Theorem addlocprlemeq 6723

Description: Lemma for addlocpr 6726. 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 6722	. . . . 5 |- ( ph -> ( U +Q T ) <Q ( ( D +Q E ) +Q ( P +Q P ) ) )
13	12	adantr 270	. . . 4 |- ( ( ph /\ Q = ( D +Q E ) ) -> ( U +Q T ) <Q ( ( D +Q E ) +Q ( P +Q P ) ) )
14		oveq1 5539	. . . . 5 |- ( Q = ( D +Q E ) -> ( Q +Q ( P +Q P ) ) = ( ( D +Q E ) +Q ( P +Q P ) ) )
15	5, 14	sylan9req 2134	. . . 4 |- ( ( ph /\ Q = ( D +Q E ) ) -> R = ( ( D +Q E ) +Q ( P +Q P ) ) )
16	13, 15	breqtrrd 3811	. . 3 |- ( ( ph /\ Q = ( D +Q E ) ) -> ( U +Q T ) <Q R )
17	1, 7	jca 300	. . . . 5 |- ( ph -> ( A e. P. /\ U e. ( 2nd ` A ) ) )
18	2, 10	jca 300	. . . . 5 |- ( ph -> ( B e. P. /\ T e. ( 2nd ` B ) ) )
19		ltrelnq 6555	. . . . . . . 8 |- <Q C_ ( Q. X. Q. )
20	19	brel 4410	. . . . . . 7 |- ( Q <Q R -> ( Q e. Q. /\ R e. Q. ) )
21	20	simprd 112	. . . . . 6 |- ( Q <Q R -> R e. Q. )
22	3, 21	syl 14	. . . . 5 |- ( ph -> R e. Q. )
23		addnqpru 6720	. . . . 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 1168	. . . 4 |- ( ph -> ( ( U +Q T ) <Q R -> R e. ( 2nd ` ( A +P. B ) ) ) )
25	24	adantr 270	. . 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 113	1 |- ( ph -> ( Q = ( D +Q E ) -> R e. ( 2nd ` ( A +P. B ) ) ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   class class class wbr 3785   ` cfv 4922  (class class class)co 5532   1stc1st 5785   2ndc2nd 5786   Q.cnq 6470   +Q cplq 6472   <Q cltq 6475   P.cnp 6481   +P. cpp 6483

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-eprel 4044  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-1o 6024  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-pli 6495  df-mi 6496  df-lti 6497  df-plpq 6534  df-mpq 6535  df-enq 6537  df-nqqs 6538  df-plqqs 6539  df-mqqs 6540  df-1nqqs 6541  df-rq 6542  df-ltnqqs 6543  df-inp 6656  df-iplp 6658

This theorem is referenced by:  addlocprlem  6725