Theorem ltexpri 7445

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 2209	. . . . . . 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 5800	. . . . . . . 8 |- ( ( y = u /\ z = v ) -> ( z +Q y ) = ( v +Q u ) )
5	4	eleq1d 2209	. . . . . . 7 |- ( ( y = u /\ z = v ) -> ( ( z +Q y ) e. ( 1st ` B ) <-> ( v +Q u ) e. ( 1st ` B ) ) )
6	2, 5	anbi12d 465	. . . . . 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 1902	. . . . 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 2688	. . . 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 2209	. . . . . . 7 |- ( ( y = u /\ z = v ) -> ( z e. ( 1st ` A ) <-> v e. ( 1st ` A ) ) )
10	4	eleq1d 2209	. . . . . . 7 |- ( ( y = u /\ z = v ) -> ( ( z +Q y ) e. ( 2nd ` B ) <-> ( v +Q u ) e. ( 2nd ` B ) ) )
11	9, 10	anbi12d 465	. . . . . 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 1902	. . . . 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 2688	. . . 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 3718	. . 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 7440	. 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 7441	. . . 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 7442	. . . 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 3119	. . 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 7443	. . . 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 7444	. . . 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 3119	. . 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 7337	. . . . . . 7 |- <P C_ ( P. X. P. )
23	22	brel 4599	. . . . . 6 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
24	23	simpld 111	. . . . 5 |- ( A <P B -> A e. P. )
25		addclpr 7369	. . . . 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 7304	. . . 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 929	. 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 5790	. . . 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 2149	. . 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 2793	. 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 1332   E.wex 1469   e. wcel 1481   E.wrex 2418   {crab 2421   <.cop 3535   class class class wbr 3937   ` cfv 5131  (class class class)co 5782   1stc1st 6044   2ndc2nd 6045   Q.cnq 7112   +Q cplq 7114   P.cnp 7123   +P. cpp 7125   <P cltp 7127

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-eprel 4219  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-1o 6321  df-2o 6322  df-oadd 6325  df-omul 6326  df-er 6437  df-ec 6439  df-qs 6443  df-ni 7136  df-pli 7137  df-mi 7138  df-lti 7139  df-plpq 7176  df-mpq 7177  df-enq 7179  df-nqqs 7180  df-plqqs 7181  df-mqqs 7182  df-1nqqs 7183  df-rq 7184  df-ltnqqs 7185  df-enq0 7256  df-nq0 7257  df-0nq0 7258  df-plq0 7259  df-mq0 7260  df-inp 7298  df-iplp 7300  df-iltp 7302

This theorem is referenced by:  lteupri  7449  ltaprlem  7450  ltaprg  7451  ltmprr  7474  recexgt0sr  7605  mulgt0sr  7610  map2psrprg  7637