Theorem addlocprlemeqgt 7494

Description: Lemma for addlocpr 7498. 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 7437	. . . . . 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 7444	. . . . 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 7443	. . . . . 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 7337	. . . . 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 7437	. . . . . 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 7444	. . . . 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 7443	. . . . . 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 7337	. . . . 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 7366	. . . 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 1234	. . 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 431	. 2 |- ( ph -> ( U +Q T ) <Q ( ( D +Q P ) +Q ( E +Q P ) ) )
29		addcomnqg 7343	. . . 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 7344	. . . 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 7337	. . . 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 6037	. 2 |- ( ph -> ( ( D +Q P ) +Q ( E +Q P ) ) = ( ( D +Q E ) +Q ( P +Q P ) ) )
36	28, 35	breqtrd 4015	1 |- ( ph -> ( U +Q T ) <Q ( ( D +Q E ) +Q ( P +Q P ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 973   = wceq 1348   e. wcel 2141   <.cop 3586   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   1stc1st 6117   2ndc2nd 6118   Q.cnq 7242   +Q cplq 7244   <Q cltq 7247   P.cnp 7253

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-ltnqqs 7315  df-inp 7428

This theorem is referenced by:  addlocprlemeq  7495  addlocprlemgt  7496