Theorem mulextsr1 7991

Description: Strong extensionality of multiplication of signed reals. (Contributed by Jim Kingdon, 18-Feb-2020.)

Assertion

Ref	Expression
mulextsr1	|- ( ( A e. R. /\ B e. R. /\ C e. R. ) -> ( ( A .R C ) <R ( B .R C ) -> ( A <R B \/ B <R A ) ) )

Proof of Theorem mulextsr1

Dummy variables u v w x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nr 7937	. 2 |- R. = ( ( P. X. P. ) /. ~R )
2		oveq1 6020	. . . 4 |- ( [ <. x , y >. ] ~R = A -> ( [ <. x , y >. ] ~R .R [ <. u , v >. ] ~R ) = ( A .R [ <. u , v >. ] ~R ) )
3	2	breq1d 4096	. . 3 |- ( [ <. x , y >. ] ~R = A -> ( ( [ <. x , y >. ] ~R .R [ <. u , v >. ] ~R ) <R ( [ <. z , w >. ] ~R .R [ <. u , v >. ] ~R ) <-> ( A .R [ <. u , v >. ] ~R ) <R ( [ <. z , w >. ] ~R .R [ <. u , v >. ] ~R ) ) )
4		breq1 4089	. . . 4 |- ( [ <. x , y >. ] ~R = A -> ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R <-> A <R [ <. z , w >. ] ~R ) )
5		breq2 4090	. . . 4 |- ( [ <. x , y >. ] ~R = A -> ( [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R <-> [ <. z , w >. ] ~R <R A ) )
6	4, 5	orbi12d 798	. . 3 |- ( [ <. x , y >. ] ~R = A -> ( ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R ) <-> ( A <R [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R A ) ) )
7	3, 6	imbi12d 234	. 2 |- ( [ <. x , y >. ] ~R = A -> ( ( ( [ <. x , y >. ] ~R .R [ <. u , v >. ] ~R ) <R ( [ <. z , w >. ] ~R .R [ <. u , v >. ] ~R ) -> ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R ) ) <-> ( ( A .R [ <. u , v >. ] ~R ) <R ( [ <. z , w >. ] ~R .R [ <. u , v >. ] ~R ) -> ( A <R [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R A ) ) ) )
8		oveq1 6020	. . . 4 |- ( [ <. z , w >. ] ~R = B -> ( [ <. z , w >. ] ~R .R [ <. u , v >. ] ~R ) = ( B .R [ <. u , v >. ] ~R ) )
9	8	breq2d 4098	. . 3 |- ( [ <. z , w >. ] ~R = B -> ( ( A .R [ <. u , v >. ] ~R ) <R ( [ <. z , w >. ] ~R .R [ <. u , v >. ] ~R ) <-> ( A .R [ <. u , v >. ] ~R ) <R ( B .R [ <. u , v >. ] ~R ) ) )
10		breq2 4090	. . . 4 |- ( [ <. z , w >. ] ~R = B -> ( A <R [ <. z , w >. ] ~R <-> A <R B ) )
11		breq1 4089	. . . 4 |- ( [ <. z , w >. ] ~R = B -> ( [ <. z , w >. ] ~R <R A <-> B <R A ) )
12	10, 11	orbi12d 798	. . 3 |- ( [ <. z , w >. ] ~R = B -> ( ( A <R [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R A ) <-> ( A <R B \/ B <R A ) ) )
13	9, 12	imbi12d 234	. 2 |- ( [ <. z , w >. ] ~R = B -> ( ( ( A .R [ <. u , v >. ] ~R ) <R ( [ <. z , w >. ] ~R .R [ <. u , v >. ] ~R ) -> ( A <R [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R A ) ) <-> ( ( A .R [ <. u , v >. ] ~R ) <R ( B .R [ <. u , v >. ] ~R ) -> ( A <R B \/ B <R A ) ) ) )
14		oveq2 6021	. . . 4 |- ( [ <. u , v >. ] ~R = C -> ( A .R [ <. u , v >. ] ~R ) = ( A .R C ) )
15		oveq2 6021	. . . 4 |- ( [ <. u , v >. ] ~R = C -> ( B .R [ <. u , v >. ] ~R ) = ( B .R C ) )
16	14, 15	breq12d 4099	. . 3 |- ( [ <. u , v >. ] ~R = C -> ( ( A .R [ <. u , v >. ] ~R ) <R ( B .R [ <. u , v >. ] ~R ) <-> ( A .R C ) <R ( B .R C ) ) )
17	16	imbi1d 231	. 2 |- ( [ <. u , v >. ] ~R = C -> ( ( ( A .R [ <. u , v >. ] ~R ) <R ( B .R [ <. u , v >. ] ~R ) -> ( A <R B \/ B <R A ) ) <-> ( ( A .R C ) <R ( B .R C ) -> ( A <R B \/ B <R A ) ) ) )
18		mulextsr1lem 7990	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> ( ( ( ( x .P. u ) +P. ( y .P. v ) ) +P. ( ( z .P. v ) +P. ( w .P. u ) ) ) <P ( ( ( x .P. v ) +P. ( y .P. u ) ) +P. ( ( z .P. u ) +P. ( w .P. v ) ) ) -> ( ( x +P. w ) <P ( y +P. z ) \/ ( z +P. y ) <P ( w +P. x ) ) ) )
19		mulsrpr 7956	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> ( [ <. x , y >. ] ~R .R [ <. u , v >. ] ~R ) = [ <. ( ( x .P. u ) +P. ( y .P. v ) ) , ( ( x .P. v ) +P. ( y .P. u ) ) >. ] ~R )
20	19	3adant2 1040	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> ( [ <. x , y >. ] ~R .R [ <. u , v >. ] ~R ) = [ <. ( ( x .P. u ) +P. ( y .P. v ) ) , ( ( x .P. v ) +P. ( y .P. u ) ) >. ] ~R )
21		mulsrpr 7956	. . . . . 6 |- ( ( ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> ( [ <. z , w >. ] ~R .R [ <. u , v >. ] ~R ) = [ <. ( ( z .P. u ) +P. ( w .P. v ) ) , ( ( z .P. v ) +P. ( w .P. u ) ) >. ] ~R )
22	21	3adant1 1039	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> ( [ <. z , w >. ] ~R .R [ <. u , v >. ] ~R ) = [ <. ( ( z .P. u ) +P. ( w .P. v ) ) , ( ( z .P. v ) +P. ( w .P. u ) ) >. ] ~R )
23	20, 22	breq12d 4099	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> ( ( [ <. x , y >. ] ~R .R [ <. u , v >. ] ~R ) <R ( [ <. z , w >. ] ~R .R [ <. u , v >. ] ~R ) <-> [ <. ( ( x .P. u ) +P. ( y .P. v ) ) , ( ( x .P. v ) +P. ( y .P. u ) ) >. ] ~R <R [ <. ( ( z .P. u ) +P. ( w .P. v ) ) , ( ( z .P. v ) +P. ( w .P. u ) ) >. ] ~R ) )
24		simp1l 1045	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> x e. P. )
25		simp3l 1049	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> u e. P. )
26		mulclpr 7782	. . . . . . 7 |- ( ( x e. P. /\ u e. P. ) -> ( x .P. u ) e. P. )
27	24, 25, 26	syl2anc 411	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> ( x .P. u ) e. P. )
28		simp1r 1046	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> y e. P. )
29		simp3r 1050	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> v e. P. )
30		mulclpr 7782	. . . . . . 7 |- ( ( y e. P. /\ v e. P. ) -> ( y .P. v ) e. P. )
31	28, 29, 30	syl2anc 411	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> ( y .P. v ) e. P. )
32		addclpr 7747	. . . . . 6 |- ( ( ( x .P. u ) e. P. /\ ( y .P. v ) e. P. ) -> ( ( x .P. u ) +P. ( y .P. v ) ) e. P. )
33	27, 31, 32	syl2anc 411	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> ( ( x .P. u ) +P. ( y .P. v ) ) e. P. )
34		mulclpr 7782	. . . . . . 7 |- ( ( x e. P. /\ v e. P. ) -> ( x .P. v ) e. P. )
35	24, 29, 34	syl2anc 411	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> ( x .P. v ) e. P. )
36		mulclpr 7782	. . . . . . 7 |- ( ( y e. P. /\ u e. P. ) -> ( y .P. u ) e. P. )
37	28, 25, 36	syl2anc 411	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> ( y .P. u ) e. P. )
38		addclpr 7747	. . . . . 6 |- ( ( ( x .P. v ) e. P. /\ ( y .P. u ) e. P. ) -> ( ( x .P. v ) +P. ( y .P. u ) ) e. P. )
39	35, 37, 38	syl2anc 411	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> ( ( x .P. v ) +P. ( y .P. u ) ) e. P. )
40		simp2l 1047	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> z e. P. )
41		mulclpr 7782	. . . . . . 7 |- ( ( z e. P. /\ u e. P. ) -> ( z .P. u ) e. P. )
42	40, 25, 41	syl2anc 411	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> ( z .P. u ) e. P. )
43		simp2r 1048	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> w e. P. )
44		mulclpr 7782	. . . . . . 7 |- ( ( w e. P. /\ v e. P. ) -> ( w .P. v ) e. P. )
45	43, 29, 44	syl2anc 411	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> ( w .P. v ) e. P. )
46		addclpr 7747	. . . . . 6 |- ( ( ( z .P. u ) e. P. /\ ( w .P. v ) e. P. ) -> ( ( z .P. u ) +P. ( w .P. v ) ) e. P. )
47	42, 45, 46	syl2anc 411	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> ( ( z .P. u ) +P. ( w .P. v ) ) e. P. )
48		mulclpr 7782	. . . . . . 7 |- ( ( z e. P. /\ v e. P. ) -> ( z .P. v ) e. P. )
49	40, 29, 48	syl2anc 411	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> ( z .P. v ) e. P. )
50		mulclpr 7782	. . . . . . 7 |- ( ( w e. P. /\ u e. P. ) -> ( w .P. u ) e. P. )
51	43, 25, 50	syl2anc 411	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> ( w .P. u ) e. P. )
52		addclpr 7747	. . . . . 6 |- ( ( ( z .P. v ) e. P. /\ ( w .P. u ) e. P. ) -> ( ( z .P. v ) +P. ( w .P. u ) ) e. P. )
53	49, 51, 52	syl2anc 411	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> ( ( z .P. v ) +P. ( w .P. u ) ) e. P. )
54		ltsrprg 7957	. . . . 5 |- ( ( ( ( ( x .P. u ) +P. ( y .P. v ) ) e. P. /\ ( ( x .P. v ) +P. ( y .P. u ) ) e. P. ) /\ ( ( ( z .P. u ) +P. ( w .P. v ) ) e. P. /\ ( ( z .P. v ) +P. ( w .P. u ) ) e. P. ) ) -> ( [ <. ( ( x .P. u ) +P. ( y .P. v ) ) , ( ( x .P. v ) +P. ( y .P. u ) ) >. ] ~R <R [ <. ( ( z .P. u ) +P. ( w .P. v ) ) , ( ( z .P. v ) +P. ( w .P. u ) ) >. ] ~R <-> ( ( ( x .P. u ) +P. ( y .P. v ) ) +P. ( ( z .P. v ) +P. ( w .P. u ) ) ) <P ( ( ( x .P. v ) +P. ( y .P. u ) ) +P. ( ( z .P. u ) +P. ( w .P. v ) ) ) ) )
55	33, 39, 47, 53, 54	syl22anc 1272	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> ( [ <. ( ( x .P. u ) +P. ( y .P. v ) ) , ( ( x .P. v ) +P. ( y .P. u ) ) >. ] ~R <R [ <. ( ( z .P. u ) +P. ( w .P. v ) ) , ( ( z .P. v ) +P. ( w .P. u ) ) >. ] ~R <-> ( ( ( x .P. u ) +P. ( y .P. v ) ) +P. ( ( z .P. v ) +P. ( w .P. u ) ) ) <P ( ( ( x .P. v ) +P. ( y .P. u ) ) +P. ( ( z .P. u ) +P. ( w .P. v ) ) ) ) )
56	23, 55	bitrd 188	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> ( ( [ <. x , y >. ] ~R .R [ <. u , v >. ] ~R ) <R ( [ <. z , w >. ] ~R .R [ <. u , v >. ] ~R ) <-> ( ( ( x .P. u ) +P. ( y .P. v ) ) +P. ( ( z .P. v ) +P. ( w .P. u ) ) ) <P ( ( ( x .P. v ) +P. ( y .P. u ) ) +P. ( ( z .P. u ) +P. ( w .P. v ) ) ) ) )
57		ltsrprg 7957	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R <-> ( x +P. w ) <P ( y +P. z ) ) )
58	57	3adant3 1041	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R <-> ( x +P. w ) <P ( y +P. z ) ) )
59		ltsrprg 7957	. . . . . 6 |- ( ( ( z e. P. /\ w e. P. ) /\ ( x e. P. /\ y e. P. ) ) -> ( [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R <-> ( z +P. y ) <P ( w +P. x ) ) )
60	59	ancoms 268	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R <-> ( z +P. y ) <P ( w +P. x ) ) )
61	60	3adant3 1041	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> ( [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R <-> ( z +P. y ) <P ( w +P. x ) ) )
62	58, 61	orbi12d 798	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> ( ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R ) <-> ( ( x +P. w ) <P ( y +P. z ) \/ ( z +P. y ) <P ( w +P. x ) ) ) )
63	18, 56, 62	3imtr4d 203	. 2 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> ( ( [ <. x , y >. ] ~R .R [ <. u , v >. ] ~R ) <R ( [ <. z , w >. ] ~R .R [ <. u , v >. ] ~R ) -> ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R ) ) )
64	1, 7, 13, 17, 63	3ecoptocl 6788	1 |- ( ( A e. R. /\ B e. R. /\ C e. R. ) -> ( ( A .R C ) <R ( B .R C ) -> ( A <R B \/ B <R A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   <.cop 3670   class class class wbr 4086  (class class class)co 6013   [cec 6695   P.cnp 7501   +P. cpp 7503   .P. cmp 7504   <P cltp 7505   ~R cer 7506   R.cnr 7507   .R cmr 7512   <R cltr 7513

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-1o 6577  df-2o 6578  df-oadd 6581  df-omul 6582  df-er 6697  df-ec 6699  df-qs 6703  df-ni 7514  df-pli 7515  df-mi 7516  df-lti 7517  df-plpq 7554  df-mpq 7555  df-enq 7557  df-nqqs 7558  df-plqqs 7559  df-mqqs 7560  df-1nqqs 7561  df-rq 7562  df-ltnqqs 7563  df-enq0 7634  df-nq0 7635  df-0nq0 7636  df-plq0 7637  df-mq0 7638  df-inp 7676  df-i1p 7677  df-iplp 7678  df-imp 7679  df-iltp 7680  df-enr 7936  df-nr 7937  df-mr 7939  df-ltr 7940

This theorem is referenced by:  axpre-mulext  8098