvald 10815	. . . . . . . 8 |- ( ph -> ( A ^ 2 ) = ( A x. A ) )
24	23	oveq1d 5959	. . . . . . 7 |- ( ph -> ( ( A ^ 2 ) + ( 2 x. ( A x. 1 ) ) ) = ( ( A x. A ) + ( 2 x. ( A x. 1 ) ) ) )
25	18	sqvald 10815	. . . . . . 7 |- ( ph -> ( 1 ^ 2 ) = ( 1 x. 1 ) )
26	24, 25	oveq12d 5962	. . . . . 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 2246	. . . . 5 |- ( ph -> ( ( A + 1 ) x. ( A + 1 ) ) = ( ( ( A x. A ) + ( 2 x. ( A x. 1 ) ) ) + ( 1 x. 1 ) ) )
28	17	mulridd 8089	. . . . . . . 8 |- ( ph -> ( A x. 1 ) = A )
29	28	oveq2d 5960	. . . . . . 7 |- ( ph -> ( 2 x. ( A x. 1 ) ) = ( 2 x. A ) )
30	29	oveq2d 5960	. . . . . 6 |- ( ph -> ( ( A x. A ) + ( 2 x. ( A x. 1 ) ) ) = ( ( A x. A ) + ( 2 x. A ) ) )
31	18	mulridd 8089	. . . . . 6 |- ( ph -> ( 1 x. 1 ) = 1 )
32	30, 31	oveq12d 5962	. . . . 5 |- ( ph -> ( ( ( A x. A ) + ( 2 x. ( A x. 1 ) ) ) + ( 1 x. 1 ) ) = ( ( ( A x. A ) + ( 2 x. A ) ) + 1 ) )
33	27, 32	eqtrd 2238	. . . 4 |- ( ph -> ( ( A + 1 ) x. ( A + 1 ) ) = ( ( ( A x. A ) + ( 2 x. A ) ) + 1 ) )
34	33	breq1d 4054	. . 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 1373   e. wcel 2176   class class class wbr 4044  (class class class)co 5944   CCcc 7923   1c1 7926   + caddc 7928   x. cmul 7930   < clt 8107   <_ cle 8108   2c2 9087   NN0cn0 9295   ^cexp 10683

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-frec 6477  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-inn 9037  df-2 9095  df-n0 9296  df-z 9373  df-uz 9649  df-seqfrec 10593  df-exp 10684

This theorem is referenced by:  nn0opthlem2d  10866