Theorem nn0opthlem1d 11180

Description: A rather pretty lemma for nn0opth2 11184. (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 9590	. . . . 5 |- 1 e. NN0
3	2	a1i 9	. . . 4 |- ( ph -> 1 e. NN0 )
4	1, 3	nn0addcld 9635	. . 3 |- ( ph -> ( A + 1 ) e. NN0 )
5		nn0opthlem1d.2	. . 3 |- ( ph -> C e. NN0 )
6	4, 5	nn0le2msqd 11179	. 2 |- ( ph -> ( ( A + 1 ) <_ C <-> ( ( A + 1 ) x. ( A + 1 ) ) <_ ( C x. C ) ) )
7		nn0ltp1le 9718	. . 3 |- ( ( A e. NN0 /\ C e. NN0 ) -> ( A < C <-> ( A + 1 ) <_ C ) )
8	1, 5, 7	syl2anc 415	. 2 |- ( ph -> ( A < C <-> ( A + 1 ) <_ C ) )
9	1, 1	nn0mulcld 9636	. . . . 5 |- ( ph -> ( A x. A ) e. NN0 )
10		2nn0 9591	. . . . . . 7 |- 2 e. NN0
11	10	a1i 9	. . . . . 6 |- ( ph -> 2 e. NN0 )
12	11, 1	nn0mulcld 9636	. . . . 5 |- ( ph -> ( 2 x. A ) e. NN0 )
13	9, 12	nn0addcld 9635	. . . 4 |- ( ph -> ( ( A x. A ) + ( 2 x. A ) ) e. NN0 )
14	5, 5	nn0mulcld 9636	. . . 4 |- ( ph -> ( C x. C ) e. NN0 )
15		nn0ltp1le 9718	. . . 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 415	. . 3 |- ( ph -> ( ( ( A x. A ) + ( 2 x. A ) ) < ( C x. C ) <-> ( ( ( A x. A ) + ( 2 x. A ) ) + 1 ) <_ ( C x. C ) ) )
17	1	nn0cnd 9633	. . . . . . 7 |- ( ph -> A e. CC )
18		1cnd 8348	. . . . . . 7 |- ( ph -> 1 e. CC )
19		binom2 11109	. . . . . . 7 |- ( ( A e. CC /\ 1 e. CC ) -> ( ( A + 1 ) ^ 2 ) = ( ( ( A ^ 2 ) + ( 2 x. ( A x. 1 ) ) ) + ( 1 ^ 2 ) ) )
20	17, 18, 19	syl2anc 415	. . . . . 6 |- ( ph -> ( ( A + 1 ) ^ 2 ) = ( ( ( A ^ 2 ) + ( 2 x. ( A x. 1 ) ) ) + ( 1 ^ 2 ) ) )
21	17, 18	addcld 8351	. . . . . . 7 |- ( ph -> ( A + 1 ) e. CC )
22	21	sqvald 11129	. . . . . 6 |- ( ph -> ( ( A + 1 ) ^ 2 ) = ( ( A + 1 ) x. ( A + 1 ) ) )
23	17	sqvald 11129	. . . . . . . 8 |- ( ph -> ( A ^ 2 ) = ( A x. A ) )
24	23	oveq1d 6103	. . . . . . 7 |- ( ph -> ( ( A ^ 2 ) + ( 2 x. ( A x. 1 ) ) ) = ( ( A x. A ) + ( 2 x. ( A x. 1 ) ) ) )
25	18	sqvald 11129	. . . . . . 7 |- ( ph -> ( 1 ^ 2 ) = ( 1 x. 1 ) )
26	24, 25	oveq12d 6106	. . . . . 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 2279	. . . . 5 |- ( ph -> ( ( A + 1 ) x. ( A + 1 ) ) = ( ( ( A x. A ) + ( 2 x. ( A x. 1 ) ) ) + ( 1 x. 1 ) ) )
28	17	mulridd 8349	. . . . . . . 8 |- ( ph -> ( A x. 1 ) = A )
29	28	oveq2d 6104	. . . . . . 7 |- ( ph -> ( 2 x. ( A x. 1 ) ) = ( 2 x. A ) )
30	29	oveq2d 6104	. . . . . 6 |- ( ph -> ( ( A x. A ) + ( 2 x. ( A x. 1 ) ) ) = ( ( A x. A ) + ( 2 x. A ) ) )
31	18	mulridd 8349	. . . . . 6 |- ( ph -> ( 1 x. 1 ) = 1 )
32	30, 31	oveq12d 6106	. . . . 5 |- ( ph -> ( ( ( A x. A ) + ( 2 x. ( A x. 1 ) ) ) + ( 1 x. 1 ) ) = ( ( ( A x. A ) + ( 2 x. A ) ) + 1 ) )
33	27, 32	eqtrd 2271	. . . 4 |- ( ph -> ( ( A + 1 ) x. ( A + 1 ) ) = ( ( ( A x. A ) + ( 2 x. A ) ) + 1 ) )
34	33	breq1d 4141	. . 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 ) ) )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   = wceq 1402   e. wcel 2209   class class class wbr 4131  (class class class)co 6088   CCcc 8183   1c1 8186   + caddc 8188   x. cmul 8190   < clt 8366   <_ cle 8367   2c2 9364   NN0cn0 9574   ^cexp 10996

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-n0 9575  df-z 9656  df-uz 9938  df-seqfrec 10906  df-exp 10997

This theorem is used by:  nn0opthlem2d  11181