Theorem nn0opthlem2d 11159

Description: Lemma for nn0opth2 11162. (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 9624	. . . . . . 7 |- ( ph -> ( A + B ) e. NN0 )
4	3	nn0red 9621	. . . . . 6 |- ( ph -> ( A + B ) e. RR )
5	4, 4	remulcld 8356	. . . . 5 |- ( ph -> ( ( A + B ) x. ( A + B ) ) e. RR )
6	2	nn0red 9621	. . . . 5 |- ( ph -> B e. RR )
7	5, 6	readdcld 8355	. . . 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 9621	. . . . . 6 |- ( ph -> C e. RR )
11	10, 10	remulcld 8356	. . . . 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 9621	. . . . . 6 |- ( ph -> D e. RR )
15	11, 14	readdcld 8355	. . . . 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 9374	. . . . . . . . 9 |- 2 e. RR
18	17	a1i 9	. . . . . . . 8 |- ( ph -> 2 e. RR )
19	18, 4	remulcld 8356	. . . . . . 7 |- ( ph -> ( 2 x. ( A + B ) ) e. RR )
20	5, 19	readdcld 8355	. . . . . 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 9610	. . . . . . . . 9 |- ( ( B e. RR /\ A e. NN0 ) -> B <_ ( A + B ) )
23	6, 1, 22	syl2anc 415	. . . . . . . 8 |- ( ph -> B <_ ( A + B ) )
24		nn0addge1 9609	. . . . . . . . . 10 |- ( ( ( A + B ) e. RR /\ ( A + B ) e. NN0 ) -> ( A + B ) <_ ( ( A + B ) + ( A + B ) ) )
25	4, 3, 24	syl2anc 415	. . . . . . . . 9 |- ( ph -> ( A + B ) <_ ( ( A + B ) + ( A + B ) ) )
26	4	recnd 8354	. . . . . . . . . 10 |- ( ph -> ( A + B ) e. CC )
27	26	2timesd 9548	. . . . . . . . 9 |- ( ph -> ( 2 x. ( A + B ) ) = ( ( A + B ) + ( A + B ) ) )
28	25, 27	breqtrrd 4158	. . . . . . . 8 |- ( ph -> ( A + B ) <_ ( 2 x. ( A + B ) ) )
29	6, 4, 19, 23, 28	letrd 8450	. . . . . . 7 |- ( ph -> B <_ ( 2 x. ( A + B ) ) )
30	6, 19, 5, 29	leadd2dd 8888	. . . . . 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 11158	. . . . . 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 8451	. . . 4 |- ( ( ph /\ ( A + B ) < C ) -> ( ( ( A + B ) x. ( A + B ) ) + B ) < ( C x. C ) )
35		nn0addge1 9609	. . . . . 6 |- ( ( ( C x. C ) e. RR /\ D e. NN0 ) -> ( C x. C ) <_ ( ( C x. C ) + D ) )
36	11, 13, 35	syl2anc 415	. . . . 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 8751	. . 3 |- ( ( ph /\ ( A + B ) < C ) -> ( ( ( A + B ) x. ( A + B ) ) + B ) < ( ( C x. C ) + D ) )
39	8, 38	gtned 8438	. 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 ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   e. wcel 2209   =/= wne 2420   class class class wbr 4130  (class class class)co 6085   RRcr 8178   + caddc 8182   x. cmul 8184   < clt 8360   <_ cle 8361   2c2 9355   NN0cn0 9563

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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297

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 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-n0 9564  df-z 9645  df-uz 9922  df-seqfrec 10885  df-exp 10976

This theorem is used by:  nn0opthd  11160