Theorem ltmprr 7755

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 7751	. . . . 5 |- ( C e. P. -> E. y e. P. ( C .P. y ) = 1P )
2	1	3ad2ant3 1023	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> E. y e. P. ( C .P. y ) = 1P )
3	2	adantr 276	. . 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 7726	. . . . 5 |- ( ( C .P. A ) <P ( C .P. B ) -> E. x e. P. ( ( C .P. A ) +P. x ) = ( C .P. B ) )
5	4	ad2antlr 489	. . . 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 533	. . . . . . 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 1012	. . . . . 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 535	. . . . . . 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 529	. . . . . . 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 7685	. . . . . . 7 |- ( ( y e. P. /\ x e. P. ) -> ( y .P. x ) e. P. )
11	8, 9, 10	syl2anc 411	. . . . . 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 7710	. . . . . 6 |- ( ( A e. P. /\ ( y .P. x ) e. P. ) -> A <P ( A +P. ( y .P. x ) ) )
13	7, 11, 12	syl2anc 411	. . . . 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 531	. . . . . . 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 5960	. . . . . 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 1014	. . . . . . . . 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 7685	. . . . . . . . 9 |- ( ( C e. P. /\ A e. P. ) -> ( C .P. A ) e. P. )
18	16, 7, 17	syl2anc 411	. . . . . . . 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 7701	. . . . . . . 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 1250	. . . . . . 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 7694	. . . . . . . . 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 1250	. . . . . . . 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 5959	. . . . . . 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 7693	. . . . . . . . . . . 12 |- ( ( y e. P. /\ C e. P. ) -> ( y .P. C ) = ( C .P. y ) )
25	8, 16, 24	syl2anc 411	. . . . . . . . . . 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 536	. . . . . . . . . . 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 2238	. . . . . . . . . 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 5959	. . . . . . . . 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 7667	. . . . . . . . . . . 12 |- 1P e. P.
30		mulcomprg 7693	. . . . . . . . . . . 12 |- ( ( A e. P. /\ 1P e. P. ) -> ( A .P. 1P ) = ( 1P .P. A ) )
31	29, 30	mpan2 425	. . . . . . . . . . 11 |- ( A e. P. -> ( A .P. 1P ) = ( 1P .P. A ) )
32		1idpr 7705	. . . . . . . . . . 11 |- ( A e. P. -> ( A .P. 1P ) = A )
33	31, 32	eqtr3d 2240	. . . . . . . . . 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 2238	. . . . . . . 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 5959	. . . . . . 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 2244	. . . . . 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 5959	. . . . . . 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 1013	. . . . . . . 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 7694	. . . . . . . 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 1250	. . . . . . 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 7693	. . . . . . . . . 10 |- ( ( B e. P. /\ 1P e. P. ) -> ( B .P. 1P ) = ( 1P .P. B ) )
43	29, 42	mpan2 425	. . . . . . . . 9 |- ( B e. P. -> ( B .P. 1P ) = ( 1P .P. B ) )
44		1idpr 7705	. . . . . . . . 9 |- ( B e. P. -> ( B .P. 1P ) = B )
45	43, 44	eqtr3d 2240	. . . . . . . 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 2246	. . . . . 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 2246	. . . . 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 4070	. . . 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 2628	. . 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 2628	. 2 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C .P. A ) <P ( C .P. B ) ) -> A <P B )
52	51	ex 115	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 104   /\ w3a 981   = wceq 1373   e. wcel 2176   E.wrex 2485   class class class wbr 4044  (class class class)co 5944   P.cnp 7404   1Pc1p 7405   +P. cpp 7406   .P. cmp 7407   <P cltp 7408

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-eprel 4336  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-irdg 6456  df-1o 6502  df-2o 6503  df-oadd 6506  df-omul 6507  df-er 6620  df-ec 6622  df-qs 6626  df-ni 7417  df-pli 7418  df-mi 7419  df-lti 7420  df-plpq 7457  df-mpq 7458  df-enq 7460  df-nqqs 7461  df-plqqs 7462  df-mqqs 7463  df-1nqqs 7464  df-rq 7465  df-ltnqqs 7466  df-enq0 7537  df-nq0 7538  df-0nq0 7539  df-plq0 7540  df-mq0 7541  df-inp 7579  df-i1p 7580  df-iplp 7581  df-imp 7582  df-iltp 7583

This theorem is referenced by:  mulextsr1lem  7893