Theorem mulextsr1 7779

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 7725	. 2 |- R. = ( ( P. X. P. ) /. ~R )
2		oveq1 5881	. . . 4 |- ( [ <. x , y >. ] ~R = A -> ( [ <. x , y >. ] ~R .R [ <. u , v >. ] ~R ) = ( A .R [ <. u , v >. ] ~R ) )
3	2	breq1d 4013	. . 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 4006	. . . 4 |- ( [ <. x , y >. ] ~R = A -> ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R <-> A <R [ <. z , w >. ] ~R ) )
5		breq2 4007	. . . 4 |- ( [ <. x , y >. ] ~R = A -> ( [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R <-> [ <. z , w >. ] ~R <R A ) )
6	4, 5	orbi12d 793	. . 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 5881	. . . 4 |- ( [ <. z , w >. ] ~R = B -> ( [ <. z , w >. ] ~R .R [ <. u , v >. ] ~R ) = ( B .R [ <. u , v >. ] ~R ) )
9	8	breq2d 4015	. . 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 4007	. . . 4 |- ( [ <. z , w >. ] ~R = B -> ( A <R [ <. z , w >. ] ~R <-> A <R B ) )
11		breq1 4006	. . . 4 |- ( [ <. z , w >. ] ~R = B -> ( [ <. z , w >. ] ~R <R A <-> B <R A ) )
12	10, 11	orbi12d 793	. . 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 5882	. . . 4 |- ( [ <. u , v >. ] ~R = C -> ( A .R [ <. u , v >. ] ~R ) = ( A .R C ) )
15		oveq2 5882	. . . 4 |- ( [ <. u , v >. ] ~R = C -> ( B .R [ <. u , v >. ] ~R ) = ( B .R C ) )
16	14, 15	breq12d 4016	. . 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 7778	. . 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 7744	. . . . . 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 1016	. . . . 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 7744	. . . . . 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 1015	. . . . 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 4016	. . . 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 1021	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> x e. P. )
25		simp3l 1025	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> u e. P. )
26		mulclpr 7570	. . . . . . 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 1022	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> y e. P. )
29		simp3r 1026	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> v e. P. )
30		mulclpr 7570	. . . . . . 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 7535	. . . . . 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 7570	. . . . . . 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 7570	. . . . . . 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 7535	. . . . . 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 1023	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> z e. P. )
41		mulclpr 7570	. . . . . . 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 1024	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( u e. P. /\ v e. P. ) ) -> w e. P. )
44		mulclpr 7570	. . . . . . 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 7535	. . . . . 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 7570	. . . . . . 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 7570	. . . . . . 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 7535	. . . . . 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 7745	. . . . 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 1239	. . . 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 7745	. . . . 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 1017	. . . 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 7745	. . . . . 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 1017	. . . 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 793	. . 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 6623	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 708   /\ w3a 978   = wceq 1353   e. wcel 2148   <.cop 3595   class class class wbr 4003  (class class class)co 5874   [cec 6532   P.cnp 7289   +P. cpp 7291   .P. cmp 7292   <P cltp 7293   ~R cer 7294   R.cnr 7295   .R cmr 7300   <R cltr 7301

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-2o 6417  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-pli 7303  df-mi 7304  df-lti 7305  df-plpq 7342  df-mpq 7343  df-enq 7345  df-nqqs 7346  df-plqqs 7347  df-mqqs 7348  df-1nqqs 7349  df-rq 7350  df-ltnqqs 7351  df-enq0 7422  df-nq0 7423  df-0nq0 7424  df-plq0 7425  df-mq0 7426  df-inp 7464  df-i1p 7465  df-iplp 7466  df-imp 7467  df-iltp 7468  df-enr 7724  df-nr 7725  df-mr 7727  df-ltr 7728

This theorem is referenced by:  axpre-mulext  7886