Theorem addlocprlemeq 7552

Description: Lemma for addlocpr 7555. 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 7551	. . . . 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 5899	. . . . 5 |- ( Q = ( D +Q E ) -> ( Q +Q ( P +Q P ) ) = ( ( D +Q E ) +Q ( P +Q P ) ) )
15	5, 14	sylan9req 2243	. . . 4 |- ( ( ph /\ Q = ( D +Q E ) ) -> R = ( ( D +Q E ) +Q ( P +Q P ) ) )
16	13, 15	breqtrrd 4046	. . 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 7384	. . . . . . . 8 |- <Q C_ ( Q. X. Q. )
20	19	brel 4693	. . . . . . 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 7549	. . . . 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 1248	. . . 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 1364   e. wcel 2160   class class class wbr 4018   ` cfv 5232  (class class class)co 5892   1stc1st 6158   2ndc2nd 6159   Q.cnq 7299   +Q cplq 7301   <Q cltq 7304   P.cnp 7310   +P. cpp 7312

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4304  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-ov 5895  df-oprab 5896  df-mpo 5897  df-1st 6160  df-2nd 6161  df-recs 6325  df-irdg 6390  df-1o 6436  df-oadd 6440  df-omul 6441  df-er 6554  df-ec 6556  df-qs 6560  df-ni 7323  df-pli 7324  df-mi 7325  df-lti 7326  df-plpq 7363  df-mpq 7364  df-enq 7366  df-nqqs 7367  df-plqqs 7368  df-mqqs 7369  df-1nqqs 7370  df-rq 7371  df-ltnqqs 7372  df-inp 7485  df-iplp 7487

This theorem is referenced by:  addlocprlem  7554