Theorem mulextsr1 7848

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 7794	. 2 |- R. = ( ( P. X. P. ) /. ~R )
2		oveq1 5929	. . . 4 |- ( [ <. x , y >. ] ~R = A -> ( [ <. x , y >. ] ~R .R [ <. u , v >. ] ~R ) = ( A .R [ <. u , v >. ] ~R ) )
3	2	breq1d 4043	. . 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 4036	. . . 4 |- ( [ <. x , y >. ] ~R = A -> ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R <-> A <R [ <. z , w >. ] ~R ) )
5		breq2 4037	. . . 4 |- ( [ <. x , y >. ] ~R = A -> ( [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R <-> [ <. z , w >. ] ~R <R A ) )
6	4, 5	orbi12d 794	. . 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 5929	. . . 4 |- ( [ <. z , w >. ] ~R = B -> ( [ <. z , w >. ] ~R .R [ <. u , v >. ] ~R ) = ( B .R [ <. u , v >. ] ~R ) )
9	8	breq2d 4045	. . 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 4037	. . . 4 |- ( [ <. z , w >. ] ~R = B -> ( A <R [ <. z , w >. ] ~R <-> A <R B ) )
11		breq1 4036	. . . 4 |- ( [ <. z , w >. ] ~R = B -> ( [ <. z , w >. ] ~R <R A <-> B <R A ) )
12	10, 11	orbi12d 794	. . 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 5930	. . . 4 |- ( [ <. u , v >. ] ~R = C -> ( A .R [ <. u , v >. ] ~R ) = ( A .R C ) )
15		oveq2 5930	. . . 4 |- ( [ <. u , v >. ] ~R = C -> ( B .R [ <. u , v >. ] ~R ) = ( B .R C ) )
16	14, 15	breq12d 4046	. . 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 7847	. . 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 7813	. . . . . 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 1018	. . . . 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 7813	. . . . . 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 1017	. . . . 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 4046	. . . 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 1023	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> x e. P. )
25		simp3l 1027	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> u e. P. )
26		mulclpr 7639	. . . . . . 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 1024	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> y e. P. )
29		simp3r 1028	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> v e. P. )
30		mulclpr 7639	. . . . . . 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 7604	. . . . . 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 7639	. . . . . . 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 7639	. . . . . . 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 7604	. . . . . 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 1025	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> z e. P. )
41		mulclpr 7639	. . . . . . 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 1026	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> w e. P. )
44		mulclpr 7639	. . . . . . 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 7604	. . . . . 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 7639	. . . . . . 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 7639	. . . . . . 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 7604	. . . . . 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 7814	. . . . 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 1250	. . . 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 7814	. . . . 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 1019	. . . 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 7814	. . . . . 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 1019	. . . 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 794	. . 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 6683	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 709   /\ w3a 980   = wceq 1364   e. wcel 2167   <.cop 3625   class class class wbr 4033  (class class class)co 5922   [cec 6590   P.cnp 7358   +P. cpp 7360   .P. cmp 7361   <P cltp 7362   ~R cer 7363   R.cnr 7364   .R cmr 7369   <R cltr 7370

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 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

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 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-eprel 4324  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-1o 6474  df-2o 6475  df-oadd 6478  df-omul 6479  df-er 6592  df-ec 6594  df-qs 6598  df-ni 7371  df-pli 7372  df-mi 7373  df-lti 7374  df-plpq 7411  df-mpq 7412  df-enq 7414  df-nqqs 7415  df-plqqs 7416  df-mqqs 7417  df-1nqqs 7418  df-rq 7419  df-ltnqqs 7420  df-enq0 7491  df-nq0 7492  df-0nq0 7493  df-plq0 7494  df-mq0 7495  df-inp 7533  df-i1p 7534  df-iplp 7535  df-imp 7536  df-iltp 7537  df-enr 7793  df-nr 7794  df-mr 7796  df-ltr 7797

This theorem is referenced by:  axpre-mulext  7955