Theorem addlocprlemgt 6724

Description: Lemma for addlocpr 6726. The ( D +Q E ) <Q Q 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
addlocprlemgt	|- ( ph -> ( ( D +Q E ) <Q Q -> R e. ( 2nd ` ( A +P. B ) ) ) )

Proof of Theorem addlocprlemgt

Step	Hyp	Ref	Expression
1		addlocprlem.a	. . . . . . 7 |- ( ph -> A e. P. )
2		addlocprlem.b	. . . . . . 7 |- ( ph -> B e. P. )
3		addlocprlem.qr	. . . . . . 7 |- ( ph -> Q <Q R )
4		addlocprlem.p	. . . . . . 7 |- ( ph -> P e. Q. )
5		addlocprlem.qppr	. . . . . . 7 |- ( ph -> ( Q +Q ( P +Q P ) ) = R )
6		addlocprlem.dlo	. . . . . . 7 |- ( ph -> D e. ( 1st ` A ) )
7		addlocprlem.uup	. . . . . . 7 |- ( ph -> U e. ( 2nd ` A ) )
8		addlocprlem.du	. . . . . . 7 |- ( ph -> U <Q ( D +Q P ) )
9		addlocprlem.elo	. . . . . . 7 |- ( ph -> E e. ( 1st ` B ) )
10		addlocprlem.tup	. . . . . . 7 |- ( ph -> T e. ( 2nd ` B ) )
11		addlocprlem.et	. . . . . . 7 |- ( ph -> T <Q ( E +Q P ) )
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	addlocprlemeqgt 6722	. . . . . 6 |- ( ph -> ( U +Q T ) <Q ( ( D +Q E ) +Q ( P +Q P ) ) )
13	12	adantr 270	. . . . 5 |- ( ( ph /\ ( D +Q E ) <Q Q ) -> ( U +Q T ) <Q ( ( D +Q E ) +Q ( P +Q P ) ) )
14		prop 6665	. . . . . . . . . . . 12 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
15	1, 14	syl 14	. . . . . . . . . . 11 |- ( ph -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
16		elprnql 6671	. . . . . . . . . . 11 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ D e. ( 1st ` A ) ) -> D e. Q. )
17	15, 6, 16	syl2anc 403	. . . . . . . . . 10 |- ( ph -> D e. Q. )
18		prop 6665	. . . . . . . . . . . 12 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
19	2, 18	syl 14	. . . . . . . . . . 11 |- ( ph -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
20		elprnql 6671	. . . . . . . . . . 11 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ E e. ( 1st ` B ) ) -> E e. Q. )
21	19, 9, 20	syl2anc 403	. . . . . . . . . 10 |- ( ph -> E e. Q. )
22		addclnq 6565	. . . . . . . . . 10 |- ( ( D e. Q. /\ E e. Q. ) -> ( D +Q E ) e. Q. )
23	17, 21, 22	syl2anc 403	. . . . . . . . 9 |- ( ph -> ( D +Q E ) e. Q. )
24		ltrelnq 6555	. . . . . . . . . . . 12 |- <Q C_ ( Q. X. Q. )
25	24	brel 4410	. . . . . . . . . . 11 |- ( Q <Q R -> ( Q e. Q. /\ R e. Q. ) )
26	3, 25	syl 14	. . . . . . . . . 10 |- ( ph -> ( Q e. Q. /\ R e. Q. ) )
27	26	simpld 110	. . . . . . . . 9 |- ( ph -> Q e. Q. )
28		addclnq 6565	. . . . . . . . . 10 |- ( ( P e. Q. /\ P e. Q. ) -> ( P +Q P ) e. Q. )
29	4, 4, 28	syl2anc 403	. . . . . . . . 9 |- ( ph -> ( P +Q P ) e. Q. )
30		ltanqg 6590	. . . . . . . . 9 |- ( ( ( D +Q E ) e. Q. /\ Q e. Q. /\ ( P +Q P ) e. Q. ) -> ( ( D +Q E ) <Q Q <-> ( ( P +Q P ) +Q ( D +Q E ) ) <Q ( ( P +Q P ) +Q Q ) ) )
31	23, 27, 29, 30	syl3anc 1169	. . . . . . . 8 |- ( ph -> ( ( D +Q E ) <Q Q <-> ( ( P +Q P ) +Q ( D +Q E ) ) <Q ( ( P +Q P ) +Q Q ) ) )
32		addcomnqg 6571	. . . . . . . . . 10 |- ( ( ( P +Q P ) e. Q. /\ ( D +Q E ) e. Q. ) -> ( ( P +Q P ) +Q ( D +Q E ) ) = ( ( D +Q E ) +Q ( P +Q P ) ) )
33	29, 23, 32	syl2anc 403	. . . . . . . . 9 |- ( ph -> ( ( P +Q P ) +Q ( D +Q E ) ) = ( ( D +Q E ) +Q ( P +Q P ) ) )
34		addcomnqg 6571	. . . . . . . . . 10 |- ( ( ( P +Q P ) e. Q. /\ Q e. Q. ) -> ( ( P +Q P ) +Q Q ) = ( Q +Q ( P +Q P ) ) )
35	29, 27, 34	syl2anc 403	. . . . . . . . 9 |- ( ph -> ( ( P +Q P ) +Q Q ) = ( Q +Q ( P +Q P ) ) )
36	33, 35	breq12d 3798	. . . . . . . 8 |- ( ph -> ( ( ( P +Q P ) +Q ( D +Q E ) ) <Q ( ( P +Q P ) +Q Q ) <-> ( ( D +Q E ) +Q ( P +Q P ) ) <Q ( Q +Q ( P +Q P ) ) ) )
37	31, 36	bitrd 186	. . . . . . 7 |- ( ph -> ( ( D +Q E ) <Q Q <-> ( ( D +Q E ) +Q ( P +Q P ) ) <Q ( Q +Q ( P +Q P ) ) ) )
38	37	biimpa 290	. . . . . 6 |- ( ( ph /\ ( D +Q E ) <Q Q ) -> ( ( D +Q E ) +Q ( P +Q P ) ) <Q ( Q +Q ( P +Q P ) ) )
39	5	breq2d 3797	. . . . . . 7 |- ( ph -> ( ( ( D +Q E ) +Q ( P +Q P ) ) <Q ( Q +Q ( P +Q P ) ) <-> ( ( D +Q E ) +Q ( P +Q P ) ) <Q R ) )
40	39	adantr 270	. . . . . 6 |- ( ( ph /\ ( D +Q E ) <Q Q ) -> ( ( ( D +Q E ) +Q ( P +Q P ) ) <Q ( Q +Q ( P +Q P ) ) <-> ( ( D +Q E ) +Q ( P +Q P ) ) <Q R ) )
41	38, 40	mpbid 145	. . . . 5 |- ( ( ph /\ ( D +Q E ) <Q Q ) -> ( ( D +Q E ) +Q ( P +Q P ) ) <Q R )
42	13, 41	jca 300	. . . 4 |- ( ( ph /\ ( D +Q E ) <Q Q ) -> ( ( U +Q T ) <Q ( ( D +Q E ) +Q ( P +Q P ) ) /\ ( ( D +Q E ) +Q ( P +Q P ) ) <Q R ) )
43		ltsonq 6588	. . . . 5 |- <Q Or Q.
44	43, 24	sotri 4740	. . . 4 |- ( ( ( U +Q T ) <Q ( ( D +Q E ) +Q ( P +Q P ) ) /\ ( ( D +Q E ) +Q ( P +Q P ) ) <Q R ) -> ( U +Q T ) <Q R )
45	42, 44	syl 14	. . 3 |- ( ( ph /\ ( D +Q E ) <Q Q ) -> ( U +Q T ) <Q R )
46	1, 7	jca 300	. . . . 5 |- ( ph -> ( A e. P. /\ U e. ( 2nd ` A ) ) )
47	2, 10	jca 300	. . . . 5 |- ( ph -> ( B e. P. /\ T e. ( 2nd ` B ) ) )
48	26	simprd 112	. . . . 5 |- ( ph -> R e. Q. )
49		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 ) ) ) )
50	46, 47, 48, 49	syl21anc 1168	. . . 4 |- ( ph -> ( ( U +Q T ) <Q R -> R e. ( 2nd ` ( A +P. B ) ) ) )
51	50	adantr 270	. . 3 |- ( ( ph /\ ( D +Q E ) <Q Q ) -> ( ( U +Q T ) <Q R -> R e. ( 2nd ` ( A +P. B ) ) ) )
52	45, 51	mpd 13	. 2 |- ( ( ph /\ ( D +Q E ) <Q Q ) -> R e. ( 2nd ` ( A +P. B ) ) )
53	52	ex 113	1 |- ( ph -> ( ( D +Q E ) <Q Q -> R e. ( 2nd ` ( A +P. B ) ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   <.cop 3401   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