Theorem addlocprlemeq 6516

Description: Lemma for addlocpr 6519. 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 6515	. . . . 5 |- (ph -> ( U +Q T) <Q (( D +Q E) +Q ( P +Q P)))
13	12	adantr 261	. . . 4 |- ((ph /\ Q = ( D +Q E)) -> ( U +Q T) <Q (( D +Q E) +Q ( P +Q P)))
14		oveq1 5462	. . . . 5 |- ( Q = ( D +Q E) -> ( Q +Q ( P +Q P)) = (( D +Q E) +Q ( P +Q P)))
15	5, 14	sylan9req 2090	. . . 4 |- ((ph /\ Q = ( D +Q E)) -> R = (( D +Q E) +Q ( P +Q P)))
16	13, 15	breqtrrd 3781	. . 3 |- ((ph /\ Q = ( D +Q E)) -> ( U +Q T) <Q R)
17	1, 7	jca 290	. . . . 5 |- (ph -> (A e. P. /\ U e. ( 2nd ` A)))
18	2, 10	jca 290	. . . . 5 |- (ph -> (B e. P. /\ T e. ( 2nd ` B)))
19		ltrelnq 6349	. . . . . . . 8 |- <Q C_ ( Q. X. Q.)
20	19	brel 4335	. . . . . . 7 |- ( Q <Q R -> ( Q e. Q. /\ R e. Q.))
21	20	simprd 107	. . . . . 6 |- ( Q <Q R -> R e. Q.)
22	3, 21	syl 14	. . . . 5 |- (ph -> R e. Q.)
23		addnqpru 6513	. . . . 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 1133	. . . 4 |- (ph -> (( U +Q T) <Q R -> R e. ( 2nd ` (A +P. B))))
25	24	adantr 261	. . 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 108	1 |- (ph -> ( Q = ( D +Q E) -> R e. ( 2nd ` (A +P. B))))

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1242   e. wcel 1390   class class class wbr 3755   ` cfv 4845  (class class class)co 5455   1stc1st 5707   2ndc2nd 5708   Q.cnq 6264   +Q cplq 6266   <Q cltq 6269   P.cnp 6275   +P. cpp 6277

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-inp 6449  df-iplp 6451

This theorem is referenced by:  addlocprlem  6518