Theorem addlocprlem 7733

Description: Lemma for addlocpr 7734. 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 7563	. . . . . 6 |- <Q C_ ( Q. X. Q. )
3	2	brel 4771	. . . . 5 |- ( Q <Q R -> ( Q e. Q. /\ R e. Q. ) )
4	3	simpld 112	. . . 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 7673	. . . . . 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 7679	. . . . 5 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ D e. ( 1st ` A ) ) -> D e. Q. )
11	8, 9, 10	syl2anc 411	. . . 4 |- ( ph -> D e. Q. )
12		addlocprlem.b	. . . . . 6 |- ( ph -> B e. P. )
13		prop 7673	. . . . . 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 7679	. . . . 5 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ E e. ( 1st ` B ) ) -> E e. Q. )
17	14, 15, 16	syl2anc 411	. . . 4 |- ( ph -> E e. Q. )
18		addclnq 7573	. . . 4 |- ( ( D e. Q. /\ E e. Q. ) -> ( D +Q E ) e. Q. )
19	11, 17, 18	syl2anc 411	. . 3 |- ( ph -> ( D +Q E ) e. Q. )
20		nqtri3or 7594	. . 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 411	. 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 7729	. . . 4 |- ( ph -> ( Q <Q ( D +Q E ) -> Q e. ( 1st ` ( A +P. B ) ) ) )
29		orc 717	. . . 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 7731	. . . 4 |- ( ph -> ( Q = ( D +Q E ) -> R e. ( 2nd ` ( A +P. B ) ) ) )
32		olc 716	. . . 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 7732	. . . 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 1338	. 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 713   \/ w3o 1001   = wceq 1395   e. wcel 2200   <.cop 3669   class class class wbr 4083   ` cfv 5318  (class class class)co 6007   1stc1st 6290   2ndc2nd 6291   Q.cnq 7478   +Q cplq 7480   <Q cltq 7483   P.cnp 7489   +P. cpp 7491

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4380  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-irdg 6522  df-1o 6568  df-oadd 6572  df-omul 6573  df-er 6688  df-ec 6690  df-qs 6694  df-ni 7502  df-pli 7503  df-mi 7504  df-lti 7505  df-plpq 7542  df-mpq 7543  df-enq 7545  df-nqqs 7546  df-plqqs 7547  df-mqqs 7548  df-1nqqs 7549  df-rq 7550  df-ltnqqs 7551  df-inp 7664  df-iplp 7666

This theorem is referenced by:  addlocpr  7734