Theorem addlocprlemeq 7474

Description: Lemma for addlocpr 7477. The Q = ( D +Q E ) case. (Contributed by Jim Kingdon, 6-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
addlocprlemeq	|- ( ph -> ( Q = ( D +Q E ) -> R e. ( 2nd ` ( A +P. B ) ) ) )

Proof of Theorem addlocprlemeq

Step	Hyp	Ref	Expression
1		addlocprlem.a	. . . . . 6 |- ( ph -> A e. P. )
2		addlocprlem.b	. . . . . 6 |- ( ph -> B e. P. )
3		addlocprlem.qr	. . . . . 6 |- ( ph -> Q <Q R )
4		addlocprlem.p	. . . . . 6 |- ( ph -> P e. Q. )
5		addlocprlem.qppr	. . . . . 6 |- ( ph -> ( Q +Q ( P +Q P ) ) = R )
6		addlocprlem.dlo	. . . . . 6 |- ( ph -> D e. ( 1st ` A ) )
7		addlocprlem.uup	. . . . . 6 |- ( ph -> U e. ( 2nd ` A ) )
8		addlocprlem.du	. . . . . 6 |- ( ph -> U <Q ( D +Q P ) )
9		addlocprlem.elo	. . . . . 6 |- ( ph -> E e. ( 1st ` B ) )
10		addlocprlem.tup	. . . . . 6 |- ( ph -> T e. ( 2nd ` B ) )
11		addlocprlem.et	. . . . . 6 |- ( ph -> T <Q ( E +Q P ) )
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	addlocprlemeqgt 7473	. . . . 5 |- ( ph -> ( U +Q T ) <Q ( ( D +Q E ) +Q ( P +Q P ) ) )
13	12	adantr 274	. . . 4 |- ( ( ph /\ Q = ( D +Q E ) ) -> ( U +Q T ) <Q ( ( D +Q E ) +Q ( P +Q P ) ) )
14		oveq1 5849	. . . . 5 |- ( Q = ( D +Q E ) -> ( Q +Q ( P +Q P ) ) = ( ( D +Q E ) +Q ( P +Q P ) ) )
15	5, 14	sylan9req 2220	. . . 4 |- ( ( ph /\ Q = ( D +Q E ) ) -> R = ( ( D +Q E ) +Q ( P +Q P ) ) )
16	13, 15	breqtrrd 4010	. . 3 |- ( ( ph /\ Q = ( D +Q E ) ) -> ( U +Q T ) <Q R )
17	1, 7	jca 304	. . . . 5 |- ( ph -> ( A e. P. /\ U e. ( 2nd ` A ) ) )
18	2, 10	jca 304	. . . . 5 |- ( ph -> ( B e. P. /\ T e. ( 2nd ` B ) ) )
19		ltrelnq 7306	. . . . . . . 8 |- <Q C_ ( Q. X. Q. )
20	19	brel 4656	. . . . . . 7 |- ( Q <Q R -> ( Q e. Q. /\ R e. Q. ) )
21	20	simprd 113	. . . . . 6 |- ( Q <Q R -> R e. Q. )
22	3, 21	syl 14	. . . . 5 |- ( ph -> R e. Q. )
23		addnqpru 7471	. . . . 5 |- ( ( ( ( A e. P. /\ U e. ( 2nd ` A ) ) /\ ( B e. P. /\ T e. ( 2nd ` B ) ) ) /\ R e. Q. ) -> ( ( U +Q T ) <Q R -> R e. ( 2nd ` ( A +P. B ) ) ) )
24	17, 18, 22, 23	syl21anc 1227	. . . 4 |- ( ph -> ( ( U +Q T ) <Q R -> R e. ( 2nd ` ( A +P. B ) ) ) )
25	24	adantr 274	. . 3 |- ( ( ph /\ Q = ( D +Q E ) ) -> ( ( U +Q T ) <Q R -> R e. ( 2nd ` ( A +P. B ) ) ) )
26	16, 25	mpd 13	. 2 |- ( ( ph /\ Q = ( D +Q E ) ) -> R e. ( 2nd ` ( A +P. B ) ) )
27	26	ex 114	1 |- ( ph -> ( Q = ( D +Q E ) -> R e. ( 2nd ` ( A +P. B ) ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   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   +P. cpp 7234

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-1o 6384  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-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294  df-inp 7407  df-iplp 7409

This theorem is referenced by:  addlocprlem  7476