Theorem addlocprlem 7536

Description: Lemma for addlocpr 7537. 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 7366	. . . . . 6 |- <Q C_ ( Q. X. Q. )
3	2	brel 4680	. . . . 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 7476	. . . . . 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 7482	. . . . 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 7476	. . . . . 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 7482	. . . . 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 7376	. . . 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 7397	. . 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 7532	. . . 4 |- ( ph -> ( Q <Q ( D +Q E ) -> Q e. ( 1st ` ( A +P. B ) ) ) )
29		orc 712	. . . 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 7534	. . . 4 |- ( ph -> ( Q = ( D +Q E ) -> R e. ( 2nd ` ( A +P. B ) ) ) )
32		olc 711	. . . 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 7535	. . . 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 1304	. 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 708   \/ w3o 977   = wceq 1353   e. wcel 2148   <.cop 3597   class class class wbr 4005   ` cfv 5218  (class class class)co 5877   1stc1st 6141   2ndc2nd 6142   Q.cnq 7281   +Q cplq 7283   <Q cltq 7286   P.cnp 7292   +P. cpp 7294

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-pli 7306  df-mi 7307  df-lti 7308  df-plpq 7345  df-mpq 7346  df-enq 7348  df-nqqs 7349  df-plqqs 7350  df-mqqs 7351  df-1nqqs 7352  df-rq 7353  df-ltnqqs 7354  df-inp 7467  df-iplp 7469

This theorem is referenced by:  addlocpr  7537