Theorem mulextsr1 7019

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 6966	. 2 |- R. = ( ( P. X. P. ) /. ~R )
2		oveq1 5550	. . . 4 |- ( [ <. x , y >. ] ~R = A -> ( [ <. x , y >. ] ~R .R [ <. u , v >. ] ~R ) = ( A .R [ <. u , v >. ] ~R ) )
3	2	breq1d 3803	. . 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 3796	. . . 4 |- ( [ <. x , y >. ] ~R = A -> ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R <-> A <R [ <. z , w >. ] ~R ) )
5		breq2 3797	. . . 4 |- ( [ <. x , y >. ] ~R = A -> ( [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R <-> [ <. z , w >. ] ~R <R A ) )
6	4, 5	orbi12d 740	. . 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 232	. 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 5550	. . . 4 |- ( [ <. z , w >. ] ~R = B -> ( [ <. z , w >. ] ~R .R [ <. u , v >. ] ~R ) = ( B .R [ <. u , v >. ] ~R ) )
9	8	breq2d 3805	. . 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 3797	. . . 4 |- ( [ <. z , w >. ] ~R = B -> ( A <R [ <. z , w >. ] ~R <-> A <R B ) )
11		breq1 3796	. . . 4 |- ( [ <. z , w >. ] ~R = B -> ( [ <. z , w >. ] ~R <R A <-> B <R A ) )
12	10, 11	orbi12d 740	. . 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 232	. 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 5551	. . . 4 |- ( [ <. u , v >. ] ~R = C -> ( A .R [ <. u , v >. ] ~R ) = ( A .R C ) )
15		oveq2 5551	. . . 4 |- ( [ <. u , v >. ] ~R = C -> ( B .R [ <. u , v >. ] ~R ) = ( B .R C ) )
16	14, 15	breq12d 3806	. . 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 229	. 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 7018	. . 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 6985	. . . . . 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 958	. . . . 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 6985	. . . . . 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 957	. . . . 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 3806	. . . 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 963	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> x e. P. )
25		simp3l 967	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> u e. P. )
26		mulclpr 6824	. . . . . . 7 |- ( ( x e. P. /\ u e. P. ) -> ( x .P. u ) e. P. )
27	24, 25, 26	syl2anc 403	. . . . . 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 964	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> y e. P. )
29		simp3r 968	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> v e. P. )
30		mulclpr 6824	. . . . . . 7 |- ( ( y e. P. /\ v e. P. ) -> ( y .P. v ) e. P. )
31	28, 29, 30	syl2anc 403	. . . . . 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 6789	. . . . . 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 403	. . . . 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 6824	. . . . . . 7 |- ( ( x e. P. /\ v e. P. ) -> ( x .P. v ) e. P. )
35	24, 29, 34	syl2anc 403	. . . . . 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 6824	. . . . . . 7 |- ( ( y e. P. /\ u e. P. ) -> ( y .P. u ) e. P. )
37	28, 25, 36	syl2anc 403	. . . . . 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 6789	. . . . . 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 403	. . . . 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 965	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> z e. P. )
41		mulclpr 6824	. . . . . . 7 |- ( ( z e. P. /\ u e. P. ) -> ( z .P. u ) e. P. )
42	40, 25, 41	syl2anc 403	. . . . . 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 966	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> w e. P. )
44		mulclpr 6824	. . . . . . 7 |- ( ( w e. P. /\ v e. P. ) -> ( w .P. v ) e. P. )
45	43, 29, 44	syl2anc 403	. . . . . 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 6789	. . . . . 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 403	. . . . 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 6824	. . . . . . 7 |- ( ( z e. P. /\ v e. P. ) -> ( z .P. v ) e. P. )
49	40, 29, 48	syl2anc 403	. . . . . 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 6824	. . . . . . 7 |- ( ( w e. P. /\ u e. P. ) -> ( w .P. u ) e. P. )
51	43, 25, 50	syl2anc 403	. . . . . 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 6789	. . . . . 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 403	. . . . 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 6986	. . . . 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 1171	. . . 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 186	. . 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 6986	. . . . 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 959	. . . 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 6986	. . . . . 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 264	. . . . 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 959	. . . 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 740	. . 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 201	. 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 6261	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 102   <-> wb 103   \/ wo 662   /\ w3a 920   = wceq 1285   e. wcel 1434   <.cop 3409   class class class wbr 3793  (class class class)co 5543   [cec 6170   P.cnp 6543   +P. cpp 6545   .P. cmp 6546   <P cltp 6547   ~R cer 6548   R.cnr 6549   .R cmr 6554   <R cltr 6555

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337

This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-2o 6066  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-lti 6559  df-plpq 6596  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-plqqs 6601  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605  df-enq0 6676  df-nq0 6677  df-0nq0 6678  df-plq0 6679  df-mq0 6680  df-inp 6718  df-i1p 6719  df-iplp 6720  df-imp 6721  df-iltp 6722  df-enr 6965  df-nr 6966  df-mr 6968  df-ltr 6969

This theorem is referenced by:  axpre-mulext  7116