Theorem addlocprlemlt 7848

Description: Lemma for addlocpr 7853. The Q <Q ( D +Q E ) case. (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
addlocprlemlt	|- ( ph -> ( Q <Q ( D +Q E ) -> Q e. ( 1st ` ( A +P. B ) ) ) )

Proof of Theorem addlocprlemlt

Step	Hyp	Ref	Expression
1		addlocprlem.a	. . 3 |- ( ph -> A e. P. )
2		addlocprlem.dlo	. . 3 |- ( ph -> D e. ( 1st ` A ) )
3	1, 2	jca 306	. 2 |- ( ph -> ( A e. P. /\ D e. ( 1st ` A ) ) )
4		addlocprlem.b	. . 3 |- ( ph -> B e. P. )
5		addlocprlem.elo	. . 3 |- ( ph -> E e. ( 1st ` B ) )
6	4, 5	jca 306	. 2 |- ( ph -> ( B e. P. /\ E e. ( 1st ` B ) ) )
7		addlocprlem.qr	. . 3 |- ( ph -> Q <Q R )
8		ltrelnq 7682	. . . . 5 |- <Q C_ ( Q. X. Q. )
9	8	brel 4804	. . . 4 |- ( Q <Q R -> ( Q e. Q. /\ R e. Q. ) )
10	9	simpld 112	. . 3 |- ( Q <Q R -> Q e. Q. )
11	7, 10	syl 14	. 2 |- ( ph -> Q e. Q. )
12		addnqprl 7846	. 2 |- ( ( ( ( A e. P. /\ D e. ( 1st ` A ) ) /\ ( B e. P. /\ E e. ( 1st ` B ) ) ) /\ Q e. Q. ) -> ( Q <Q ( D +Q E ) -> Q e. ( 1st ` ( A +P. B ) ) ) )
13	3, 6, 11, 12	syl21anc 1273	1 |- ( ph -> ( Q <Q ( D +Q E ) -> Q e. ( 1st ` ( A +P. B ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   class class class wbr 4111   ` cfv 5354  (class class class)co 6052   1stc1st 6334   2ndc2nd 6335   Q.cnq 7597   +Q cplq 7599   <Q cltq 7602   P.cnp 7608   +P. cpp 7610

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-eprel 4412  df-id 4416  df-iord 4489  df-on 4491  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-1o 6649  df-oadd 6653  df-omul 6654  df-er 6769  df-ec 6771  df-qs 6775  df-ni 7621  df-pli 7622  df-mi 7623  df-lti 7624  df-plpq 7661  df-mpq 7662  df-enq 7664  df-nqqs 7665  df-plqqs 7666  df-mqqs 7667  df-1nqqs 7668  df-rq 7669  df-ltnqqs 7670  df-inp 7783  df-iplp 7785

This theorem is referenced by:  addlocprlem  7852