Theorem addlocprlemeqgt 7464

Description: Lemma for addlocpr 7468. This is a step used in both the Q = ( D +Q E ) and ( D +Q E ) <Q Q cases. (Contributed by Jim Kingdon, 7-Dec-2019.)

Hypotheses

Ref	Expression
addlocprlem.a	|- ( ph -> A e. P. )
addlocprlem.b	|- ( ph -> B e. P. )
addlocprlem.qr	|- ( ph -> Q <Q R )
addlocprlem.p	|- ( ph -> P e. Q. )
addlocprlem.qppr	|- ( ph -> ( Q +Q ( P +Q P ) ) = R )
addlocprlem.dlo	|- ( ph -> D e. ( 1st ` A ) )
addlocprlem.uup	|- ( ph -> U e. ( 2nd ` A ) )
addlocprlem.du	|- ( ph -> U <Q ( D +Q P ) )
addlocprlem.elo	|- ( ph -> E e. ( 1st ` B ) )
addlocprlem.tup	|- ( ph -> T e. ( 2nd ` B ) )
addlocprlem.et	|- ( ph -> T <Q ( E +Q P ) )

Assertion

Ref	Expression
addlocprlemeqgt	|- ( ph -> ( U +Q T ) <Q ( ( D +Q E ) +Q ( P +Q P ) ) )

Proof of Theorem addlocprlemeqgt

Dummy variables f g h are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		addlocprlem.du	. . 3 |- ( ph -> U <Q ( D +Q P ) )
2		addlocprlem.et	. . 3 |- ( ph -> T <Q ( E +Q P ) )
3		addlocprlem.a	. . . . . 6 |- ( ph -> A e. P. )
4		prop 7407	. . . . . 6 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
5	3, 4	syl 14	. . . . 5 |- ( ph -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
6		addlocprlem.uup	. . . . 5 |- ( ph -> U e. ( 2nd ` A ) )
7		elprnqu 7414	. . . . 5 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ U e. ( 2nd ` A ) ) -> U e. Q. )
8	5, 6, 7	syl2anc 409	. . . 4 |- ( ph -> U e. Q. )
9		addlocprlem.dlo	. . . . . 6 |- ( ph -> D e. ( 1st ` A ) )
10		elprnql 7413	. . . . . 6 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ D e. ( 1st ` A ) ) -> D e. Q. )
11	5, 9, 10	syl2anc 409	. . . . 5 |- ( ph -> D e. Q. )
12		addlocprlem.p	. . . . 5 |- ( ph -> P e. Q. )
13		addclnq 7307	. . . . 5 |- ( ( D e. Q. /\ P e. Q. ) -> ( D +Q P ) e. Q. )
14	11, 12, 13	syl2anc 409	. . . 4 |- ( ph -> ( D +Q P ) e. Q. )
15		addlocprlem.b	. . . . . 6 |- ( ph -> B e. P. )
16		prop 7407	. . . . . 6 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
17	15, 16	syl 14	. . . . 5 |- ( ph -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
18		addlocprlem.tup	. . . . 5 |- ( ph -> T e. ( 2nd ` B ) )
19		elprnqu 7414	. . . . 5 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ T e. ( 2nd ` B ) ) -> T e. Q. )
20	17, 18, 19	syl2anc 409	. . . 4 |- ( ph -> T e. Q. )
21		addlocprlem.elo	. . . . . 6 |- ( ph -> E e. ( 1st ` B ) )
22		elprnql 7413	. . . . . 6 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ E e. ( 1st ` B ) ) -> E e. Q. )
23	17, 21, 22	syl2anc 409	. . . . 5 |- ( ph -> E e. Q. )
24		addclnq 7307	. . . . 5 |- ( ( E e. Q. /\ P e. Q. ) -> ( E +Q P ) e. Q. )
25	23, 12, 24	syl2anc 409	. . . 4 |- ( ph -> ( E +Q P ) e. Q. )
26		lt2addnq 7336	. . . 4 |- ( ( ( U e. Q. /\ ( D +Q P ) e. Q. ) /\ ( T e. Q. /\ ( E +Q P ) e. Q. ) ) -> ( ( U <Q ( D +Q P ) /\ T <Q ( E +Q P ) ) -> ( U +Q T ) <Q ( ( D +Q P ) +Q ( E +Q P ) ) ) )
27	8, 14, 20, 25, 26	syl22anc 1228	. . 3 |- ( ph -> ( ( U <Q ( D +Q P ) /\ T <Q ( E +Q P ) ) -> ( U +Q T ) <Q ( ( D +Q P ) +Q ( E +Q P ) ) ) )
28	1, 2, 27	mp2and 430	. 2 |- ( ph -> ( U +Q T ) <Q ( ( D +Q P ) +Q ( E +Q P ) ) )
29		addcomnqg 7313	. . . 4 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
30	29	adantl 275	. . 3 |- ( ( ph /\ ( f e. Q. /\ g e. Q. ) ) -> ( f +Q g ) = ( g +Q f ) )
31		addassnqg 7314	. . . 4 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( ( f +Q g ) +Q h ) = ( f +Q ( g +Q h ) ) )
32	31	adantl 275	. . 3 |- ( ( ph /\ ( f e. Q. /\ g e. Q. /\ h e. Q. ) ) -> ( ( f +Q g ) +Q h ) = ( f +Q ( g +Q h ) ) )
33		addclnq 7307	. . . 4 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) e. Q. )
34	33	adantl 275	. . 3 |- ( ( ph /\ ( f e. Q. /\ g e. Q. ) ) -> ( f +Q g ) e. Q. )
35	11, 12, 23, 30, 32, 12, 34	caov4d 6017	. 2 |- ( ph -> ( ( D +Q P ) +Q ( E +Q P ) ) = ( ( D +Q E ) +Q ( P +Q P ) ) )
36	28, 35	breqtrd 4002	1 |- ( ph -> ( U +Q T ) <Q ( ( D +Q E ) +Q ( P +Q P ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 967   = wceq 1342   e. wcel 2135   <.cop 3573   class class class wbr 3976   ` cfv 5182  (class class class)co 5836   1stc1st 6098   2ndc2nd 6099   Q.cnq 7212   +Q cplq 7214   <Q cltq 7217   P.cnp 7223

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4091  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-iinf 4559

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-rex 2448  df-reu 2449  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-tr 4075  df-eprel 4261  df-id 4265  df-po 4268  df-iso 4269  df-iord 4338  df-on 4340  df-suc 4343  df-iom 4562  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1st 6100  df-2nd 6101  df-recs 6264  df-irdg 6329  df-oadd 6379  df-omul 6380  df-er 6492  df-ec 6494  df-qs 6498  df-ni 7236  df-pli 7237  df-mi 7238  df-lti 7239  df-plpq 7276  df-enq 7279  df-nqqs 7280  df-plqqs 7281  df-ltnqqs 7285  df-inp 7398

This theorem is referenced by:  addlocprlemeq  7465  addlocprlemgt  7466