Theorem ltexprlemupu 6578

Description: The upper cut of our constructed difference is upper. Lemma for ltexpri 6587. (Contributed by Jim Kingdon, 21-Dec-2019.)

Hypothesis

Ref	Expression
ltexprlem.1	|- C = <. {x e. Q. | E.y(y e. ( 2nd ` A) /\ (y +Q x) e. ( 1st ` B)) } , {x e. Q. | E.y(y e. ( 1st ` A) /\ (y +Q x) e. ( 2nd ` B)) } >.

Assertion

Ref	Expression
ltexprlemupu	|- ((A <P B /\ r e. Q.) -> (E. q e. Q. ( q <Q r /\ q e. ( 2nd ` C)) -> r e. ( 2nd ` C)))

Distinct variable groups:   x,y, q, r,A   x,B,y, q, r   x, C,y, q, r

Proof of Theorem ltexprlemupu

Step	Hyp	Ref	Expression
1		simplr 482	. . . . . 6 |- (((A <P B /\ r e. Q.) /\ ( q <Q r /\ ( q e. Q. /\ (y e. ( 1st ` A) /\ (y +Q q) e. ( 2nd ` B))))) -> r e. Q.)
2		simprrr 492	. . . . . . 7 |- (((A <P B /\ r e. Q.) /\ ( q <Q r /\ ( q e. Q. /\ (y e. ( 1st ` A) /\ (y +Q q) e. ( 2nd ` B))))) -> (y e. ( 1st ` A) /\ (y +Q q) e. ( 2nd ` B)))
3	2	simpld 105	. . . . . 6 |- (((A <P B /\ r e. Q.) /\ ( q <Q r /\ ( q e. Q. /\ (y e. ( 1st ` A) /\ (y +Q q) e. ( 2nd ` B))))) -> y e. ( 1st ` A))
4		simprl 483	. . . . . . . 8 |- (((A <P B /\ r e. Q.) /\ ( q <Q r /\ ( q e. Q. /\ (y e. ( 1st ` A) /\ (y +Q q) e. ( 2nd ` B))))) -> q <Q r)
5		simpll 481	. . . . . . . . 9 |- (((A <P B /\ r e. Q.) /\ ( q <Q r /\ ( q e. Q. /\ (y e. ( 1st ` A) /\ (y +Q q) e. ( 2nd ` B))))) -> A <P B)
6		simprrl 491	. . . . . . . . . 10 |- (( q <Q r /\ ( q e. Q. /\ (y e. ( 1st ` A) /\ (y +Q q) e. ( 2nd ` B)))) -> y e. ( 1st ` A))
7	6	adantl 262	. . . . . . . . 9 |- (((A <P B /\ r e. Q.) /\ ( q <Q r /\ ( q e. Q. /\ (y e. ( 1st ` A) /\ (y +Q q) e. ( 2nd ` B))))) -> y e. ( 1st ` A))
8		ltrelpr 6488	. . . . . . . . . . . . 13 |- <P C_ ( P. X. P.)
9	8	brel 4335	. . . . . . . . . . . 12 |- (A <P B -> (A e. P. /\ B e. P.))
10	9	simpld 105	. . . . . . . . . . 11 |- (A <P B -> A e. P.)
11		prop 6458	. . . . . . . . . . 11 |- (A e. P. -> <.( 1st ` A) , ( 2nd ` A) >. e. P.)
12	10, 11	syl 14	. . . . . . . . . 10 |- (A <P B -> <.( 1st ` A) , ( 2nd ` A) >. e. P.)
13		elprnql 6464	. . . . . . . . . 10 |- (( <.( 1st ` A) , ( 2nd ` A) >. e. P. /\ y e. ( 1st ` A)) -> y e. Q.)
14	12, 13	sylan 267	. . . . . . . . 9 |- ((A <P B /\ y e. ( 1st ` A)) -> y e. Q.)
15	5, 7, 14	syl2anc 391	. . . . . . . 8 |- (((A <P B /\ r e. Q.) /\ ( q <Q r /\ ( q e. Q. /\ (y e. ( 1st ` A) /\ (y +Q q) e. ( 2nd ` B))))) -> y e. Q.)
16		ltanqi 6386	. . . . . . . 8 |- (( q <Q r /\ y e. Q.) -> (y +Q q) <Q (y +Q r))
17	4, 15, 16	syl2anc 391	. . . . . . 7 |- (((A <P B /\ r e. Q.) /\ ( q <Q r /\ ( q e. Q. /\ (y e. ( 1st ` A) /\ (y +Q q) e. ( 2nd ` B))))) -> (y +Q q) <Q (y +Q r))
18	9	simprd 107	. . . . . . . . 9 |- (A <P B -> B e. P.)
19	5, 18	syl 14	. . . . . . . 8 |- (((A <P B /\ r e. Q.) /\ ( q <Q r /\ ( q e. Q. /\ (y e. ( 1st ` A) /\ (y +Q q) e. ( 2nd ` B))))) -> B e. P.)
20	2	simprd 107	. . . . . . . 8 |- (((A <P B /\ r e. Q.) /\ ( q <Q r /\ ( q e. Q. /\ (y e. ( 1st ` A) /\ (y +Q q) e. ( 2nd ` B))))) -> (y +Q q) e. ( 2nd ` B))
21		prop 6458	. . . . . . . . 9 |- (B e. P. -> <.( 1st ` B) , ( 2nd ` B) >. e. P.)
22		prcunqu 6468	. . . . . . . . 9 |- (( <.( 1st ` B) , ( 2nd ` B) >. e. P. /\ (y +Q q) e. ( 2nd ` B)) -> ((y +Q q) <Q (y +Q r) -> (y +Q r) e. ( 2nd ` B)))
23	21, 22	sylan 267	. . . . . . . 8 |- ((B e. P. /\ (y +Q q) e. ( 2nd ` B)) -> ((y +Q q) <Q (y +Q r) -> (y +Q r) e. ( 2nd ` B)))
24	19, 20, 23	syl2anc 391	. . . . . . 7 |- (((A <P B /\ r e. Q.) /\ ( q <Q r /\ ( q e. Q. /\ (y e. ( 1st ` A) /\ (y +Q q) e. ( 2nd ` B))))) -> ((y +Q q) <Q (y +Q r) -> (y +Q r) e. ( 2nd ` B)))
25	17, 24	mpd 13	. . . . . 6 |- (((A <P B /\ r e. Q.) /\ ( q <Q r /\ ( q e. Q. /\ (y e. ( 1st ` A) /\ (y +Q q) e. ( 2nd ` B))))) -> (y +Q r) e. ( 2nd ` B))
26	1, 3, 25	jca32 293	. . . . 5 |- (((A <P B /\ r e. Q.) /\ ( q <Q r /\ ( q e. Q. /\ (y e. ( 1st ` A) /\ (y +Q q) e. ( 2nd ` B))))) -> ( r e. Q. /\ (y e. ( 1st ` A) /\ (y +Q r) e. ( 2nd ` B))))
27	26	eximi 1488	. . . 4 |- (E.y((A <P B /\ r e. Q.) /\ ( q <Q r /\ ( q e. Q. /\ (y e. ( 1st ` A) /\ (y +Q q) e. ( 2nd ` B))))) -> E.y( r e. Q. /\ (y e. ( 1st ` A) /\ (y +Q r) e. ( 2nd ` B))))
28		ltexprlem.1	. . . . . . . . . 10 |- C = <. {x e. Q. | E.y(y e. ( 2nd ` A) /\ (y +Q x) e. ( 1st ` B)) } , {x e. Q. | E.y(y e. ( 1st ` A) /\ (y +Q x) e. ( 2nd ` B)) } >.
29	28	ltexprlemelu 6573	. . . . . . . . 9 |- ( q e. ( 2nd ` C) <-> ( q e. Q. /\ E.y(y e. ( 1st ` A) /\ (y +Q q) e. ( 2nd ` B))))
30		19.42v 1783	. . . . . . . . 9 |- (E.y( q e. Q. /\ (y e. ( 1st ` A) /\ (y +Q q) e. ( 2nd ` B))) <-> ( q e. Q. /\ E.y(y e. ( 1st ` A) /\ (y +Q q) e. ( 2nd ` B))))
31	29, 30	bitr4i 176	. . . . . . . 8 |- ( q e. ( 2nd ` C) <-> E.y( q e. Q. /\ (y e. ( 1st ` A) /\ (y +Q q) e. ( 2nd ` B))))
32	31	anbi2i 430	. . . . . . 7 |- (( q <Q r /\ q e. ( 2nd ` C)) <-> ( q <Q r /\ E.y( q e. Q. /\ (y e. ( 1st ` A) /\ (y +Q q) e. ( 2nd ` B)))))
33		19.42v 1783	. . . . . . 7 |- (E.y( q <Q r /\ ( q e. Q. /\ (y e. ( 1st ` A) /\ (y +Q q) e. ( 2nd ` B)))) <-> ( q <Q r /\ E.y( q e. Q. /\ (y e. ( 1st ` A) /\ (y +Q q) e. ( 2nd ` B)))))
34	32, 33	bitr4i 176	. . . . . 6 |- (( q <Q r /\ q e. ( 2nd ` C)) <-> E.y( q <Q r /\ ( q e. Q. /\ (y e. ( 1st ` A) /\ (y +Q q) e. ( 2nd ` B)))))
35	34	anbi2i 430	. . . . 5 |- (((A <P B /\ r e. Q.) /\ ( q <Q r /\ q e. ( 2nd ` C))) <-> ((A <P B /\ r e. Q.) /\ E.y( q <Q r /\ ( q e. Q. /\ (y e. ( 1st ` A) /\ (y +Q q) e. ( 2nd ` B))))))
36		19.42v 1783	. . . . 5 |- (E.y((A <P B /\ r e. Q.) /\ ( q <Q r /\ ( q e. Q. /\ (y e. ( 1st ` A) /\ (y +Q q) e. ( 2nd ` B))))) <-> ((A <P B /\ r e. Q.) /\ E.y( q <Q r /\ ( q e. Q. /\ (y e. ( 1st ` A) /\ (y +Q q) e. ( 2nd ` B))))))
37	35, 36	bitr4i 176	. . . 4 |- (((A <P B /\ r e. Q.) /\ ( q <Q r /\ q e. ( 2nd ` C))) <-> E.y((A <P B /\ r e. Q.) /\ ( q <Q r /\ ( q e. Q. /\ (y e. ( 1st ` A) /\ (y +Q q) e. ( 2nd ` B))))))
38	28	ltexprlemelu 6573	. . . . 5 |- ( r e. ( 2nd ` C) <-> ( r e. Q. /\ E.y(y e. ( 1st ` A) /\ (y +Q r) e. ( 2nd ` B))))
39		19.42v 1783	. . . . 5 |- (E.y( r e. Q. /\ (y e. ( 1st ` A) /\ (y +Q r) e. ( 2nd ` B))) <-> ( r e. Q. /\ E.y(y e. ( 1st ` A) /\ (y +Q r) e. ( 2nd ` B))))
40	38, 39	bitr4i 176	. . . 4 |- ( r e. ( 2nd ` C) <-> E.y( r e. Q. /\ (y e. ( 1st ` A) /\ (y +Q r) e. ( 2nd ` B))))
41	27, 37, 40	3imtr4i 190	. . 3 |- (((A <P B /\ r e. Q.) /\ ( q <Q r /\ q e. ( 2nd ` C))) -> r e. ( 2nd ` C))
42	41	ex 108	. 2 |- ((A <P B /\ r e. Q.) -> (( q <Q r /\ q e. ( 2nd ` C)) -> r e. ( 2nd ` C)))
43	42	rexlimdvw 2430	1 |- ((A <P B /\ r e. Q.) -> (E. q e. Q. ( q <Q r /\ q e. ( 2nd ` C)) -> r e. ( 2nd ` C)))

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1242  E.wex 1378   e. wcel 1390  E.wrex 2301   {crab 2304   <.cop 3370   class class class wbr 3755   ` cfv 4845  (class class class)co 5455   1stc1st 5707   2ndc2nd 5708   Q.cnq 6264   +Q cplq 6266   <Q cltq 6269   P.cnp 6275   <P cltp 6279

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-ltnqqs 6337  df-inp 6449  df-iltp 6453

This theorem is referenced by:  ltexprlemrnd  6579