Theorem ltexpri 7970

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 2307	. . . . . . 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 6093	. . . . . . . 8 |- ( ( y = u /\ z = v ) -> ( z +Q y ) = ( v +Q u ) )
5	4	eleq1d 2307	. . . . . . 7 |- ( ( y = u /\ z = v ) -> ( ( z +Q y ) e. ( 1st ` B ) <-> ( v +Q u ) e. ( 1st ` B ) ) )
6	2, 5	anbi12d 477	. . . . . 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 1985	. . . . 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 2820	. . . 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 2307	. . . . . . 7 |- ( ( y = u /\ z = v ) -> ( z e. ( 1st ` A ) <-> v e. ( 1st ` A ) ) )
10	4	eleq1d 2307	. . . . . . 7 |- ( ( y = u /\ z = v ) -> ( ( z +Q y ) e. ( 2nd ` B ) <-> ( v +Q u ) e. ( 2nd ` B ) ) )
11	9, 10	anbi12d 477	. . . . . 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 1985	. . . . 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 2820	. . . 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 3904	. . 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 7965	. 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 7966	. . . 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 7967	. . . 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 3265	. . 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 7968	. . . 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 7969	. . . 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 3265	. . 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 7862	. . . . . . 7 |- <P C_ ( P. X. P. )
23	22	brel 4822	. . . . . 6 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
24	23	simpld 112	. . . . 5 |- ( A <P B -> A e. P. )
25		addclpr 7894	. . . . 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 415	. . . 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 7829	. . . 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 415	. . 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 957	. 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 6083	. . . 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 2247	. . 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 2929	. 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 415	1 |- ( A <P B -> E. x e. P. ( A +P. x ) = B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   E.wex 1545   e. wcel 2209   E.wrex 2529   {crab 2532   <.cop 3708   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   1stc1st 6362   2ndc2nd 6363   Q.cnq 7637   +Q cplq 7639   P.cnp 7648   +P. cpp 7650   <P cltp 7652

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-eprel 4429  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-1o 6677  df-2o 6678  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-pli 7662  df-mi 7663  df-lti 7664  df-plpq 7701  df-mpq 7702  df-enq 7704  df-nqqs 7705  df-plqqs 7706  df-mqqs 7707  df-1nqqs 7708  df-rq 7709  df-ltnqqs 7710  df-enq0 7781  df-nq0 7782  df-0nq0 7783  df-plq0 7784  df-mq0 7785  df-inp 7823  df-iplp 7825  df-iltp 7827

This theorem is referenced by:  lteupri  7974  ltaprlem  7975  ltaprg  7976  ltmprr  7999  recexgt0sr  8130  mulgt0sr  8135  map2psrprg  8162