Theorem addlocprlem 7678

Description: Lemma for addlocpr 7679. 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 7508	. . . . . 6 |- <Q C_ ( Q. X. Q. )
3	2	brel 4740	. . . . 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 7618	. . . . . 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 7624	. . . . 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 7618	. . . . . 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 7624	. . . . 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 7518	. . . 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 7539	. . 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 7674	. . . 4 |- ( ph -> ( Q <Q ( D +Q E ) -> Q e. ( 1st ` ( A +P. B ) ) ) )
29		orc 714	. . . 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 7676	. . . 4 |- ( ph -> ( Q = ( D +Q E ) -> R e. ( 2nd ` ( A +P. B ) ) ) )
32		olc 713	. . . 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 7677	. . . 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 1317	. 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 710   \/ w3o 980   = wceq 1373   e. wcel 2177   <.cop 3641   class class class wbr 4054   ` cfv 5285  (class class class)co 5962   1stc1st 6242   2ndc2nd 6243   Q.cnq 7423   +Q cplq 7425   <Q cltq 7428   P.cnp 7434   +P. cpp 7436

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4170  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-iinf 4649

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-tr 4154  df-eprel 4349  df-id 4353  df-po 4356  df-iso 4357  df-iord 4426  df-on 4428  df-suc 4431  df-iom 4652  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-ov 5965  df-oprab 5966  df-mpo 5967  df-1st 6244  df-2nd 6245  df-recs 6409  df-irdg 6474  df-1o 6520  df-oadd 6524  df-omul 6525  df-er 6638  df-ec 6640  df-qs 6644  df-ni 7447  df-pli 7448  df-mi 7449  df-lti 7450  df-plpq 7487  df-mpq 7488  df-enq 7490  df-nqqs 7491  df-plqqs 7492  df-mqqs 7493  df-1nqqs 7494  df-rq 7495  df-ltnqqs 7496  df-inp 7609  df-iplp 7611

This theorem is referenced by:  addlocpr  7679