Theorem ltexprlemrnd 7689

Description: Our constructed difference is rounded. Lemma for ltexpri 7697. (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 7685	. . . . 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 1207	. . . 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 7686	. . . 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 2570	. 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 7687	. . . . 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 1207	. . . 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 7688	. . . 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 2570	. 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 1364   E.wex 1506   e. wcel 2167   A.wral 2475   E.wrex 2476   {crab 2479   <.cop 3626   class class class wbr 4034   ` cfv 5259  (class class class)co 5925   1stc1st 6205   2ndc2nd 6206   Q.cnq 7364   +Q cplq 7366   <Q cltq 7369   <P cltp 7379

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-eprel 4325  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-1o 6483  df-oadd 6487  df-omul 6488  df-er 6601  df-ec 6603  df-qs 6607  df-ni 7388  df-pli 7389  df-mi 7390  df-lti 7391  df-plpq 7428  df-mpq 7429  df-enq 7431  df-nqqs 7432  df-plqqs 7433  df-mqqs 7434  df-1nqqs 7435  df-ltnqqs 7437  df-inp 7550  df-iltp 7554

This theorem is referenced by:  ltexprlempr  7692