Theorem addlocprlem 7754

Description: Lemma for addlocpr 7755. 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 7584	. . . . . 6 |- <Q C_ ( Q. X. Q. )
3	2	brel 4778	. . . . 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 7694	. . . . . 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 7700	. . . . 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 7694	. . . . . 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 7700	. . . . 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 7594	. . . 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 7615	. . 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 7750	. . . 4 |- ( ph -> ( Q <Q ( D +Q E ) -> Q e. ( 1st ` ( A +P. B ) ) ) )
29		orc 719	. . . 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 7752	. . . 4 |- ( ph -> ( Q = ( D +Q E ) -> R e. ( 2nd ` ( A +P. B ) ) ) )
32		olc 718	. . . 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 7753	. . . 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 1340	. 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 715   \/ w3o 1003   = wceq 1397   e. wcel 2202   <.cop 3672   class class class wbr 4088   ` cfv 5326  (class class class)co 6017   1stc1st 6300   2ndc2nd 6301   Q.cnq 7499   +Q cplq 7501   <Q cltq 7504   P.cnp 7510   +P. cpp 7512

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-eprel 4386  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-1o 6581  df-oadd 6585  df-omul 6586  df-er 6701  df-ec 6703  df-qs 6707  df-ni 7523  df-pli 7524  df-mi 7525  df-lti 7526  df-plpq 7563  df-mpq 7564  df-enq 7566  df-nqqs 7567  df-plqqs 7568  df-mqqs 7569  df-1nqqs 7570  df-rq 7571  df-ltnqqs 7572  df-inp 7685  df-iplp 7687

This theorem is referenced by:  addlocpr  7755