Theorem ltexprlemrnd 7738

Description: Our constructed difference is rounded. Lemma for ltexpri 7746. (Contributed by Jim Kingdon, 17-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
ltexprlemrnd	|- ( A <P B -> ( A. q e. Q. ( q e. ( 1st ` C ) <-> E. r e. Q. ( q <Q r /\ r e. ( 1st ` C ) ) ) /\ A. r e. Q. ( r e. ( 2nd ` C ) <-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` C ) ) ) ) )

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

Proof of Theorem ltexprlemrnd

Step	Hyp	Ref	Expression
1		ltexprlem.1	. . . . . 6 |- 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 ) ) } >.
2	1	ltexprlemopl 7734	. . . . 5 |- ( ( A <P B /\ q e. Q. /\ q e. ( 1st ` C ) ) -> E. r e. Q. ( q <Q r /\ r e. ( 1st ` C ) ) )
3	2	3expia 1208	. . . 4 |- ( ( A <P B /\ q e. Q. ) -> ( q e. ( 1st ` C ) -> E. r e. Q. ( q <Q r /\ r e. ( 1st ` C ) ) ) )
4	1	ltexprlemlol 7735	. . . 4 |- ( ( A <P B /\ q e. Q. ) -> ( E. r e. Q. ( q <Q r /\ r e. ( 1st ` C ) ) -> q e. ( 1st ` C ) ) )
5	3, 4	impbid 129	. . 3 |- ( ( A <P B /\ q e. Q. ) -> ( q e. ( 1st ` C ) <-> E. r e. Q. ( q <Q r /\ r e. ( 1st ` C ) ) ) )
6	5	ralrimiva 2580	. 2 |- ( A <P B -> A. q e. Q. ( q e. ( 1st ` C ) <-> E. r e. Q. ( q <Q r /\ r e. ( 1st ` C ) ) ) )
7	1	ltexprlemopu 7736	. . . . 5 |- ( ( A <P B /\ r e. Q. /\ r e. ( 2nd ` C ) ) -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` C ) ) )
8	7	3expia 1208	. . . 4 |- ( ( A <P B /\ r e. Q. ) -> ( r e. ( 2nd ` C ) -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` C ) ) ) )
9	1	ltexprlemupu 7737	. . . 4 |- ( ( A <P B /\ r e. Q. ) -> ( E. q e. Q. ( q <Q r /\ q e. ( 2nd ` C ) ) -> r e. ( 2nd ` C ) ) )
10	8, 9	impbid 129	. . 3 |- ( ( A <P B /\ r e. Q. ) -> ( r e. ( 2nd ` C ) <-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` C ) ) ) )
11	10	ralrimiva 2580	. 2 |- ( A <P B -> A. r e. Q. ( r e. ( 2nd ` C ) <-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` C ) ) ) )
12	6, 11	jca 306	1 |- ( A <P B -> ( A. q e. Q. ( q e. ( 1st ` C ) <-> E. r e. Q. ( q <Q r /\ r e. ( 1st ` C ) ) ) /\ A. r e. Q. ( r e. ( 2nd ` C ) <-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` C ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   E.wex 1516   e. wcel 2177   A.wral 2485   E.wrex 2486   {crab 2489   <.cop 3641   class class class wbr 4051   ` cfv 5280  (class class class)co 5957   1stc1st 6237   2ndc2nd 6238   Q.cnq 7413   +Q cplq 7415   <Q cltq 7418   <P cltp 7428

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-eprel 4344  df-id 4348  df-po 4351  df-iso 4352  df-iord 4421  df-on 4423  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-recs 6404  df-irdg 6469  df-1o 6515  df-oadd 6519  df-omul 6520  df-er 6633  df-ec 6635  df-qs 6639  df-ni 7437  df-pli 7438  df-mi 7439  df-lti 7440  df-plpq 7477  df-mpq 7478  df-enq 7480  df-nqqs 7481  df-plqqs 7482  df-mqqs 7483  df-1nqqs 7484  df-ltnqqs 7486  df-inp 7599  df-iltp 7603

This theorem is referenced by:  ltexprlempr  7741