Theorem nn0opthlem2d 10733

Description: Lemma for nn0opth2 10736. (Contributed by Jim Kingdon, 31-Oct-2021.)

Hypotheses

Ref	Expression
nn0opthd.1	|- ( ph -> A e. NN0 )
nn0opthd.2	|- ( ph -> B e. NN0 )
nn0opthd.3	|- ( ph -> C e. NN0 )
nn0opthd.4	|- ( ph -> D e. NN0 )

Assertion

Ref	Expression
nn0opthlem2d	|- ( ph -> ( ( A + B ) < C -> ( ( C x. C ) + D ) =/= ( ( ( A + B ) x. ( A + B ) ) + B ) ) )

Proof of Theorem nn0opthlem2d

Step	Hyp	Ref	Expression
1		nn0opthd.1	. . . . . . . 8 |- ( ph -> A e. NN0 )
2		nn0opthd.2	. . . . . . . 8 |- ( ph -> B e. NN0 )
3	1, 2	nn0addcld 9263	. . . . . . 7 |- ( ph -> ( A + B ) e. NN0 )
4	3	nn0red 9260	. . . . . 6 |- ( ph -> ( A + B ) e. RR )
5	4, 4	remulcld 8018	. . . . 5 |- ( ph -> ( ( A + B ) x. ( A + B ) ) e. RR )
6	2	nn0red 9260	. . . . 5 |- ( ph -> B e. RR )
7	5, 6	readdcld 8017	. . . 4 |- ( ph -> ( ( ( A + B ) x. ( A + B ) ) + B ) e. RR )
8	7	adantr 276	. . 3 |- ( ( ph /\ ( A + B ) < C ) -> ( ( ( A + B ) x. ( A + B ) ) + B ) e. RR )
9		nn0opthd.3	. . . . . . 7 |- ( ph -> C e. NN0 )
10	9	nn0red 9260	. . . . . 6 |- ( ph -> C e. RR )
11	10, 10	remulcld 8018	. . . . 5 |- ( ph -> ( C x. C ) e. RR )
12	11	adantr 276	. . . 4 |- ( ( ph /\ ( A + B ) < C ) -> ( C x. C ) e. RR )
13		nn0opthd.4	. . . . . . 7 |- ( ph -> D e. NN0 )
14	13	nn0red 9260	. . . . . 6 |- ( ph -> D e. RR )
15	11, 14	readdcld 8017	. . . . 5 |- ( ph -> ( ( C x. C ) + D ) e. RR )
16	15	adantr 276	. . . 4 |- ( ( ph /\ ( A + B ) < C ) -> ( ( C x. C ) + D ) e. RR )
17		2re 9019	. . . . . . . . 9 |- 2 e. RR
18	17	a1i 9	. . . . . . . 8 |- ( ph -> 2 e. RR )
19	18, 4	remulcld 8018	. . . . . . 7 |- ( ph -> ( 2 x. ( A + B ) ) e. RR )
20	5, 19	readdcld 8017	. . . . . 6 |- ( ph -> ( ( ( A + B ) x. ( A + B ) ) + ( 2 x. ( A + B ) ) ) e. RR )
21	20	adantr 276	. . . . 5 |- ( ( ph /\ ( A + B ) < C ) -> ( ( ( A + B ) x. ( A + B ) ) + ( 2 x. ( A + B ) ) ) e. RR )
22		nn0addge2 9253	. . . . . . . . 9 |- ( ( B e. RR /\ A e. NN0 ) -> B <_ ( A + B ) )
23	6, 1, 22	syl2anc 411	. . . . . . . 8 |- ( ph -> B <_ ( A + B ) )
24		nn0addge1 9252	. . . . . . . . . 10 |- ( ( ( A + B ) e. RR /\ ( A + B ) e. NN0 ) -> ( A + B ) <_ ( ( A + B ) + ( A + B ) ) )
25	4, 3, 24	syl2anc 411	. . . . . . . . 9 |- ( ph -> ( A + B ) <_ ( ( A + B ) + ( A + B ) ) )
26	4	recnd 8016	. . . . . . . . . 10 |- ( ph -> ( A + B ) e. CC )
27	26	2timesd 9191	. . . . . . . . 9 |- ( ph -> ( 2 x. ( A + B ) ) = ( ( A + B ) + ( A + B ) ) )
28	25, 27	breqtrrd 4046	. . . . . . . 8 |- ( ph -> ( A + B ) <_ ( 2 x. ( A + B ) ) )
29	6, 4, 19, 23, 28	letrd 8111	. . . . . . 7 |- ( ph -> B <_ ( 2 x. ( A + B ) ) )
30	6, 19, 5, 29	leadd2dd 8547	. . . . . 6 |- ( ph -> ( ( ( A + B ) x. ( A + B ) ) + B ) <_ ( ( ( A + B ) x. ( A + B ) ) + ( 2 x. ( A + B ) ) ) )
31	30	adantr 276	. . . . 5 |- ( ( ph /\ ( A + B ) < C ) -> ( ( ( A + B ) x. ( A + B ) ) + B ) <_ ( ( ( A + B ) x. ( A + B ) ) + ( 2 x. ( A + B ) ) ) )
32	3, 9	nn0opthlem1d 10732	. . . . . 6 |- ( ph -> ( ( A + B ) < C <-> ( ( ( A + B ) x. ( A + B ) ) + ( 2 x. ( A + B ) ) ) < ( C x. C ) ) )
33	32	biimpa 296	. . . . 5 |- ( ( ph /\ ( A + B ) < C ) -> ( ( ( A + B ) x. ( A + B ) ) + ( 2 x. ( A + B ) ) ) < ( C x. C ) )
34	8, 21, 12, 31, 33	lelttrd 8112	. . . 4 |- ( ( ph /\ ( A + B ) < C ) -> ( ( ( A + B ) x. ( A + B ) ) + B ) < ( C x. C ) )
35		nn0addge1 9252	. . . . . 6 |- ( ( ( C x. C ) e. RR /\ D e. NN0 ) -> ( C x. C ) <_ ( ( C x. C ) + D ) )
36	11, 13, 35	syl2anc 411	. . . . 5 |- ( ph -> ( C x. C ) <_ ( ( C x. C ) + D ) )
37	36	adantr 276	. . . 4 |- ( ( ph /\ ( A + B ) < C ) -> ( C x. C ) <_ ( ( C x. C ) + D ) )
38	8, 12, 16, 34, 37	ltletrd 8410	. . 3 |- ( ( ph /\ ( A + B ) < C ) -> ( ( ( A + B ) x. ( A + B ) ) + B ) < ( ( C x. C ) + D ) )
39	8, 38	gtned 8100	. 2 |- ( ( ph /\ ( A + B ) < C ) -> ( ( C x. C ) + D ) =/= ( ( ( A + B ) x. ( A + B ) ) + B ) )
40	39	ex 115	1 |- ( ph -> ( ( A + B ) < C -> ( ( C x. C ) + D ) =/= ( ( ( A + B ) x. ( A + B ) ) + B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2160   =/= wne 2360   class class class wbr 4018  (class class class)co 5896   RRcr 7840   + caddc 7844   x. cmul 7846   < clt 8022   <_ cle 8023   2c2 9000   NN0cn0 9206

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-mulrcl 7940  ax-addcom 7941  ax-mulcom 7942  ax-addass 7943  ax-mulass 7944  ax-distr 7945  ax-i2m1 7946  ax-0lt1 7947  ax-1rid 7948  ax-0id 7949  ax-rnegex 7950  ax-precex 7951  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-ltwlin 7954  ax-pre-lttrn 7955  ax-pre-apti 7956  ax-pre-ltadd 7957  ax-pre-mulgt0 7958  ax-pre-mulext 7959

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-frec 6416  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028  df-sub 8160  df-neg 8161  df-reap 8562  df-ap 8569  df-div 8660  df-inn 8950  df-2 9008  df-n0 9207  df-z 9284  df-uz 9559  df-seqfrec 10477  df-exp 10551

This theorem is referenced by:  nn0opthd  10734