Theorem ltmprr 7474

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 7470	. . . . 5 |- ( C e. P. -> E. y e. P. ( C .P. y ) = 1P )
2	1	3ad2ant3 1005	. . . 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 7445	. . . . 5 |- ( ( C .P. A ) <P ( C .P. B ) -> E. x e. P. ( ( C .P. A ) +P. x ) = ( C .P. B ) )
5	4	ad2antlr 481	. . . 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 523	. . . . . . 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 994	. . . . . 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 525	. . . . . . 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 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 ) ) ) -> x e. P. )
10		mulclpr 7404	. . . . . . 7 |- ( ( y e. P. /\ x e. P. ) -> ( y .P. x ) e. P. )
11	8, 9, 10	syl2anc 409	. . . . . 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 7429	. . . . . 6 |- ( ( A e. P. /\ ( y .P. x ) e. P. ) -> A <P ( A +P. ( y .P. x ) ) )
13	7, 11, 12	syl2anc 409	. . . . 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 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 ) ) ) -> ( ( C .P. A ) +P. x ) = ( C .P. B ) )
15	14	oveq2d 5798	. . . . . 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 996	. . . . . . . . 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 7404	. . . . . . . . 9 |- ( ( C e. P. /\ A e. P. ) -> ( C .P. A ) e. P. )
18	16, 7, 17	syl2anc 409	. . . . . . . 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 7420	. . . . . . . 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 1217	. . . . . . 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 7413	. . . . . . . . 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 1217	. . . . . . . 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 5797	. . . . . . 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 7412	. . . . . . . . . . . 12 |- ( ( y e. P. /\ C e. P. ) -> ( y .P. C ) = ( C .P. y ) )
25	8, 16, 24	syl2anc 409	. . . . . . . . . . 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 526	. . . . . . . . . . 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 2173	. . . . . . . . . 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 5797	. . . . . . . . 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 7386	. . . . . . . . . . . 12 |- 1P e. P.
30		mulcomprg 7412	. . . . . . . . . . . 12 |- ( ( A e. P. /\ 1P e. P. ) -> ( A .P. 1P ) = ( 1P .P. A ) )
31	29, 30	mpan2 422	. . . . . . . . . . 11 |- ( A e. P. -> ( A .P. 1P ) = ( 1P .P. A ) )
32		1idpr 7424	. . . . . . . . . . 11 |- ( A e. P. -> ( A .P. 1P ) = A )
33	31, 32	eqtr3d 2175	. . . . . . . . . 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 2173	. . . . . . . 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 5797	. . . . . . 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 2179	. . . . . 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 5797	. . . . . . 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 995	. . . . . . . 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 7413	. . . . . . . 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 1217	. . . . . . 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 7412	. . . . . . . . . 10 |- ( ( B e. P. /\ 1P e. P. ) -> ( B .P. 1P ) = ( 1P .P. B ) )
43	29, 42	mpan2 422	. . . . . . . . 9 |- ( B e. P. -> ( B .P. 1P ) = ( 1P .P. B ) )
44		1idpr 7424	. . . . . . . . 9 |- ( B e. P. -> ( B .P. 1P ) = B )
45	43, 44	eqtr3d 2175	. . . . . . . 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 2181	. . . . . 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 2181	. . . . 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 3962	. . . 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 2557	. . 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 2557	. 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 963   = wceq 1332   e. wcel 1481   E.wrex 2418   class class class wbr 3937  (class class class)co 5782   P.cnp 7123   1Pc1p 7124   +P. cpp 7125   .P. cmp 7126   <P cltp 7127

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-eprel 4219  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-1o 6321  df-2o 6322  df-oadd 6325  df-omul 6326  df-er 6437  df-ec 6439  df-qs 6443  df-ni 7136  df-pli 7137  df-mi 7138  df-lti 7139  df-plpq 7176  df-mpq 7177  df-enq 7179  df-nqqs 7180  df-plqqs 7181  df-mqqs 7182  df-1nqqs 7183  df-rq 7184  df-ltnqqs 7185  df-enq0 7256  df-nq0 7257  df-0nq0 7258  df-plq0 7259  df-mq0 7260  df-inp 7298  df-i1p 7299  df-iplp 7300  df-imp 7301  df-iltp 7302

This theorem is referenced by:  mulextsr1lem  7612