Theorem addlocprlemeq 7681

Description: Lemma for addlocpr 7684. 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 7680	. . . . 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 5974	. . . . 5 |- ( Q = ( D +Q E ) -> ( Q +Q ( P +Q P ) ) = ( ( D +Q E ) +Q ( P +Q P ) ) )
15	5, 14	sylan9req 2261	. . . 4 |- ( ( ph /\ Q = ( D +Q E ) ) -> R = ( ( D +Q E ) +Q ( P +Q P ) ) )
16	13, 15	breqtrrd 4087	. . 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 7513	. . . . . . . 8 |- <Q C_ ( Q. X. Q. )
20	19	brel 4745	. . . . . . 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 7678	. . . . 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 1249	. . . 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 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   class class class wbr 4059   ` cfv 5290  (class class class)co 5967   1stc1st 6247   2ndc2nd 6248   Q.cnq 7428   +Q cplq 7430   <Q cltq 7433   P.cnp 7439   +P. cpp 7441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-eprel 4354  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-irdg 6479  df-1o 6525  df-oadd 6529  df-omul 6530  df-er 6643  df-ec 6645  df-qs 6649  df-ni 7452  df-pli 7453  df-mi 7454  df-lti 7455  df-plpq 7492  df-mpq 7493  df-enq 7495  df-nqqs 7496  df-plqqs 7497  df-mqqs 7498  df-1nqqs 7499  df-rq 7500  df-ltnqqs 7501  df-inp 7614  df-iplp 7616

This theorem is referenced by:  addlocprlem  7683