Theorem ltmprr 7450

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 7446	. . . . 5 |- ( C e. P. -> E. y e. P. ( C .P. y ) = 1P )
2	1	3ad2ant3 1004	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> E. y e. P. ( C .P. y ) = 1P )
3	2	adantr 274	. . 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 7421	. . . . 5 |- ( ( C .P. A ) <P ( C .P. B ) -> E. x e. P. ( ( C .P. A ) +P. x ) = ( C .P. B ) )
5	4	ad2antlr 480	. . . 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 522	. . . . . . 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 993	. . . . . 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 524	. . . . . . 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 520	. . . . . . 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 7380	. . . . . . 7 |- ( ( y e. P. /\ x e. P. ) -> ( y .P. x ) e. P. )
11	8, 9, 10	syl2anc 408	. . . . . 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 7405	. . . . . 6 |- ( ( A e. P. /\ ( y .P. x ) e. P. ) -> A <P ( A +P. ( y .P. x ) ) )
13	7, 11, 12	syl2anc 408	. . . . 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 521	. . . . . . 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 5790	. . . . . 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 995	. . . . . . . . 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 7380	. . . . . . . . 9 |- ( ( C e. P. /\ A e. P. ) -> ( C .P. A ) e. P. )
18	16, 7, 17	syl2anc 408	. . . . . . . 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 7396	. . . . . . . 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 1216	. . . . . . 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 7389	. . . . . . . . 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 1216	. . . . . . . 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 5789	. . . . . . 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 7388	. . . . . . . . . . . 12 |- ( ( y e. P. /\ C e. P. ) -> ( y .P. C ) = ( C .P. y ) )
25	8, 16, 24	syl2anc 408	. . . . . . . . . . 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 525	. . . . . . . . . . 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 2172	. . . . . . . . . 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 5789	. . . . . . . . 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 7362	. . . . . . . . . . . 12 |- 1P e. P.
30		mulcomprg 7388	. . . . . . . . . . . 12 |- ( ( A e. P. /\ 1P e. P. ) -> ( A .P. 1P ) = ( 1P .P. A ) )
31	29, 30	mpan2 421	. . . . . . . . . . 11 |- ( A e. P. -> ( A .P. 1P ) = ( 1P .P. A ) )
32		1idpr 7400	. . . . . . . . . . 11 |- ( A e. P. -> ( A .P. 1P ) = A )
33	31, 32	eqtr3d 2174	. . . . . . . . . 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 2172	. . . . . . . 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 5789	. . . . . . 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 2178	. . . . . 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 5789	. . . . . . 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 994	. . . . . . . 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 7389	. . . . . . . 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 1216	. . . . . . 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 7388	. . . . . . . . . 10 |- ( ( B e. P. /\ 1P e. P. ) -> ( B .P. 1P ) = ( 1P .P. B ) )
43	29, 42	mpan2 421	. . . . . . . . 9 |- ( B e. P. -> ( B .P. 1P ) = ( 1P .P. B ) )
44		1idpr 7400	. . . . . . . . 9 |- ( B e. P. -> ( B .P. 1P ) = B )
45	43, 44	eqtr3d 2174	. . . . . . . 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 2180	. . . . . 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 2180	. . . . 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 3954	. . . 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 2554	. . 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 2554	. 2 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C .P. A ) <P ( C .P. B ) ) -> A <P B )
52	51	ex 114	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 103   /\ w3a 962   = wceq 1331   e. wcel 1480   E.wrex 2417   class class class wbr 3929  (class class class)co 5774   P.cnp 7099   1Pc1p 7100   +P. cpp 7101   .P. cmp 7102   <P cltp 7103

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-plpq 7152  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-plqqs 7157  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161  df-enq0 7232  df-nq0 7233  df-0nq0 7234  df-plq0 7235  df-mq0 7236  df-inp 7274  df-i1p 7275  df-iplp 7276  df-imp 7277  df-iltp 7278

This theorem is referenced by:  mulextsr1lem  7588