Theorem mulextsr1 7743

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 7689	. 2 |- R. = ( ( P. X. P. ) /. ~R )
2		oveq1 5860	. . . 4 |- ( [ <. x , y >. ] ~R = A -> ( [ <. x , y >. ] ~R .R [ <. u , v >. ] ~R ) = ( A .R [ <. u , v >. ] ~R ) )
3	2	breq1d 3999	. . 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 3992	. . . 4 |- ( [ <. x , y >. ] ~R = A -> ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R <-> A <R [ <. z , w >. ] ~R ) )
5		breq2 3993	. . . 4 |- ( [ <. x , y >. ] ~R = A -> ( [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R <-> [ <. z , w >. ] ~R <R A ) )
6	4, 5	orbi12d 788	. . 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 233	. 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 5860	. . . 4 |- ( [ <. z , w >. ] ~R = B -> ( [ <. z , w >. ] ~R .R [ <. u , v >. ] ~R ) = ( B .R [ <. u , v >. ] ~R ) )
9	8	breq2d 4001	. . 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 3993	. . . 4 |- ( [ <. z , w >. ] ~R = B -> ( A <R [ <. z , w >. ] ~R <-> A <R B ) )
11		breq1 3992	. . . 4 |- ( [ <. z , w >. ] ~R = B -> ( [ <. z , w >. ] ~R <R A <-> B <R A ) )
12	10, 11	orbi12d 788	. . 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 233	. 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 5861	. . . 4 |- ( [ <. u , v >. ] ~R = C -> ( A .R [ <. u , v >. ] ~R ) = ( A .R C ) )
15		oveq2 5861	. . . 4 |- ( [ <. u , v >. ] ~R = C -> ( B .R [ <. u , v >. ] ~R ) = ( B .R C ) )
16	14, 15	breq12d 4002	. . 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 230	. 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 7742	. . 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 7708	. . . . . 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 1011	. . . . 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 7708	. . . . . 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 1010	. . . . 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 4002	. . . 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 1016	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> x e. P. )
25		simp3l 1020	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> u e. P. )
26		mulclpr 7534	. . . . . . 7 |- ( ( x e. P. /\ u e. P. ) -> ( x .P. u ) e. P. )
27	24, 25, 26	syl2anc 409	. . . . . 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 1017	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> y e. P. )
29		simp3r 1021	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> v e. P. )
30		mulclpr 7534	. . . . . . 7 |- ( ( y e. P. /\ v e. P. ) -> ( y .P. v ) e. P. )
31	28, 29, 30	syl2anc 409	. . . . . 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 7499	. . . . . 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 409	. . . . 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 7534	. . . . . . 7 |- ( ( x e. P. /\ v e. P. ) -> ( x .P. v ) e. P. )
35	24, 29, 34	syl2anc 409	. . . . . 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 7534	. . . . . . 7 |- ( ( y e. P. /\ u e. P. ) -> ( y .P. u ) e. P. )
37	28, 25, 36	syl2anc 409	. . . . . 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 7499	. . . . . 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 409	. . . . 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 1018	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> z e. P. )
41		mulclpr 7534	. . . . . . 7 |- ( ( z e. P. /\ u e. P. ) -> ( z .P. u ) e. P. )
42	40, 25, 41	syl2anc 409	. . . . . 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 1019	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> w e. P. )
44		mulclpr 7534	. . . . . . 7 |- ( ( w e. P. /\ v e. P. ) -> ( w .P. v ) e. P. )
45	43, 29, 44	syl2anc 409	. . . . . 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 7499	. . . . . 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 409	. . . . 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 7534	. . . . . . 7 |- ( ( z e. P. /\ v e. P. ) -> ( z .P. v ) e. P. )
49	40, 29, 48	syl2anc 409	. . . . . 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 7534	. . . . . . 7 |- ( ( w e. P. /\ u e. P. ) -> ( w .P. u ) e. P. )
51	43, 25, 50	syl2anc 409	. . . . . 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 7499	. . . . . 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 409	. . . . 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 7709	. . . . 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 1234	. . . 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 187	. . 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 7709	. . . . 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 1012	. . . 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 7709	. . . . . 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 266	. . . . 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 1012	. . . 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 788	. . 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 202	. 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 6602	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 103   <-> wb 104   \/ wo 703   /\ w3a 973   = wceq 1348   e. wcel 2141   <.cop 3586   class class class wbr 3989  (class class class)co 5853   [cec 6511   P.cnp 7253   +P. cpp 7255   .P. cmp 7256   <P cltp 7257   ~R cer 7258   R.cnr 7259   .R cmr 7264   <R cltr 7265

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-i1p 7429  df-iplp 7430  df-imp 7431  df-iltp 7432  df-enr 7688  df-nr 7689  df-mr 7691  df-ltr 7692

This theorem is referenced by:  axpre-mulext  7850