Theorem nn0opthlem2d 10973

Description: Lemma for nn0opth2 10976. (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 9449	. . . . . . 7 |- ( ph -> ( A + B ) e. NN0 )
4	3	nn0red 9446	. . . . . 6 |- ( ph -> ( A + B ) e. RR )
5	4, 4	remulcld 8200	. . . . 5 |- ( ph -> ( ( A + B ) x. ( A + B ) ) e. RR )
6	2	nn0red 9446	. . . . 5 |- ( ph -> B e. RR )
7	5, 6	readdcld 8199	. . . 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 9446	. . . . . 6 |- ( ph -> C e. RR )
11	10, 10	remulcld 8200	. . . . 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 9446	. . . . . 6 |- ( ph -> D e. RR )
15	11, 14	readdcld 8199	. . . . 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 9203	. . . . . . . . 9 |- 2 e. RR
18	17	a1i 9	. . . . . . . 8 |- ( ph -> 2 e. RR )
19	18, 4	remulcld 8200	. . . . . . 7 |- ( ph -> ( 2 x. ( A + B ) ) e. RR )
20	5, 19	readdcld 8199	. . . . . 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 9439	. . . . . . . . 9 |- ( ( B e. RR /\ A e. NN0 ) -> B <_ ( A + B ) )
23	6, 1, 22	syl2anc 411	. . . . . . . 8 |- ( ph -> B <_ ( A + B ) )
24		nn0addge1 9438	. . . . . . . . . 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 8198	. . . . . . . . . 10 |- ( ph -> ( A + B ) e. CC )
27	26	2timesd 9377	. . . . . . . . 9 |- ( ph -> ( 2 x. ( A + B ) ) = ( ( A + B ) + ( A + B ) ) )
28	25, 27	breqtrrd 4114	. . . . . . . 8 |- ( ph -> ( A + B ) <_ ( 2 x. ( A + B ) ) )
29	6, 4, 19, 23, 28	letrd 8293	. . . . . . 7 |- ( ph -> B <_ ( 2 x. ( A + B ) ) )
30	6, 19, 5, 29	leadd2dd 8730	. . . . . 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 10972	. . . . . 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 8294	. . . 4 |- ( ( ph /\ ( A + B ) < C ) -> ( ( ( A + B ) x. ( A + B ) ) + B ) < ( C x. C ) )
35		nn0addge1 9438	. . . . . 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 8593	. . 3 |- ( ( ph /\ ( A + B ) < C ) -> ( ( ( A + B ) x. ( A + B ) ) + B ) < ( ( C x. C ) + D ) )
39	8, 38	gtned 8282	. 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 2200   =/= wne 2400   class class class wbr 4086  (class class class)co 6013   RRcr 8021   + caddc 8025   x. cmul 8027   < clt 8204   <_ cle 8205   2c2 9184   NN0cn0 9392

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-2 9192  df-n0 9393  df-z 9470  df-uz 9746  df-seqfrec 10700  df-exp 10791

This theorem is referenced by:  nn0opthd  10974