Theorem addlocprlem 7346

Description: Lemma for addlocpr 7347. The result, in deduction form. (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
addlocprlem	|- ( ph -> ( Q e. ( 1st ` ( A +P. B ) ) \/ R e. ( 2nd ` ( A +P. B ) ) ) )

Proof of Theorem addlocprlem

Step	Hyp	Ref	Expression
1		addlocprlem.qr	. . . 4 |- ( ph -> Q <Q R )
2		ltrelnq 7176	. . . . . 6 |- <Q C_ ( Q. X. Q. )
3	2	brel 4591	. . . . 5 |- ( Q <Q R -> ( Q e. Q. /\ R e. Q. ) )
4	3	simpld 111	. . . 4 |- ( Q <Q R -> Q e. Q. )
5	1, 4	syl 14	. . 3 |- ( ph -> Q e. Q. )
6		addlocprlem.a	. . . . . 6 |- ( ph -> A e. P. )
7		prop 7286	. . . . . 6 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
8	6, 7	syl 14	. . . . 5 |- ( ph -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
9		addlocprlem.dlo	. . . . 5 |- ( ph -> D e. ( 1st ` A ) )
10		elprnql 7292	. . . . 5 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ D e. ( 1st ` A ) ) -> D e. Q. )
11	8, 9, 10	syl2anc 408	. . . 4 |- ( ph -> D e. Q. )
12		addlocprlem.b	. . . . . 6 |- ( ph -> B e. P. )
13		prop 7286	. . . . . 6 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
14	12, 13	syl 14	. . . . 5 |- ( ph -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
15		addlocprlem.elo	. . . . 5 |- ( ph -> E e. ( 1st ` B ) )
16		elprnql 7292	. . . . 5 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ E e. ( 1st ` B ) ) -> E e. Q. )
17	14, 15, 16	syl2anc 408	. . . 4 |- ( ph -> E e. Q. )
18		addclnq 7186	. . . 4 |- ( ( D e. Q. /\ E e. Q. ) -> ( D +Q E ) e. Q. )
19	11, 17, 18	syl2anc 408	. . 3 |- ( ph -> ( D +Q E ) e. Q. )
20		nqtri3or 7207	. . 3 |- ( ( Q e. Q. /\ ( D +Q E ) e. Q. ) -> ( Q <Q ( D +Q E ) \/ Q = ( D +Q E ) \/ ( D +Q E ) <Q Q ) )
21	5, 19, 20	syl2anc 408	. 2 |- ( ph -> ( Q <Q ( D +Q E ) \/ Q = ( D +Q E ) \/ ( D +Q E ) <Q Q ) )
22		addlocprlem.p	. . . . 5 |- ( ph -> P e. Q. )
23		addlocprlem.qppr	. . . . 5 |- ( ph -> ( Q +Q ( P +Q P ) ) = R )
24		addlocprlem.uup	. . . . 5 |- ( ph -> U e. ( 2nd ` A ) )
25		addlocprlem.du	. . . . 5 |- ( ph -> U <Q ( D +Q P ) )
26		addlocprlem.tup	. . . . 5 |- ( ph -> T e. ( 2nd ` B ) )
27		addlocprlem.et	. . . . 5 |- ( ph -> T <Q ( E +Q P ) )
28	6, 12, 1, 22, 23, 9, 24, 25, 15, 26, 27	addlocprlemlt 7342	. . . 4 |- ( ph -> ( Q <Q ( D +Q E ) -> Q e. ( 1st ` ( A +P. B ) ) ) )
29		orc 701	. . . 4 |- ( Q e. ( 1st ` ( A +P. B ) ) -> ( Q e. ( 1st ` ( A +P. B ) ) \/ R e. ( 2nd ` ( A +P. B ) ) ) )
30	28, 29	syl6 33	. . 3 |- ( ph -> ( Q <Q ( D +Q E ) -> ( Q e. ( 1st ` ( A +P. B ) ) \/ R e. ( 2nd ` ( A +P. B ) ) ) ) )
31	6, 12, 1, 22, 23, 9, 24, 25, 15, 26, 27	addlocprlemeq 7344	. . . 4 |- ( ph -> ( Q = ( D +Q E ) -> R e. ( 2nd ` ( A +P. B ) ) ) )
32		olc 700	. . . 4 |- ( R e. ( 2nd ` ( A +P. B ) ) -> ( Q e. ( 1st ` ( A +P. B ) ) \/ R e. ( 2nd ` ( A +P. B ) ) ) )
33	31, 32	syl6 33	. . 3 |- ( ph -> ( Q = ( D +Q E ) -> ( Q e. ( 1st ` ( A +P. B ) ) \/ R e. ( 2nd ` ( A +P. B ) ) ) ) )
34	6, 12, 1, 22, 23, 9, 24, 25, 15, 26, 27	addlocprlemgt 7345	. . . 4 |- ( ph -> ( ( D +Q E ) <Q Q -> R e. ( 2nd ` ( A +P. B ) ) ) )
35	34, 32	syl6 33	. . 3 |- ( ph -> ( ( D +Q E ) <Q Q -> ( Q e. ( 1st ` ( A +P. B ) ) \/ R e. ( 2nd ` ( A +P. B ) ) ) ) )
36	30, 33, 35	3jaod 1282	. 2 |- ( ph -> ( ( Q <Q ( D +Q E ) \/ Q = ( D +Q E ) \/ ( D +Q E ) <Q Q ) -> ( Q e. ( 1st ` ( A +P. B ) ) \/ R e. ( 2nd ` ( A +P. B ) ) ) ) )
37	21, 36	mpd 13	1 |- ( ph -> ( Q e. ( 1st ` ( A +P. B ) ) \/ R e. ( 2nd ` ( A +P. B ) ) ) )

Syntax hints:   -> wi 4   \/ wo 697   \/ w3o 961   = wceq 1331   e. wcel 1480   <.cop 3530   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   1stc1st 6036   2ndc2nd 6037   Q.cnq 7091   +Q cplq 7093   <Q cltq 7096   P.cnp 7102   +P. cpp 7104

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7115  df-pli 7116  df-mi 7117  df-lti 7118  df-plpq 7155  df-mpq 7156  df-enq 7158  df-nqqs 7159  df-plqqs 7160  df-mqqs 7161  df-1nqqs 7162  df-rq 7163  df-ltnqqs 7164  df-inp 7277  df-iplp 7279

This theorem is referenced by:  addlocpr  7347