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Theorem ltmprr 6614
Description: Ordering property of multiplication. (Contributed by Jim Kingdon, 18-Feb-2020.)
Assertion
Ref Expression
ltmprr ((A P B P 𝐶 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.
StepHypRef Expression
1 recexpr 6610 . . . . 5 (𝐶 Py P (𝐶 ·P y) = 1P)
213ad2ant3 926 . . . 4 ((A P B P 𝐶 P) → y P (𝐶 ·P y) = 1P)
32adantr 261 . . 3 (((A P B P 𝐶 P) (𝐶 ·P A)<P (𝐶 ·P B)) → y P (𝐶 ·P y) = 1P)
4 ltexpri 6587 . . . . 5 ((𝐶 ·P A)<P (𝐶 ·P B) → x P ((𝐶 ·P A) +P x) = (𝐶 ·P B))
54ad2antlr 458 . . . 4 ((((A P B P 𝐶 P) (𝐶 ·P A)<P (𝐶 ·P B)) (y P (𝐶 ·P y) = 1P)) → x P ((𝐶 ·P A) +P x) = (𝐶 ·P B))
6 simplll 485 . . . . . . 7 (((((A P B P 𝐶 P) (𝐶 ·P A)<P (𝐶 ·P B)) (y P (𝐶 ·P y) = 1P)) (x P ((𝐶 ·P A) +P x) = (𝐶 ·P B))) → (A P B P 𝐶 P))
76simp1d 915 . . . . . 6 (((((A P B P 𝐶 P) (𝐶 ·P A)<P (𝐶 ·P B)) (y P (𝐶 ·P y) = 1P)) (x P ((𝐶 ·P A) +P x) = (𝐶 ·P B))) → A P)
8 simplrl 487 . . . . . . 7 (((((A P B P 𝐶 P) (𝐶 ·P A)<P (𝐶 ·P B)) (y P (𝐶 ·P y) = 1P)) (x P ((𝐶 ·P A) +P x) = (𝐶 ·P B))) → y P)
9 simprl 483 . . . . . . 7 (((((A P B P 𝐶 P) (𝐶 ·P A)<P (𝐶 ·P B)) (y P (𝐶 ·P y) = 1P)) (x P ((𝐶 ·P A) +P x) = (𝐶 ·P B))) → x P)
10 mulclpr 6553 . . . . . . 7 ((y P x P) → (y ·P x) P)
118, 9, 10syl2anc 391 . . . . . 6 (((((A P B P 𝐶 P) (𝐶 ·P A)<P (𝐶 ·P B)) (y P (𝐶 ·P y) = 1P)) (x P ((𝐶 ·P A) +P x) = (𝐶 ·P B))) → (y ·P x) P)
12 ltaddpr 6571 . . . . . 6 ((A P (y ·P x) P) → A<P (A +P (y ·P x)))
137, 11, 12syl2anc 391 . . . . 5 (((((A P B P 𝐶 P) (𝐶 ·P A)<P (𝐶 ·P B)) (y P (𝐶 ·P y) = 1P)) (x P ((𝐶 ·P A) +P x) = (𝐶 ·P B))) → A<P (A +P (y ·P x)))
14 simprr 484 . . . . . . 7 (((((A P B P 𝐶 P) (𝐶 ·P A)<P (𝐶 ·P B)) (y P (𝐶 ·P y) = 1P)) (x P ((𝐶 ·P A) +P x) = (𝐶 ·P B))) → ((𝐶 ·P A) +P x) = (𝐶 ·P B))
1514oveq2d 5471 . . . . . 6 (((((A P B P 𝐶 P) (𝐶 ·P A)<P (𝐶 ·P B)) (y P (𝐶 ·P y) = 1P)) (x P ((𝐶 ·P A) +P x) = (𝐶 ·P B))) → (y ·P ((𝐶 ·P A) +P x)) = (y ·P (𝐶 ·P B)))
166simp3d 917 . . . . . . . . 9 (((((A P B P 𝐶 P) (𝐶 ·P A)<P (𝐶 ·P B)) (y P (𝐶 ·P y) = 1P)) (x P ((𝐶 ·P A) +P x) = (𝐶 ·P B))) → 𝐶 P)
17 mulclpr 6553 . . . . . . . . 9 ((𝐶 P A P) → (𝐶 ·P A) P)
1816, 7, 17syl2anc 391 . . . . . . . 8 (((((A P B P 𝐶 P) (𝐶 ·P A)<P (𝐶 ·P B)) (y P (𝐶 ·P y) = 1P)) (x P ((𝐶 ·P A) +P x) = (𝐶 ·P B))) → (𝐶 ·P A) P)
19 distrprg 6564 . . . . . . . 8 ((y P (𝐶 ·P A) P x P) → (y ·P ((𝐶 ·P A) +P x)) = ((y ·P (𝐶 ·P A)) +P (y ·P x)))
208, 18, 9, 19syl3anc 1134 . . . . . . 7 (((((A P B P 𝐶 P) (𝐶 ·P A)<P (𝐶 ·P B)) (y P (𝐶 ·P y) = 1P)) (x 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 6557 . . . . . . . . 9 ((y P 𝐶 P A P) → ((y ·P 𝐶) ·P A) = (y ·P (𝐶 ·P A)))
228, 16, 7, 21syl3anc 1134 . . . . . . . 8 (((((A P B P 𝐶 P) (𝐶 ·P A)<P (𝐶 ·P B)) (y P (𝐶 ·P y) = 1P)) (x P ((𝐶 ·P A) +P x) = (𝐶 ·P B))) → ((y ·P 𝐶) ·P A) = (y ·P (𝐶 ·P A)))
2322oveq1d 5470 . . . . . . 7 (((((A P B P 𝐶 P) (𝐶 ·P A)<P (𝐶 ·P B)) (y P (𝐶 ·P y) = 1P)) (x 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 6556 . . . . . . . . . . . 12 ((y P 𝐶 P) → (y ·P 𝐶) = (𝐶 ·P y))
258, 16, 24syl2anc 391 . . . . . . . . . . 11 (((((A P B P 𝐶 P) (𝐶 ·P A)<P (𝐶 ·P B)) (y P (𝐶 ·P y) = 1P)) (x P ((𝐶 ·P A) +P x) = (𝐶 ·P B))) → (y ·P 𝐶) = (𝐶 ·P y))
26 simplrr 488 . . . . . . . . . . 11 (((((A P B P 𝐶 P) (𝐶 ·P A)<P (𝐶 ·P B)) (y P (𝐶 ·P y) = 1P)) (x P ((𝐶 ·P A) +P x) = (𝐶 ·P B))) → (𝐶 ·P y) = 1P)
2725, 26eqtrd 2069 . . . . . . . . . 10 (((((A P B P 𝐶 P) (𝐶 ·P A)<P (𝐶 ·P B)) (y P (𝐶 ·P y) = 1P)) (x P ((𝐶 ·P A) +P x) = (𝐶 ·P B))) → (y ·P 𝐶) = 1P)
2827oveq1d 5470 . . . . . . . . 9 (((((A P B P 𝐶 P) (𝐶 ·P A)<P (𝐶 ·P B)) (y P (𝐶 ·P y) = 1P)) (x P ((𝐶 ·P A) +P x) = (𝐶 ·P B))) → ((y ·P 𝐶) ·P A) = (1P ·P A))
29 1pr 6535 . . . . . . . . . . . 12 1P P
30 mulcomprg 6556 . . . . . . . . . . . 12 ((A P 1P P) → (A ·P 1P) = (1P ·P A))
3129, 30mpan2 401 . . . . . . . . . . 11 (A P → (A ·P 1P) = (1P ·P A))
32 1idpr 6568 . . . . . . . . . . 11 (A P → (A ·P 1P) = A)
3331, 32eqtr3d 2071 . . . . . . . . . 10 (A P → (1P ·P A) = A)
347, 33syl 14 . . . . . . . . 9 (((((A P B P 𝐶 P) (𝐶 ·P A)<P (𝐶 ·P B)) (y P (𝐶 ·P y) = 1P)) (x P ((𝐶 ·P A) +P x) = (𝐶 ·P B))) → (1P ·P A) = A)
3528, 34eqtrd 2069 . . . . . . . 8 (((((A P B P 𝐶 P) (𝐶 ·P A)<P (𝐶 ·P B)) (y P (𝐶 ·P y) = 1P)) (x P ((𝐶 ·P A) +P x) = (𝐶 ·P B))) → ((y ·P 𝐶) ·P A) = A)
3635oveq1d 5470 . . . . . . 7 (((((A P B P 𝐶 P) (𝐶 ·P A)<P (𝐶 ·P B)) (y P (𝐶 ·P y) = 1P)) (x P ((𝐶 ·P A) +P x) = (𝐶 ·P B))) → (((y ·P 𝐶) ·P A) +P (y ·P x)) = (A +P (y ·P x)))
3720, 23, 363eqtr2d 2075 . . . . . 6 (((((A P B P 𝐶 P) (𝐶 ·P A)<P (𝐶 ·P B)) (y P (𝐶 ·P y) = 1P)) (x P ((𝐶 ·P A) +P x) = (𝐶 ·P B))) → (y ·P ((𝐶 ·P A) +P x)) = (A +P (y ·P x)))
3827oveq1d 5470 . . . . . . 7 (((((A P B P 𝐶 P) (𝐶 ·P A)<P (𝐶 ·P B)) (y P (𝐶 ·P y) = 1P)) (x P ((𝐶 ·P A) +P x) = (𝐶 ·P B))) → ((y ·P 𝐶) ·P B) = (1P ·P B))
396simp2d 916 . . . . . . . 8 (((((A P B P 𝐶 P) (𝐶 ·P A)<P (𝐶 ·P B)) (y P (𝐶 ·P y) = 1P)) (x P ((𝐶 ·P A) +P x) = (𝐶 ·P B))) → B P)
40 mulassprg 6557 . . . . . . . 8 ((y P 𝐶 P B P) → ((y ·P 𝐶) ·P B) = (y ·P (𝐶 ·P B)))
418, 16, 39, 40syl3anc 1134 . . . . . . 7 (((((A P B P 𝐶 P) (𝐶 ·P A)<P (𝐶 ·P B)) (y P (𝐶 ·P y) = 1P)) (x P ((𝐶 ·P A) +P x) = (𝐶 ·P B))) → ((y ·P 𝐶) ·P B) = (y ·P (𝐶 ·P B)))
42 mulcomprg 6556 . . . . . . . . . 10 ((B P 1P P) → (B ·P 1P) = (1P ·P B))
4329, 42mpan2 401 . . . . . . . . 9 (B P → (B ·P 1P) = (1P ·P B))
44 1idpr 6568 . . . . . . . . 9 (B P → (B ·P 1P) = B)
4543, 44eqtr3d 2071 . . . . . . . 8 (B P → (1P ·P B) = B)
4639, 45syl 14 . . . . . . 7 (((((A P B P 𝐶 P) (𝐶 ·P A)<P (𝐶 ·P B)) (y P (𝐶 ·P y) = 1P)) (x P ((𝐶 ·P A) +P x) = (𝐶 ·P B))) → (1P ·P B) = B)
4738, 41, 463eqtr3d 2077 . . . . . 6 (((((A P B P 𝐶 P) (𝐶 ·P A)<P (𝐶 ·P B)) (y P (𝐶 ·P y) = 1P)) (x P ((𝐶 ·P A) +P x) = (𝐶 ·P B))) → (y ·P (𝐶 ·P B)) = B)
4815, 37, 473eqtr3d 2077 . . . . 5 (((((A P B P 𝐶 P) (𝐶 ·P A)<P (𝐶 ·P B)) (y P (𝐶 ·P y) = 1P)) (x P ((𝐶 ·P A) +P x) = (𝐶 ·P B))) → (A +P (y ·P x)) = B)
4913, 48breqtrd 3779 . . . 4 (((((A P B P 𝐶 P) (𝐶 ·P A)<P (𝐶 ·P B)) (y P (𝐶 ·P y) = 1P)) (x P ((𝐶 ·P A) +P x) = (𝐶 ·P B))) → A<P B)
505, 49rexlimddv 2431 . . 3 ((((A P B P 𝐶 P) (𝐶 ·P A)<P (𝐶 ·P B)) (y P (𝐶 ·P y) = 1P)) → A<P B)
513, 50rexlimddv 2431 . 2 (((A P B P 𝐶 P) (𝐶 ·P A)<P (𝐶 ·P B)) → A<P B)
5251ex 108 1 ((A P B P 𝐶 P) → ((𝐶 ·P A)<P (𝐶 ·P B) → A<P B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 884   = wceq 1242   wcel 1390  wrex 2301   class class class wbr 3755  (class class class)co 5455  Pcnp 6275  1Pc1p 6276   +P cpp 6277   ·P cmp 6278  <P cltp 6279
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6407  df-nq0 6408  df-0nq0 6409  df-plq0 6410  df-mq0 6411  df-inp 6449  df-i1p 6450  df-iplp 6451  df-imp 6452  df-iltp 6453
This theorem is referenced by:  mulextsr1lem  6706
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