Theorem genplt2i 7720

Description: Operating on both sides of two inequalities, when the operation is consistent with <Q. (Contributed by Jim Kingdon, 6-Oct-2019.)

Hypotheses

Ref	Expression
genplt2i.ord	|- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( x <Q y <-> ( z G x ) <Q ( z G y ) ) )
genplt2i.com	|- ( ( x e. Q. /\ y e. Q. ) -> ( x G y ) = ( y G x ) )

Assertion

Ref	Expression
genplt2i	|- ( ( A <Q B /\ C <Q D ) -> ( A G C ) <Q ( B G D ) )

Distinct variable groups:   x, A, y, z   x, B, y, z   x, C, y, z   x, D, y, z   x, G, y, z

Proof of Theorem genplt2i

Step	Hyp	Ref	Expression
1		simpl 109	. . 3 |- ( ( A <Q B /\ C <Q D ) -> A <Q B )
2		genplt2i.ord	. . . . 5 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( x <Q y <-> ( z G x ) <Q ( z G y ) ) )
3	2	adantl 277	. . . 4 |- ( ( ( A <Q B /\ C <Q D ) /\ ( x e. Q. /\ y e. Q. /\ z e. Q. ) ) -> ( x <Q y <-> ( z G x ) <Q ( z G y ) ) )
4		ltrelnq 7575	. . . . . 6 |- <Q C_ ( Q. X. Q. )
5	4	brel 4776	. . . . 5 |- ( A <Q B -> ( A e. Q. /\ B e. Q. ) )
6	4	brel 4776	. . . . 5 |- ( C <Q D -> ( C e. Q. /\ D e. Q. ) )
7		simpll 527	. . . . 5 |- ( ( ( A e. Q. /\ B e. Q. ) /\ ( C e. Q. /\ D e. Q. ) ) -> A e. Q. )
8	5, 6, 7	syl2an 289	. . . 4 |- ( ( A <Q B /\ C <Q D ) -> A e. Q. )
9		simplr 528	. . . . 5 |- ( ( ( A e. Q. /\ B e. Q. ) /\ ( C e. Q. /\ D e. Q. ) ) -> B e. Q. )
10	5, 6, 9	syl2an 289	. . . 4 |- ( ( A <Q B /\ C <Q D ) -> B e. Q. )
11		simprl 529	. . . . 5 |- ( ( ( A e. Q. /\ B e. Q. ) /\ ( C e. Q. /\ D e. Q. ) ) -> C e. Q. )
12	5, 6, 11	syl2an 289	. . . 4 |- ( ( A <Q B /\ C <Q D ) -> C e. Q. )
13		genplt2i.com	. . . . 5 |- ( ( x e. Q. /\ y e. Q. ) -> ( x G y ) = ( y G x ) )
14	13	adantl 277	. . . 4 |- ( ( ( A <Q B /\ C <Q D ) /\ ( x e. Q. /\ y e. Q. ) ) -> ( x G y ) = ( y G x ) )
15	3, 8, 10, 12, 14	caovord2d 6187	. . 3 |- ( ( A <Q B /\ C <Q D ) -> ( A <Q B <-> ( A G C ) <Q ( B G C ) ) )
16	1, 15	mpbid 147	. 2 |- ( ( A <Q B /\ C <Q D ) -> ( A G C ) <Q ( B G C ) )
17		simpr 110	. . 3 |- ( ( A <Q B /\ C <Q D ) -> C <Q D )
18		simprr 531	. . . . 5 |- ( ( ( A e. Q. /\ B e. Q. ) /\ ( C e. Q. /\ D e. Q. ) ) -> D e. Q. )
19	5, 6, 18	syl2an 289	. . . 4 |- ( ( A <Q B /\ C <Q D ) -> D e. Q. )
20	3, 12, 19, 10	caovordd 6186	. . 3 |- ( ( A <Q B /\ C <Q D ) -> ( C <Q D <-> ( B G C ) <Q ( B G D ) ) )
21	17, 20	mpbid 147	. 2 |- ( ( A <Q B /\ C <Q D ) -> ( B G C ) <Q ( B G D ) )
22		ltsonq 7608	. . 3 |- <Q Or Q.
23	22, 4	sotri 5130	. 2 |- ( ( ( A G C ) <Q ( B G C ) /\ ( B G C ) <Q ( B G D ) ) -> ( A G C ) <Q ( B G D ) )
24	16, 21, 23	syl2anc 411	1 |- ( ( A <Q B /\ C <Q D ) -> ( A G C ) <Q ( B G D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   class class class wbr 4086  (class class class)co 6013   Q.cnq 7490   <Q cltq 7495

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 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

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 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-eprel 4384  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-oadd 6581  df-omul 6582  df-er 6697  df-ec 6699  df-qs 6703  df-ni 7514  df-mi 7516  df-lti 7517  df-enq 7557  df-nqqs 7558  df-ltnqqs 7563

This theorem is referenced by:  genprndl  7731  genprndu  7732  genpdisj  7733