Theorem ltexpri 7603

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 110	. . . . . . . 8 |- ( ( y = u /\ z = v ) -> z = v )
2	1	eleq1d 2246	. . . . . . 7 |- ( ( y = u /\ z = v ) -> ( z e. ( 2nd ` A ) <-> v e. ( 2nd ` A ) ) )
3		simpl 109	. . . . . . . . 9 |- ( ( y = u /\ z = v ) -> y = u )
4	1, 3	oveq12d 5887	. . . . . . . 8 |- ( ( y = u /\ z = v ) -> ( z +Q y ) = ( v +Q u ) )
5	4	eleq1d 2246	. . . . . . 7 |- ( ( y = u /\ z = v ) -> ( ( z +Q y ) e. ( 1st ` B ) <-> ( v +Q u ) e. ( 1st ` B ) ) )
6	2, 5	anbi12d 473	. . . . . 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 1929	. . . . 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 2736	. . . 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 2246	. . . . . . 7 |- ( ( y = u /\ z = v ) -> ( z e. ( 1st ` A ) <-> v e. ( 1st ` A ) ) )
10	4	eleq1d 2246	. . . . . . 7 |- ( ( y = u /\ z = v ) -> ( ( z +Q y ) e. ( 2nd ` B ) <-> ( v +Q u ) e. ( 2nd ` B ) ) )
11	9, 10	anbi12d 473	. . . . . 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 1929	. . . . 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 2736	. . . 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 3781	. . 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 7598	. 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 7599	. . . 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 7600	. . . 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 3172	. . 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 7601	. . . 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 7602	. . . 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 3172	. . 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 7495	. . . . . . 7 |- <P C_ ( P. X. P. )
23	22	brel 4675	. . . . . 6 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
24	23	simpld 112	. . . . 5 |- ( A <P B -> A e. P. )
25		addclpr 7527	. . . . 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 411	. . . 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 114	. . . 4 |- ( A <P B -> B e. P. )
28		preqlu 7462	. . . 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 411	. . 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 944	. 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 5877	. . . 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 2186	. . 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 2841	. 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 411	1 |- ( A <P B -> E. x e. P. ( A +P. x ) = B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   E.wex 1492   e. wcel 2148   E.wrex 2456   {crab 2459   <.cop 3594   class class class wbr 4000   ` cfv 5212  (class class class)co 5869   1stc1st 6133   2ndc2nd 6134   Q.cnq 7270   +Q cplq 7272   P.cnp 7281   +P. cpp 7283   <P cltp 7285

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-2o 6412  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343  df-enq0 7414  df-nq0 7415  df-0nq0 7416  df-plq0 7417  df-mq0 7418  df-inp 7456  df-iplp 7458  df-iltp 7460

This theorem is referenced by:  lteupri  7607  ltaprlem  7608  ltaprg  7609  ltmprr  7632  recexgt0sr  7763  mulgt0sr  7768  map2psrprg  7795