Theorem addlocprlemeq 7906

Description: Lemma for addlocpr 7909. 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 7905	. . . . 5 |- ( ph -> ( U +Q T ) <Q ( ( D +Q E ) +Q ( P +Q P ) ) )
13	12	adantr 276	. . . 4 |- ( ( ph /\ Q = ( D +Q E ) ) -> ( U +Q T ) <Q ( ( D +Q E ) +Q ( P +Q P ) ) )
14		oveq1 6095	. . . . 5 |- ( Q = ( D +Q E ) -> ( Q +Q ( P +Q P ) ) = ( ( D +Q E ) +Q ( P +Q P ) ) )
15	5, 14	sylan9req 2292	. . . 4 |- ( ( ph /\ Q = ( D +Q E ) ) -> R = ( ( D +Q E ) +Q ( P +Q P ) ) )
16	13, 15	breqtrrd 4159	. . 3 |- ( ( ph /\ Q = ( D +Q E ) ) -> ( U +Q T ) <Q R )
17	1, 7	jca 306	. . . . 5 |- ( ph -> ( A e. P. /\ U e. ( 2nd ` A ) ) )
18	2, 10	jca 306	. . . . 5 |- ( ph -> ( B e. P. /\ T e. ( 2nd ` B ) ) )
19		ltrelnq 7738	. . . . . . . 8 |- <Q C_ ( Q. X. Q. )
20	19	brel 4829	. . . . . . 7 |- ( Q <Q R -> ( Q e. Q. /\ R e. Q. ) )
21	20	simprd 114	. . . . . 6 |- ( Q <Q R -> R e. Q. )
22	3, 21	syl 14	. . . . 5 |- ( ph -> R e. Q. )
23		addnqpru 7903	. . . . 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 1277	. . . 4 |- ( ph -> ( ( U +Q T ) <Q R -> R e. ( 2nd ` ( A +P. B ) ) ) )
25	24	adantr 276	. . 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 115	1 |- ( ph -> ( Q = ( D +Q E ) -> R e. ( 2nd ` ( A +P. B ) ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   class class class wbr 4131   ` cfv 5380  (class class class)co 6088   1stc1st 6376   2ndc2nd 6377   Q.cnq 7653   +Q cplq 7655   <Q cltq 7658   P.cnp 7664   +P. cpp 7666

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-eprel 4436  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-1o 6691  df-oadd 6695  df-omul 6696  df-er 6811  df-ec 6813  df-qs 6817  df-ni 7677  df-pli 7678  df-mi 7679  df-lti 7680  df-plpq 7717  df-mpq 7718  df-enq 7720  df-nqqs 7721  df-plqqs 7722  df-mqqs 7723  df-1nqqs 7724  df-rq 7725  df-ltnqqs 7726  df-inp 7839  df-iplp 7841

This theorem is used by:  addlocprlem  7908