Theorem ltmprr 6832

Description: Ordering property of multiplication. (Contributed by Jim Kingdon, 18-Feb-2020.)

Assertion

Ref	Expression
ltmprr	|- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( C .P. A ) <P ( C .P. B ) -> A <P B ) )

Proof of Theorem ltmprr

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		recexpr 6828	. . . . 5 |- ( C e. P. -> E. y e. P. ( C .P. y ) = 1P )
2	1	3ad2ant3 961	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> E. y e. P. ( C .P. y ) = 1P )
3	2	adantr 270	. . 3 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C .P. A ) <P ( C .P. B ) ) -> E. y e. P. ( C .P. y ) = 1P )
4		ltexpri 6803	. . . . 5 |- ( ( C .P. A ) <P ( C .P. B ) -> E. x e. P. ( ( C .P. A ) +P. x ) = ( C .P. B ) )
5	4	ad2antlr 472	. . . 4 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C .P. A ) <P ( C .P. B ) ) /\ ( y e. P. /\ ( C .P. y ) = 1P ) ) -> E. x e. P. ( ( C .P. A ) +P. x ) = ( C .P. B ) )
6		simplll 499	. . . . . . 7 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C .P. A ) <P ( C .P. B ) ) /\ ( y e. P. /\ ( C .P. y ) = 1P ) ) /\ ( x e. P. /\ ( ( C .P. A ) +P. x ) = ( C .P. B ) ) ) -> ( A e. P. /\ B e. P. /\ C e. P. ) )
7	6	simp1d 950	. . . . . 6 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C .P. A ) <P ( C .P. B ) ) /\ ( y e. P. /\ ( C .P. y ) = 1P ) ) /\ ( x e. P. /\ ( ( C .P. A ) +P. x ) = ( C .P. B ) ) ) -> A e. P. )
8		simplrl 501	. . . . . . 7 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C .P. A ) <P ( C .P. B ) ) /\ ( y e. P. /\ ( C .P. y ) = 1P ) ) /\ ( x e. P. /\ ( ( C .P. A ) +P. x ) = ( C .P. B ) ) ) -> y e. P. )
9		simprl 497	. . . . . . 7 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C .P. A ) <P ( C .P. B ) ) /\ ( y e. P. /\ ( C .P. y ) = 1P ) ) /\ ( x e. P. /\ ( ( C .P. A ) +P. x ) = ( C .P. B ) ) ) -> x e. P. )
10		mulclpr 6762	. . . . . . 7 |- ( ( y e. P. /\ x e. P. ) -> ( y .P. x ) e. P. )
11	8, 9, 10	syl2anc 403	. . . . . 6 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C .P. A ) <P ( C .P. B ) ) /\ ( y e. P. /\ ( C .P. y ) = 1P ) ) /\ ( x e. P. /\ ( ( C .P. A ) +P. x ) = ( C .P. B ) ) ) -> ( y .P. x ) e. P. )
12		ltaddpr 6787	. . . . . 6 |- ( ( A e. P. /\ ( y .P. x ) e. P. ) -> A <P ( A +P. ( y .P. x ) ) )
13	7, 11, 12	syl2anc 403	. . . . 5 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C .P. A ) <P ( C .P. B ) ) /\ ( y e. P. /\ ( C .P. y ) = 1P ) ) /\ ( x e. P. /\ ( ( C .P. A ) +P. x ) = ( C .P. B ) ) ) -> A <P ( A +P. ( y .P. x ) ) )
14		simprr 498	. . . . . . 7 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C .P. A ) <P ( C .P. B ) ) /\ ( y e. P. /\ ( C .P. y ) = 1P ) ) /\ ( x e. P. /\ ( ( C .P. A ) +P. x ) = ( C .P. B ) ) ) -> ( ( C .P. A ) +P. x ) = ( C .P. B ) )
15	14	oveq2d 5548	. . . . . 6 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C .P. A ) <P ( C .P. B ) ) /\ ( y e. P. /\ ( C .P. y ) = 1P ) ) /\ ( x e. P. /\ ( ( C .P. A ) +P. x ) = ( C .P. B ) ) ) -> ( y .P. ( ( C .P. A ) +P. x ) ) = ( y .P. ( C .P. B ) ) )
16	6	simp3d 952	. . . . . . . . 9 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C .P. A ) <P ( C .P. B ) ) /\ ( y e. P. /\ ( C .P. y ) = 1P ) ) /\ ( x e. P. /\ ( ( C .P. A ) +P. x ) = ( C .P. B ) ) ) -> C e. P. )
17		mulclpr 6762	. . . . . . . . 9 |- ( ( C e. P. /\ A e. P. ) -> ( C .P. A ) e. P. )
18	16, 7, 17	syl2anc 403	. . . . . . . 8 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C .P. A ) <P ( C .P. B ) ) /\ ( y e. P. /\ ( C .P. y ) = 1P ) ) /\ ( x e. P. /\ ( ( C .P. A ) +P. x ) = ( C .P. B ) ) ) -> ( C .P. A ) e. P. )
19		distrprg 6778	. . . . . . . 8 |- ( ( y e. P. /\ ( C .P. A ) e. P. /\ x e. P. ) -> ( y .P. ( ( C .P. A ) +P. x ) ) = ( ( y .P. ( C .P. A ) ) +P. ( y .P. x ) ) )
20	8, 18, 9, 19	syl3anc 1169	. . . . . . 7 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C .P. A ) <P ( C .P. B ) ) /\ ( y e. P. /\ ( C .P. y ) = 1P ) ) /\ ( x e. P. /\ ( ( C .P. A ) +P. x ) = ( C .P. B ) ) ) -> ( y .P. ( ( C .P. A ) +P. x ) ) = ( ( y .P. ( C .P. A ) ) +P. ( y .P. x ) ) )
21		mulassprg 6771	. . . . . . . . 9 |- ( ( y e. P. /\ C e. P. /\ A e. P. ) -> ( ( y .P. C ) .P. A ) = ( y .P. ( C .P. A ) ) )
22	8, 16, 7, 21	syl3anc 1169	. . . . . . . 8 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C .P. A ) <P ( C .P. B ) ) /\ ( y e. P. /\ ( C .P. y ) = 1P ) ) /\ ( x e. P. /\ ( ( C .P. A ) +P. x ) = ( C .P. B ) ) ) -> ( ( y .P. C ) .P. A ) = ( y .P. ( C .P. A ) ) )
23	22	oveq1d 5547	. . . . . . 7 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C .P. A ) <P ( C .P. B ) ) /\ ( y e. P. /\ ( C .P. y ) = 1P ) ) /\ ( x e. P. /\ ( ( C .P. A ) +P. x ) = ( C .P. B ) ) ) -> ( ( ( y .P. C ) .P. A ) +P. ( y .P. x ) ) = ( ( y .P. ( C .P. A ) ) +P. ( y .P. x ) ) )
24		mulcomprg 6770	. . . . . . . . . . . 12 |- ( ( y e. P. /\ C e. P. ) -> ( y .P. C ) = ( C .P. y ) )
25	8, 16, 24	syl2anc 403	. . . . . . . . . . 11 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C .P. A ) <P ( C .P. B ) ) /\ ( y e. P. /\ ( C .P. y ) = 1P ) ) /\ ( x e. P. /\ ( ( C .P. A ) +P. x ) = ( C .P. B ) ) ) -> ( y .P. C ) = ( C .P. y ) )
26		simplrr 502	. . . . . . . . . . 11 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C .P. A ) <P ( C .P. B ) ) /\ ( y e. P. /\ ( C .P. y ) = 1P ) ) /\ ( x e. P. /\ ( ( C .P. A ) +P. x ) = ( C .P. B ) ) ) -> ( C .P. y ) = 1P )
27	25, 26	eqtrd 2113	. . . . . . . . . 10 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C .P. A ) <P ( C .P. B ) ) /\ ( y e. P. /\ ( C .P. y ) = 1P ) ) /\ ( x e. P. /\ ( ( C .P. A ) +P. x ) = ( C .P. B ) ) ) -> ( y .P. C ) = 1P )
28	27	oveq1d 5547	. . . . . . . . 9 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C .P. A ) <P ( C .P. B ) ) /\ ( y e. P. /\ ( C .P. y ) = 1P ) ) /\ ( x e. P. /\ ( ( C .P. A ) +P. x ) = ( C .P. B ) ) ) -> ( ( y .P. C ) .P. A ) = ( 1P .P. A ) )
29		1pr 6744	. . . . . . . . . . . 12 |- 1P e. P.
30		mulcomprg 6770	. . . . . . . . . . . 12 |- ( ( A e. P. /\ 1P e. P. ) -> ( A .P. 1P ) = ( 1P .P. A ) )
31	29, 30	mpan2 415	. . . . . . . . . . 11 |- ( A e. P. -> ( A .P. 1P ) = ( 1P .P. A ) )
32		1idpr 6782	. . . . . . . . . . 11 |- ( A e. P. -> ( A .P. 1P ) = A )
33	31, 32	eqtr3d 2115	. . . . . . . . . 10 |- ( A e. P. -> ( 1P .P. A ) = A )
34	7, 33	syl 14	. . . . . . . . 9 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C .P. A ) <P ( C .P. B ) ) /\ ( y e. P. /\ ( C .P. y ) = 1P ) ) /\ ( x e. P. /\ ( ( C .P. A ) +P. x ) = ( C .P. B ) ) ) -> ( 1P .P. A ) = A )
35	28, 34	eqtrd 2113	. . . . . . . 8 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C .P. A ) <P ( C .P. B ) ) /\ ( y e. P. /\ ( C .P. y ) = 1P ) ) /\ ( x e. P. /\ ( ( C .P. A ) +P. x ) = ( C .P. B ) ) ) -> ( ( y .P. C ) .P. A ) = A )
36	35	oveq1d 5547	. . . . . . 7 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C .P. A ) <P ( C .P. B ) ) /\ ( y e. P. /\ ( C .P. y ) = 1P ) ) /\ ( x e. P. /\ ( ( C .P. A ) +P. x ) = ( C .P. B ) ) ) -> ( ( ( y .P. C ) .P. A ) +P. ( y .P. x ) ) = ( A +P. ( y .P. x ) ) )
37	20, 23, 36	3eqtr2d 2119	. . . . . 6 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C .P. A ) <P ( C .P. B ) ) /\ ( y e. P. /\ ( C .P. y ) = 1P ) ) /\ ( x e. P. /\ ( ( C .P. A ) +P. x ) = ( C .P. B ) ) ) -> ( y .P. ( ( C .P. A ) +P. x ) ) = ( A +P. ( y .P. x ) ) )
38	27	oveq1d 5547	. . . . . . 7 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C .P. A ) <P ( C .P. B ) ) /\ ( y e. P. /\ ( C .P. y ) = 1P ) ) /\ ( x e. P. /\ ( ( C .P. A ) +P. x ) = ( C .P. B ) ) ) -> ( ( y .P. C ) .P. B ) = ( 1P .P. B ) )
39	6	simp2d 951	. . . . . . . 8 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C .P. A ) <P ( C .P. B ) ) /\ ( y e. P. /\ ( C .P. y ) = 1P ) ) /\ ( x e. P. /\ ( ( C .P. A ) +P. x ) = ( C .P. B ) ) ) -> B e. P. )
40		mulassprg 6771	. . . . . . . 8 |- ( ( y e. P. /\ C e. P. /\ B e. P. ) -> ( ( y .P. C ) .P. B ) = ( y .P. ( C .P. B ) ) )
41	8, 16, 39, 40	syl3anc 1169	. . . . . . 7 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C .P. A ) <P ( C .P. B ) ) /\ ( y e. P. /\ ( C .P. y ) = 1P ) ) /\ ( x e. P. /\ ( ( C .P. A ) +P. x ) = ( C .P. B ) ) ) -> ( ( y .P. C ) .P. B ) = ( y .P. ( C .P. B ) ) )
42		mulcomprg 6770	. . . . . . . . . 10 |- ( ( B e. P. /\ 1P e. P. ) -> ( B .P. 1P ) = ( 1P .P. B ) )
43	29, 42	mpan2 415	. . . . . . . . 9 |- ( B e. P. -> ( B .P. 1P ) = ( 1P .P. B ) )
44		1idpr 6782	. . . . . . . . 9 |- ( B e. P. -> ( B .P. 1P ) = B )
45	43, 44	eqtr3d 2115	. . . . . . . 8 |- ( B e. P. -> ( 1P .P. B ) = B )
46	39, 45	syl 14	. . . . . . 7 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C .P. A ) <P ( C .P. B ) ) /\ ( y e. P. /\ ( C .P. y ) = 1P ) ) /\ ( x e. P. /\ ( ( C .P. A ) +P. x ) = ( C .P. B ) ) ) -> ( 1P .P. B ) = B )
47	38, 41, 46	3eqtr3d 2121	. . . . . 6 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C .P. A ) <P ( C .P. B ) ) /\ ( y e. P. /\ ( C .P. y ) = 1P ) ) /\ ( x e. P. /\ ( ( C .P. A ) +P. x ) = ( C .P. B ) ) ) -> ( y .P. ( C .P. B ) ) = B )
48	15, 37, 47	3eqtr3d 2121	. . . . 5 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C .P. A ) <P ( C .P. B ) ) /\ ( y e. P. /\ ( C .P. y ) = 1P ) ) /\ ( x e. P. /\ ( ( C .P. A ) +P. x ) = ( C .P. B ) ) ) -> ( A +P. ( y .P. x ) ) = B )
49	13, 48	breqtrd 3809	. . . 4 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C .P. A ) <P ( C .P. B ) ) /\ ( y e. P. /\ ( C .P. y ) = 1P ) ) /\ ( x e. P. /\ ( ( C .P. A ) +P. x ) = ( C .P. B ) ) ) -> A <P B )
50	5, 49	rexlimddv 2481	. . 3 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C .P. A ) <P ( C .P. B ) ) /\ ( y e. P. /\ ( C .P. y ) = 1P ) ) -> A <P B )
51	3, 50	rexlimddv 2481	. 2 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C .P. A ) <P ( C .P. B ) ) -> A <P B )
52	51	ex 113	1 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( C .P. A ) <P ( C .P. B ) -> A <P B ) )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 919   = wceq 1284   e. wcel 1433   E.wrex 2349   class class class wbr 3785  (class class class)co 5532   P.cnp 6481   1Pc1p 6482   +P. cpp 6483   .P. cmp 6484   <P cltp 6485

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-eprel 4044  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-1o 6024  df-2o 6025  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-pli 6495  df-mi 6496  df-lti 6497  df-plpq 6534  df-mpq 6535  df-enq 6537  df-nqqs 6538  df-plqqs 6539  df-mqqs 6540  df-1nqqs 6541  df-rq 6542  df-ltnqqs 6543  df-enq0 6614  df-nq0 6615  df-0nq0 6616  df-plq0 6617  df-mq0 6618  df-inp 6656  df-i1p 6657  df-iplp 6658  df-imp 6659  df-iltp 6660

This theorem is referenced by:  mulextsr1lem  6956