Theorem addlocprlemeqgt 7473

Description: Lemma for addlocpr 7477. 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 7416	. . . . . 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 7423	. . . . 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 7422	. . . . . 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 7316	. . . . 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 7416	. . . . . 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 7423	. . . . 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 7422	. . . . . 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 7316	. . . . 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 7345	. . . 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 1229	. . 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 7322	. . . 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 7323	. . . 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 7316	. . . 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 6026	. 2 |- ( ph -> ( ( D +Q P ) +Q ( E +Q P ) ) = ( ( D +Q E ) +Q ( P +Q P ) ) )
36	28, 35	breqtrd 4008	1 |- ( ph -> ( U +Q T ) <Q ( ( D +Q E ) +Q ( P +Q P ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 968   = wceq 1343   e. wcel 2136   <.cop 3579   class class class wbr 3982   ` cfv 5188  (class class class)co 5842   1stc1st 6106   2ndc2nd 6107   Q.cnq 7221   +Q cplq 7223   <Q cltq 7226   P.cnp 7232

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-ltnqqs 7294  df-inp 7407

This theorem is referenced by:  addlocprlemeq  7474  addlocprlemgt  7475