Theorem ltexpri 7575

Description: Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 13-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.)

Assertion

Ref	Expression
ltexpri	|- ( A <P B -> E. x e. P. ( A +P. x ) = B )

Distinct variable groups:   x, A   x, B

Proof of Theorem ltexpri

Dummy variables y z u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 109	. . . . . . . 8 |- ( ( y = u /\ z = v ) -> z = v )
2	1	eleq1d 2239	. . . . . . 7 |- ( ( y = u /\ z = v ) -> ( z e. ( 2nd ` A ) <-> v e. ( 2nd ` A ) ) )
3		simpl 108	. . . . . . . . 9 |- ( ( y = u /\ z = v ) -> y = u )
4	1, 3	oveq12d 5871	. . . . . . . 8 |- ( ( y = u /\ z = v ) -> ( z +Q y ) = ( v +Q u ) )
5	4	eleq1d 2239	. . . . . . 7 |- ( ( y = u /\ z = v ) -> ( ( z +Q y ) e. ( 1st ` B ) <-> ( v +Q u ) e. ( 1st ` B ) ) )
6	2, 5	anbi12d 470	. . . . . 6 |- ( ( y = u /\ z = v ) -> ( ( z e. ( 2nd ` A ) /\ ( z +Q y ) e. ( 1st ` B ) ) <-> ( v e. ( 2nd ` A ) /\ ( v +Q u ) e. ( 1st ` B ) ) ) )
7	6	cbvexdva 1922	. . . . 5 |- ( y = u -> ( E. z ( z e. ( 2nd ` A ) /\ ( z +Q y ) e. ( 1st ` B ) ) <-> E. v ( v e. ( 2nd ` A ) /\ ( v +Q u ) e. ( 1st ` B ) ) ) )
8	7	cbvrabv 2729	. . . 4 |- { y e. Q. | E. z ( z e. ( 2nd ` A ) /\ ( z +Q y ) e. ( 1st ` B ) ) } = { u e. Q. | E. v ( v e. ( 2nd ` A ) /\ ( v +Q u ) e. ( 1st ` B ) ) }
9	1	eleq1d 2239	. . . . . . 7 |- ( ( y = u /\ z = v ) -> ( z e. ( 1st ` A ) <-> v e. ( 1st ` A ) ) )
10	4	eleq1d 2239	. . . . . . 7 |- ( ( y = u /\ z = v ) -> ( ( z +Q y ) e. ( 2nd ` B ) <-> ( v +Q u ) e. ( 2nd ` B ) ) )
11	9, 10	anbi12d 470	. . . . . 6 |- ( ( y = u /\ z = v ) -> ( ( z e. ( 1st ` A ) /\ ( z +Q y ) e. ( 2nd ` B ) ) <-> ( v e. ( 1st ` A ) /\ ( v +Q u ) e. ( 2nd ` B ) ) ) )
12	11	cbvexdva 1922	. . . . 5 |- ( y = u -> ( E. z ( z e. ( 1st ` A ) /\ ( z +Q y ) e. ( 2nd ` B ) ) <-> E. v ( v e. ( 1st ` A ) /\ ( v +Q u ) e. ( 2nd ` B ) ) ) )
13	12	cbvrabv 2729	. . . 4 |- { y e. Q. | E. z ( z e. ( 1st ` A ) /\ ( z +Q y ) e. ( 2nd ` B ) ) } = { u e. Q. | E. v ( v e. ( 1st ` A ) /\ ( v +Q u ) e. ( 2nd ` B ) ) }
14	8, 13	opeq12i 3770	. . 3 |- <. { y e. Q. | E. z ( z e. ( 2nd ` A ) /\ ( z +Q y ) e. ( 1st ` B ) ) } , { y e. Q. | E. z ( z e. ( 1st ` A ) /\ ( z +Q y ) e. ( 2nd ` B ) ) } >. = <. { u e. Q. | E. v ( v e. ( 2nd ` A ) /\ ( v +Q u ) e. ( 1st ` B ) ) } , { u e. Q. | E. v ( v e. ( 1st ` A ) /\ ( v +Q u ) e. ( 2nd ` B ) ) } >.
15	14	ltexprlempr 7570	. 2 |- ( A <P B -> <. { y e. Q. | E. z ( z e. ( 2nd ` A ) /\ ( z +Q y ) e. ( 1st ` B ) ) } , { y e. Q. | E. z ( z e. ( 1st ` A ) /\ ( z +Q y ) e. ( 2nd ` B ) ) } >. e. P. )
16	14	ltexprlemfl 7571	. . . 4 |- ( A <P B -> ( 1st ` ( A +P. <. { y e. Q. | E. z ( z e. ( 2nd ` A ) /\ ( z +Q y ) e. ( 1st ` B ) ) } , { y e. Q. | E. z ( z e. ( 1st ` A ) /\ ( z +Q y ) e. ( 2nd ` B ) ) } >. ) ) C_ ( 1st ` B ) )
17	14	ltexprlemrl 7572	. . . 4 |- ( A <P B -> ( 1st ` B ) C_ ( 1st ` ( A +P. <. { y e. Q. | E. z ( z e. ( 2nd ` A ) /\ ( z +Q y ) e. ( 1st ` B ) ) } , { y e. Q. | E. z ( z e. ( 1st ` A ) /\ ( z +Q y ) e. ( 2nd ` B ) ) } >. ) ) )
18	16, 17	eqssd 3164	. . 3 |- ( A <P B -> ( 1st ` ( A +P. <. { y e. Q. | E. z ( z e. ( 2nd ` A ) /\ ( z +Q y ) e. ( 1st ` B ) ) } , { y e. Q. | E. z ( z e. ( 1st ` A ) /\ ( z +Q y ) e. ( 2nd ` B ) ) } >. ) ) = ( 1st ` B ) )
19	14	ltexprlemfu 7573	. . . 4 |- ( A <P B -> ( 2nd ` ( A +P. <. { y e. Q. | E. z ( z e. ( 2nd ` A ) /\ ( z +Q y ) e. ( 1st ` B ) ) } , { y e. Q. | E. z ( z e. ( 1st ` A ) /\ ( z +Q y ) e. ( 2nd ` B ) ) } >. ) ) C_ ( 2nd ` B ) )
20	14	ltexprlemru 7574	. . . 4 |- ( A <P B -> ( 2nd ` B ) C_ ( 2nd ` ( A +P. <. { y e. Q. | E. z ( z e. ( 2nd ` A ) /\ ( z +Q y ) e. ( 1st ` B ) ) } , { y e. Q. | E. z ( z e. ( 1st ` A ) /\ ( z +Q y ) e. ( 2nd ` B ) ) } >. ) ) )
21	19, 20	eqssd 3164	. . 3 |- ( A <P B -> ( 2nd ` ( A +P. <. { y e. Q. | E. z ( z e. ( 2nd ` A ) /\ ( z +Q y ) e. ( 1st ` B ) ) } , { y e. Q. | E. z ( z e. ( 1st ` A ) /\ ( z +Q y ) e. ( 2nd ` B ) ) } >. ) ) = ( 2nd ` B ) )
22		ltrelpr 7467	. . . . . . 7 |- <P C_ ( P. X. P. )
23	22	brel 4663	. . . . . 6 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
24	23	simpld 111	. . . . 5 |- ( A <P B -> A e. P. )
25		addclpr 7499	. . . . 5 |- ( ( A e. P. /\ <. { y e. Q. | E. z ( z e. ( 2nd ` A ) /\ ( z +Q y ) e. ( 1st ` B ) ) } , { y e. Q. | E. z ( z e. ( 1st ` A ) /\ ( z +Q y ) e. ( 2nd ` B ) ) } >. e. P. ) -> ( A +P. <. { y e. Q. | E. z ( z e. ( 2nd ` A ) /\ ( z +Q y ) e. ( 1st ` B ) ) } , { y e. Q. | E. z ( z e. ( 1st ` A ) /\ ( z +Q y ) e. ( 2nd ` B ) ) } >. ) e. P. )
26	24, 15, 25	syl2anc 409	. . . 4 |- ( A <P B -> ( A +P. <. { y e. Q. | E. z ( z e. ( 2nd ` A ) /\ ( z +Q y ) e. ( 1st ` B ) ) } , { y e. Q. | E. z ( z e. ( 1st ` A ) /\ ( z +Q y ) e. ( 2nd ` B ) ) } >. ) e. P. )
27	23	simprd 113	. . . 4 |- ( A <P B -> B e. P. )
28		preqlu 7434	. . . 4 |- ( ( ( A +P. <. { y e. Q. | E. z ( z e. ( 2nd ` A ) /\ ( z +Q y ) e. ( 1st ` B ) ) } , { y e. Q. | E. z ( z e. ( 1st ` A ) /\ ( z +Q y ) e. ( 2nd ` B ) ) } >. ) e. P. /\ B e. P. ) -> ( ( A +P. <. { y e. Q. | E. z ( z e. ( 2nd ` A ) /\ ( z +Q y ) e. ( 1st ` B ) ) } , { y e. Q. | E. z ( z e. ( 1st ` A ) /\ ( z +Q y ) e. ( 2nd ` B ) ) } >. ) = B <-> ( ( 1st ` ( A +P. <. { y e. Q. | E. z ( z e. ( 2nd ` A ) /\ ( z +Q y ) e. ( 1st ` B ) ) } , { y e. Q. | E. z ( z e. ( 1st ` A ) /\ ( z +Q y ) e. ( 2nd ` B ) ) } >. ) ) = ( 1st ` B ) /\ ( 2nd ` ( A +P. <. { y e. Q. | E. z ( z e. ( 2nd ` A ) /\ ( z +Q y ) e. ( 1st ` B ) ) } , { y e. Q. | E. z ( z e. ( 1st ` A ) /\ ( z +Q y ) e. ( 2nd ` B ) ) } >. ) ) = ( 2nd ` B ) ) ) )
29	26, 27, 28	syl2anc 409	. . 3 |- ( A <P B -> ( ( A +P. <. { y e. Q. | E. z ( z e. ( 2nd ` A ) /\ ( z +Q y ) e. ( 1st ` B ) ) } , { y e. Q. | E. z ( z e. ( 1st ` A ) /\ ( z +Q y ) e. ( 2nd ` B ) ) } >. ) = B <-> ( ( 1st ` ( A +P. <. { y e. Q. | E. z ( z e. ( 2nd ` A ) /\ ( z +Q y ) e. ( 1st ` B ) ) } , { y e. Q. | E. z ( z e. ( 1st ` A ) /\ ( z +Q y ) e. ( 2nd ` B ) ) } >. ) ) = ( 1st ` B ) /\ ( 2nd ` ( A +P. <. { y e. Q. | E. z ( z e. ( 2nd ` A ) /\ ( z +Q y ) e. ( 1st ` B ) ) } , { y e. Q. | E. z ( z e. ( 1st ` A ) /\ ( z +Q y ) e. ( 2nd ` B ) ) } >. ) ) = ( 2nd ` B ) ) ) )
30	18, 21, 29	mpbir2and 939	. 2 |- ( A <P B -> ( A +P. <. { y e. Q. | E. z ( z e. ( 2nd ` A ) /\ ( z +Q y ) e. ( 1st ` B ) ) } , { y e. Q. | E. z ( z e. ( 1st ` A ) /\ ( z +Q y ) e. ( 2nd ` B ) ) } >. ) = B )
31		oveq2 5861	. . . 4 |- ( x = <. { y e. Q. | E. z ( z e. ( 2nd ` A ) /\ ( z +Q y ) e. ( 1st ` B ) ) } , { y e. Q. | E. z ( z e. ( 1st ` A ) /\ ( z +Q y ) e. ( 2nd ` B ) ) } >. -> ( A +P. x ) = ( A +P. <. { y e. Q. | E. z ( z e. ( 2nd ` A ) /\ ( z +Q y ) e. ( 1st ` B ) ) } , { y e. Q. | E. z ( z e. ( 1st ` A ) /\ ( z +Q y ) e. ( 2nd ` B ) ) } >. ) )
32	31	eqeq1d 2179	. . 3 |- ( x = <. { y e. Q. | E. z ( z e. ( 2nd ` A ) /\ ( z +Q y ) e. ( 1st ` B ) ) } , { y e. Q. | E. z ( z e. ( 1st ` A ) /\ ( z +Q y ) e. ( 2nd ` B ) ) } >. -> ( ( A +P. x ) = B <-> ( A +P. <. { y e. Q. | E. z ( z e. ( 2nd ` A ) /\ ( z +Q y ) e. ( 1st ` B ) ) } , { y e. Q. | E. z ( z e. ( 1st ` A ) /\ ( z +Q y ) e. ( 2nd ` B ) ) } >. ) = B ) )
33	32	rspcev 2834	. 2 |- ( ( <. { y e. Q. | E. z ( z e. ( 2nd ` A ) /\ ( z +Q y ) e. ( 1st ` B ) ) } , { y e. Q. | E. z ( z e. ( 1st ` A ) /\ ( z +Q y ) e. ( 2nd ` B ) ) } >. e. P. /\ ( A +P. <. { y e. Q. | E. z ( z e. ( 2nd ` A ) /\ ( z +Q y ) e. ( 1st ` B ) ) } , { y e. Q. | E. z ( z e. ( 1st ` A ) /\ ( z +Q y ) e. ( 2nd ` B ) ) } >. ) = B ) -> E. x e. P. ( A +P. x ) = B )
34	15, 30, 33	syl2anc 409	1 |- ( A <P B -> E. x e. P. ( A +P. x ) = B )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   E.wex 1485   e. wcel 2141   E.wrex 2449   {crab 2452   <.cop 3586   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   1stc1st 6117   2ndc2nd 6118   Q.cnq 7242   +Q cplq 7244   P.cnp 7253   +P. cpp 7255   <P cltp 7257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-iplp 7430  df-iltp 7432

This theorem is referenced by:  lteupri  7579  ltaprlem  7580  ltaprg  7581  ltmprr  7604  recexgt0sr  7735  mulgt0sr  7740  map2psrprg  7767