Theorem addlocprlemgt 7475

Description: Lemma for addlocpr 7477. 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 7473	. . . . . 6 |- ( ph -> ( U +Q T ) <Q ( ( D +Q E ) +Q ( P +Q P ) ) )
13	12	adantr 274	. . . . 5 |- ( ( ph /\ ( D +Q E ) <Q Q ) -> ( U +Q T ) <Q ( ( D +Q E ) +Q ( P +Q P ) ) )
14		prop 7416	. . . . . . . . . . . 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 7422	. . . . . . . . . . 11 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ D e. ( 1st ` A ) ) -> D e. Q. )
17	15, 6, 16	syl2anc 409	. . . . . . . . . 10 |- ( ph -> D e. Q. )
18		prop 7416	. . . . . . . . . . . 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 7422	. . . . . . . . . . 11 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ E e. ( 1st ` B ) ) -> E e. Q. )
21	19, 9, 20	syl2anc 409	. . . . . . . . . 10 |- ( ph -> E e. Q. )
22		addclnq 7316	. . . . . . . . . 10 |- ( ( D e. Q. /\ E e. Q. ) -> ( D +Q E ) e. Q. )
23	17, 21, 22	syl2anc 409	. . . . . . . . 9 |- ( ph -> ( D +Q E ) e. Q. )
24		ltrelnq 7306	. . . . . . . . . . . 12 |- <Q C_ ( Q. X. Q. )
25	24	brel 4656	. . . . . . . . . . 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 111	. . . . . . . . 9 |- ( ph -> Q e. Q. )
28		addclnq 7316	. . . . . . . . . 10 |- ( ( P e. Q. /\ P e. Q. ) -> ( P +Q P ) e. Q. )
29	4, 4, 28	syl2anc 409	. . . . . . . . 9 |- ( ph -> ( P +Q P ) e. Q. )
30		ltanqg 7341	. . . . . . . . 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 1228	. . . . . . . 8 |- ( ph -> ( ( D +Q E ) <Q Q <-> ( ( P +Q P ) +Q ( D +Q E ) ) <Q ( ( P +Q P ) +Q Q ) ) )
32		addcomnqg 7322	. . . . . . . . . 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 409	. . . . . . . . 9 |- ( ph -> ( ( P +Q P ) +Q ( D +Q E ) ) = ( ( D +Q E ) +Q ( P +Q P ) ) )
34		addcomnqg 7322	. . . . . . . . . 10 |- ( ( ( P +Q P ) e. Q. /\ Q e. Q. ) -> ( ( P +Q P ) +Q Q ) = ( Q +Q ( P +Q P ) ) )
35	29, 27, 34	syl2anc 409	. . . . . . . . 9 |- ( ph -> ( ( P +Q P ) +Q Q ) = ( Q +Q ( P +Q P ) ) )
36	33, 35	breq12d 3995	. . . . . . . 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 187	. . . . . . 7 |- ( ph -> ( ( D +Q E ) <Q Q <-> ( ( D +Q E ) +Q ( P +Q P ) ) <Q ( Q +Q ( P +Q P ) ) ) )
38	37	biimpa 294	. . . . . 6 |- ( ( ph /\ ( D +Q E ) <Q Q ) -> ( ( D +Q E ) +Q ( P +Q P ) ) <Q ( Q +Q ( P +Q P ) ) )
39	5	breq2d 3994	. . . . . . 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 274	. . . . . 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 146	. . . . 5 |- ( ( ph /\ ( D +Q E ) <Q Q ) -> ( ( D +Q E ) +Q ( P +Q P ) ) <Q R )
42	13, 41	jca 304	. . . 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 7339	. . . . 5 |- <Q Or Q.
44	43, 24	sotri 4999	. . . 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 304	. . . . 5 |- ( ph -> ( A e. P. /\ U e. ( 2nd ` A ) ) )
47	2, 10	jca 304	. . . . 5 |- ( ph -> ( B e. P. /\ T e. ( 2nd ` B ) ) )
48	26	simprd 113	. . . . 5 |- ( ph -> R e. Q. )
49		addnqpru 7471	. . . . 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 1227	. . . 4 |- ( ph -> ( ( U +Q T ) <Q R -> R e. ( 2nd ` ( A +P. B ) ) ) )
51	50	adantr 274	. . 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 114	1 |- ( ph -> ( ( D +Q E ) <Q Q -> R e. ( 2nd ` ( A +P. B ) ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   <.cop 3579   class class class wbr 3982   ` cfv 5188  (class class class)co 5842   1stc1st 6106   2ndc2nd 6107   Q.cnq 7221   +Q cplq 7223   <Q cltq 7226   P.cnp 7232   +P. cpp 7234

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294  df-inp 7407  df-iplp 7409

This theorem is referenced by:  addlocprlem  7476