Theorem nn0opthlem2d 10903

Description: Lemma for nn0opth2 10906. (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 9387	. . . . . . 7 |- ( ph -> ( A + B ) e. NN0 )
4	3	nn0red 9384	. . . . . 6 |- ( ph -> ( A + B ) e. RR )
5	4, 4	remulcld 8138	. . . . 5 |- ( ph -> ( ( A + B ) x. ( A + B ) ) e. RR )
6	2	nn0red 9384	. . . . 5 |- ( ph -> B e. RR )
7	5, 6	readdcld 8137	. . . 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 9384	. . . . . 6 |- ( ph -> C e. RR )
11	10, 10	remulcld 8138	. . . . 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 9384	. . . . . 6 |- ( ph -> D e. RR )
15	11, 14	readdcld 8137	. . . . 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 9141	. . . . . . . . 9 |- 2 e. RR
18	17	a1i 9	. . . . . . . 8 |- ( ph -> 2 e. RR )
19	18, 4	remulcld 8138	. . . . . . 7 |- ( ph -> ( 2 x. ( A + B ) ) e. RR )
20	5, 19	readdcld 8137	. . . . . 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 9377	. . . . . . . . 9 |- ( ( B e. RR /\ A e. NN0 ) -> B <_ ( A + B ) )
23	6, 1, 22	syl2anc 411	. . . . . . . 8 |- ( ph -> B <_ ( A + B ) )
24		nn0addge1 9376	. . . . . . . . . 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 8136	. . . . . . . . . 10 |- ( ph -> ( A + B ) e. CC )
27	26	2timesd 9315	. . . . . . . . 9 |- ( ph -> ( 2 x. ( A + B ) ) = ( ( A + B ) + ( A + B ) ) )
28	25, 27	breqtrrd 4087	. . . . . . . 8 |- ( ph -> ( A + B ) <_ ( 2 x. ( A + B ) ) )
29	6, 4, 19, 23, 28	letrd 8231	. . . . . . 7 |- ( ph -> B <_ ( 2 x. ( A + B ) ) )
30	6, 19, 5, 29	leadd2dd 8668	. . . . . 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 10902	. . . . . 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 8232	. . . 4 |- ( ( ph /\ ( A + B ) < C ) -> ( ( ( A + B ) x. ( A + B ) ) + B ) < ( C x. C ) )
35		nn0addge1 9376	. . . . . 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 8531	. . 3 |- ( ( ph /\ ( A + B ) < C ) -> ( ( ( A + B ) x. ( A + B ) ) + B ) < ( ( C x. C ) + D ) )
39	8, 38	gtned 8220	. 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 2178   =/= wne 2378   class class class wbr 4059  (class class class)co 5967   RRcr 7959   + caddc 7963   x. cmul 7965   < clt 8142   <_ cle 8143   2c2 9122   NN0cn0 9330

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078

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 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-n0 9331  df-z 9408  df-uz 9684  df-seqfrec 10630  df-exp 10721

This theorem is referenced by:  nn0opthd  10904