Theorem addnqpr 7769

Description: Addition of fractions embedded into positive reals. One can either add the fractions as fractions, or embed them into positive reals and add them as positive reals, and get the same result. (Contributed by Jim Kingdon, 19-Aug-2020.)

Assertion

Ref	Expression
addnqpr	|- ( ( A e. Q. /\ B e. Q. ) -> <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. = ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) )

Distinct variable groups:   A, l, u   B, l, u

Proof of Theorem addnqpr

Step	Hyp	Ref	Expression
1		addnqprlemfl 7767	. . 3 |- ( ( A e. Q. /\ B e. Q. ) -> ( 1st ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) C_ ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) )
2		addnqprlemrl 7765	. . 3 |- ( ( A e. Q. /\ B e. Q. ) -> ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) C_ ( 1st ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) )
3	1, 2	eqssd 3242	. 2 |- ( ( A e. Q. /\ B e. Q. ) -> ( 1st ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) = ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) )
4		addnqprlemfu 7768	. . 3 |- ( ( A e. Q. /\ B e. Q. ) -> ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) C_ ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) )
5		addnqprlemru 7766	. . 3 |- ( ( A e. Q. /\ B e. Q. ) -> ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) C_ ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) )
6	4, 5	eqssd 3242	. 2 |- ( ( A e. Q. /\ B e. Q. ) -> ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) = ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) )
7		addclnq 7583	. . . 4 |- ( ( A e. Q. /\ B e. Q. ) -> ( A +Q B ) e. Q. )
8		nqprlu 7755	. . . 4 |- ( ( A +Q B ) e. Q. -> <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. e. P. )
9	7, 8	syl 14	. . 3 |- ( ( A e. Q. /\ B e. Q. ) -> <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. e. P. )
10		nqprlu 7755	. . . 4 |- ( A e. Q. -> <. { l | l <Q A } , { u | A <Q u } >. e. P. )
11		nqprlu 7755	. . . 4 |- ( B e. Q. -> <. { l | l <Q B } , { u | B <Q u } >. e. P. )
12		addclpr 7745	. . . 4 |- ( ( <. { l | l <Q A } , { u | A <Q u } >. e. P. /\ <. { l | l <Q B } , { u | B <Q u } >. e. P. ) -> ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) e. P. )
13	10, 11, 12	syl2an 289	. . 3 |- ( ( A e. Q. /\ B e. Q. ) -> ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) e. P. )
14		preqlu 7680	. . 3 |- ( ( <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. e. P. /\ ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) e. P. ) -> ( <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. = ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) <-> ( ( 1st ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) = ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) /\ ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) = ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) ) )
15	9, 13, 14	syl2anc 411	. 2 |- ( ( A e. Q. /\ B e. Q. ) -> ( <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. = ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) <-> ( ( 1st ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) = ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) /\ ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) = ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) ) )
16	3, 6, 15	mpbir2and 950	1 |- ( ( A e. Q. /\ B e. Q. ) -> <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. = ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   {cab 2215   <.cop 3670   class class class wbr 4084   ` cfv 5322  (class class class)co 6011   1stc1st 6294   2ndc2nd 6295   Q.cnq 7488   +Q cplq 7490   <Q cltq 7493   P.cnp 7499   +P. cpp 7501

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4200  ax-sep 4203  ax-nul 4211  ax-pow 4260  ax-pr 4295  ax-un 4526  ax-setind 4631  ax-iinf 4682

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3890  df-int 3925  df-iun 3968  df-br 4085  df-opab 4147  df-mpt 4148  df-tr 4184  df-eprel 4382  df-id 4386  df-po 4389  df-iso 4390  df-iord 4459  df-on 4461  df-suc 4464  df-iom 4685  df-xp 4727  df-rel 4728  df-cnv 4729  df-co 4730  df-dm 4731  df-rn 4732  df-res 4733  df-ima 4734  df-iota 5282  df-fun 5324  df-fn 5325  df-f 5326  df-f1 5327  df-fo 5328  df-f1o 5329  df-fv 5330  df-ov 6014  df-oprab 6015  df-mpo 6016  df-1st 6296  df-2nd 6297  df-recs 6464  df-irdg 6529  df-1o 6575  df-2o 6576  df-oadd 6579  df-omul 6580  df-er 6695  df-ec 6697  df-qs 6701  df-ni 7512  df-pli 7513  df-mi 7514  df-lti 7515  df-plpq 7552  df-mpq 7553  df-enq 7555  df-nqqs 7556  df-plqqs 7557  df-mqqs 7558  df-1nqqs 7559  df-rq 7560  df-ltnqqs 7561  df-enq0 7632  df-nq0 7633  df-0nq0 7634  df-plq0 7635  df-mq0 7636  df-inp 7674  df-iplp 7676

This theorem is referenced by:  addnqpr1  7770  prplnqu  7828  caucvgprlemcanl  7852  caucvgprprlemloccalc  7892  caucvgprprlemnkltj  7897  caucvgprprlemnkeqj  7898  caucvgprprlemmu  7903  caucvgprprlemexbt  7914