Theorem ltmprr 7790

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 7786	. . . . 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 7761	. . . . 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 7720	. . . . . . 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 7745	. . . . . 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 5983	. . . . . 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 7720	. . . . . . . . 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 7736	. . . . . . . 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 7729	. . . . . . . . 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 5982	. . . . . . 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 7728	. . . . . . . . . . . 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 2240	. . . . . . . . . 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 5982	. . . . . . . . 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 7702	. . . . . . . . . . . 12 |- 1P e. P.
30		mulcomprg 7728	. . . . . . . . . . . 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 7740	. . . . . . . . . . 11 |- ( A e. P. -> ( A .P. 1P ) = A )
33	31, 32	eqtr3d 2242	. . . . . . . . . 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 2240	. . . . . . . 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 5982	. . . . . . 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 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. A ) +P. x ) ) = ( A +P. ( y .P. x ) ) )
38	27	oveq1d 5982	. . . . . . 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 7729	. . . . . . . 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 7728	. . . . . . . . . 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 7740	. . . . . . . . 9 |- ( B e. P. -> ( B .P. 1P ) = B )
45	43, 44	eqtr3d 2242	. . . . . . . 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 2248	. . . . . 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 2248	. . . . 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 4085	. . . 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 2630	. . 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 2630	. 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 2178   E.wrex 2487   class class class wbr 4059  (class class class)co 5967   P.cnp 7439   1Pc1p 7440   +P. cpp 7441   .P. cmp 7442   <P cltp 7443

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654

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 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-eprel 4354  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-irdg 6479  df-1o 6525  df-2o 6526  df-oadd 6529  df-omul 6530  df-er 6643  df-ec 6645  df-qs 6649  df-ni 7452  df-pli 7453  df-mi 7454  df-lti 7455  df-plpq 7492  df-mpq 7493  df-enq 7495  df-nqqs 7496  df-plqqs 7497  df-mqqs 7498  df-1nqqs 7499  df-rq 7500  df-ltnqqs 7501  df-enq0 7572  df-nq0 7573  df-0nq0 7574  df-plq0 7575  df-mq0 7576  df-inp 7614  df-i1p 7615  df-iplp 7616  df-imp 7617  df-iltp 7618

This theorem is referenced by:  mulextsr1lem  7928