Theorem addlocprlemgt 7865

Description: Lemma for addlocpr 7867. 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 7863	. . . . . 6 |- ( ph -> ( U +Q T ) <Q ( ( D +Q E ) +Q ( P +Q P ) ) )
13	12	adantr 276	. . . . 5 |- ( ( ph /\ ( D +Q E ) <Q Q ) -> ( U +Q T ) <Q ( ( D +Q E ) +Q ( P +Q P ) ) )
14		prop 7806	. . . . . . . . . . . 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 7812	. . . . . . . . . . 11 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ D e. ( 1st ` A ) ) -> D e. Q. )
17	15, 6, 16	syl2anc 411	. . . . . . . . . 10 |- ( ph -> D e. Q. )
18		prop 7806	. . . . . . . . . . . 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 7812	. . . . . . . . . . 11 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ E e. ( 1st ` B ) ) -> E e. Q. )
21	19, 9, 20	syl2anc 411	. . . . . . . . . 10 |- ( ph -> E e. Q. )
22		addclnq 7706	. . . . . . . . . 10 |- ( ( D e. Q. /\ E e. Q. ) -> ( D +Q E ) e. Q. )
23	17, 21, 22	syl2anc 411	. . . . . . . . 9 |- ( ph -> ( D +Q E ) e. Q. )
24		ltrelnq 7696	. . . . . . . . . . . 12 |- <Q C_ ( Q. X. Q. )
25	24	brel 4807	. . . . . . . . . . 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 112	. . . . . . . . 9 |- ( ph -> Q e. Q. )
28		addclnq 7706	. . . . . . . . . 10 |- ( ( P e. Q. /\ P e. Q. ) -> ( P +Q P ) e. Q. )
29	4, 4, 28	syl2anc 411	. . . . . . . . 9 |- ( ph -> ( P +Q P ) e. Q. )
30		ltanqg 7731	. . . . . . . . 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 1274	. . . . . . . 8 |- ( ph -> ( ( D +Q E ) <Q Q <-> ( ( P +Q P ) +Q ( D +Q E ) ) <Q ( ( P +Q P ) +Q Q ) ) )
32		addcomnqg 7712	. . . . . . . . . 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 411	. . . . . . . . 9 |- ( ph -> ( ( P +Q P ) +Q ( D +Q E ) ) = ( ( D +Q E ) +Q ( P +Q P ) ) )
34		addcomnqg 7712	. . . . . . . . . 10 |- ( ( ( P +Q P ) e. Q. /\ Q e. Q. ) -> ( ( P +Q P ) +Q Q ) = ( Q +Q ( P +Q P ) ) )
35	29, 27, 34	syl2anc 411	. . . . . . . . 9 |- ( ph -> ( ( P +Q P ) +Q Q ) = ( Q +Q ( P +Q P ) ) )
36	33, 35	breq12d 4127	. . . . . . . 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 188	. . . . . . 7 |- ( ph -> ( ( D +Q E ) <Q Q <-> ( ( D +Q E ) +Q ( P +Q P ) ) <Q ( Q +Q ( P +Q P ) ) ) )
38	37	biimpa 296	. . . . . 6 |- ( ( ph /\ ( D +Q E ) <Q Q ) -> ( ( D +Q E ) +Q ( P +Q P ) ) <Q ( Q +Q ( P +Q P ) ) )
39	5	breq2d 4126	. . . . . . 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 276	. . . . . 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 147	. . . . 5 |- ( ( ph /\ ( D +Q E ) <Q Q ) -> ( ( D +Q E ) +Q ( P +Q P ) ) <Q R )
42	13, 41	jca 306	. . . 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 7729	. . . . 5 |- <Q Or Q.
44	43, 24	sotri 5163	. . . 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 306	. . . . 5 |- ( ph -> ( A e. P. /\ U e. ( 2nd ` A ) ) )
47	2, 10	jca 306	. . . . 5 |- ( ph -> ( B e. P. /\ T e. ( 2nd ` B ) ) )
48	26	simprd 114	. . . . 5 |- ( ph -> R e. Q. )
49		addnqpru 7861	. . . . 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 1273	. . . 4 |- ( ph -> ( ( U +Q T ) <Q R -> R e. ( 2nd ` ( A +P. B ) ) ) )
51	50	adantr 276	. . 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 115	1 |- ( ph -> ( ( D +Q E ) <Q Q -> R e. ( 2nd ` ( A +P. B ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   <.cop 3697   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   1stc1st 6345   2ndc2nd 6346   Q.cnq 7611   +Q cplq 7613   <Q cltq 7616   P.cnp 7622   +P. cpp 7624

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-eprel 4415  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-1o 6660  df-oadd 6664  df-omul 6665  df-er 6780  df-ec 6782  df-qs 6786  df-ni 7635  df-pli 7636  df-mi 7637  df-lti 7638  df-plpq 7675  df-mpq 7676  df-enq 7678  df-nqqs 7679  df-plqqs 7680  df-mqqs 7681  df-1nqqs 7682  df-rq 7683  df-ltnqqs 7684  df-inp 7797  df-iplp 7799

This theorem is referenced by:  addlocprlem  7866