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Theorem ltmprr 6894
Description: Ordering property of multiplication. (Contributed by Jim Kingdon, 18-Feb-2020.)
Assertion
Ref Expression
ltmprr ((𝐴P𝐵P𝐶P) → ((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 6890 . . . . 5 (𝐶P → ∃𝑦P (𝐶 ·P 𝑦) = 1P)
213ad2ant3 962 . . . 4 ((𝐴P𝐵P𝐶P) → ∃𝑦P (𝐶 ·P 𝑦) = 1P)
32adantr 270 . . 3 (((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) → ∃𝑦P (𝐶 ·P 𝑦) = 1P)
4 ltexpri 6865 . . . . 5 ((𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵) → ∃𝑥P ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))
54ad2antlr 473 . . . 4 ((((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦P ∧ (𝐶 ·P 𝑦) = 1P)) → ∃𝑥P ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))
6 simplll 500 . . . . . . 7 (((((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → (𝐴P𝐵P𝐶P))
76simp1d 951 . . . . . 6 (((((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → 𝐴P)
8 simplrl 502 . . . . . . 7 (((((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → 𝑦P)
9 simprl 498 . . . . . . 7 (((((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → 𝑥P)
10 mulclpr 6824 . . . . . . 7 ((𝑦P𝑥P) → (𝑦 ·P 𝑥) ∈ P)
118, 9, 10syl2anc 403 . . . . . 6 (((((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → (𝑦 ·P 𝑥) ∈ P)
12 ltaddpr 6849 . . . . . 6 ((𝐴P ∧ (𝑦 ·P 𝑥) ∈ P) → 𝐴<P (𝐴 +P (𝑦 ·P 𝑥)))
137, 11, 12syl2anc 403 . . . . 5 (((((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → 𝐴<P (𝐴 +P (𝑦 ·P 𝑥)))
14 simprr 499 . . . . . . 7 (((((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))
1514oveq2d 5559 . . . . . 6 (((((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → (𝑦 ·P ((𝐶 ·P 𝐴) +P 𝑥)) = (𝑦 ·P (𝐶 ·P 𝐵)))
166simp3d 953 . . . . . . . . 9 (((((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → 𝐶P)
17 mulclpr 6824 . . . . . . . . 9 ((𝐶P𝐴P) → (𝐶 ·P 𝐴) ∈ P)
1816, 7, 17syl2anc 403 . . . . . . . 8 (((((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → (𝐶 ·P 𝐴) ∈ P)
19 distrprg 6840 . . . . . . . 8 ((𝑦P ∧ (𝐶 ·P 𝐴) ∈ P𝑥P) → (𝑦 ·P ((𝐶 ·P 𝐴) +P 𝑥)) = ((𝑦 ·P (𝐶 ·P 𝐴)) +P (𝑦 ·P 𝑥)))
208, 18, 9, 19syl3anc 1170 . . . . . . 7 (((((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥P ∧ ((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 6833 . . . . . . . . 9 ((𝑦P𝐶P𝐴P) → ((𝑦 ·P 𝐶) ·P 𝐴) = (𝑦 ·P (𝐶 ·P 𝐴)))
228, 16, 7, 21syl3anc 1170 . . . . . . . 8 (((((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → ((𝑦 ·P 𝐶) ·P 𝐴) = (𝑦 ·P (𝐶 ·P 𝐴)))
2322oveq1d 5558 . . . . . . 7 (((((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥P ∧ ((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 6832 . . . . . . . . . . . 12 ((𝑦P𝐶P) → (𝑦 ·P 𝐶) = (𝐶 ·P 𝑦))
258, 16, 24syl2anc 403 . . . . . . . . . . 11 (((((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → (𝑦 ·P 𝐶) = (𝐶 ·P 𝑦))
26 simplrr 503 . . . . . . . . . . 11 (((((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → (𝐶 ·P 𝑦) = 1P)
2725, 26eqtrd 2114 . . . . . . . . . 10 (((((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → (𝑦 ·P 𝐶) = 1P)
2827oveq1d 5558 . . . . . . . . 9 (((((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → ((𝑦 ·P 𝐶) ·P 𝐴) = (1P ·P 𝐴))
29 1pr 6806 . . . . . . . . . . . 12 1PP
30 mulcomprg 6832 . . . . . . . . . . . 12 ((𝐴P ∧ 1PP) → (𝐴 ·P 1P) = (1P ·P 𝐴))
3129, 30mpan2 416 . . . . . . . . . . 11 (𝐴P → (𝐴 ·P 1P) = (1P ·P 𝐴))
32 1idpr 6844 . . . . . . . . . . 11 (𝐴P → (𝐴 ·P 1P) = 𝐴)
3331, 32eqtr3d 2116 . . . . . . . . . 10 (𝐴P → (1P ·P 𝐴) = 𝐴)
347, 33syl 14 . . . . . . . . 9 (((((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → (1P ·P 𝐴) = 𝐴)
3528, 34eqtrd 2114 . . . . . . . 8 (((((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → ((𝑦 ·P 𝐶) ·P 𝐴) = 𝐴)
3635oveq1d 5558 . . . . . . 7 (((((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥P ∧ ((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 2120 . . . . . 6 (((((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → (𝑦 ·P ((𝐶 ·P 𝐴) +P 𝑥)) = (𝐴 +P (𝑦 ·P 𝑥)))
3827oveq1d 5558 . . . . . . 7 (((((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → ((𝑦 ·P 𝐶) ·P 𝐵) = (1P ·P 𝐵))
396simp2d 952 . . . . . . . 8 (((((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → 𝐵P)
40 mulassprg 6833 . . . . . . . 8 ((𝑦P𝐶P𝐵P) → ((𝑦 ·P 𝐶) ·P 𝐵) = (𝑦 ·P (𝐶 ·P 𝐵)))
418, 16, 39, 40syl3anc 1170 . . . . . . 7 (((((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → ((𝑦 ·P 𝐶) ·P 𝐵) = (𝑦 ·P (𝐶 ·P 𝐵)))
42 mulcomprg 6832 . . . . . . . . . 10 ((𝐵P ∧ 1PP) → (𝐵 ·P 1P) = (1P ·P 𝐵))
4329, 42mpan2 416 . . . . . . . . 9 (𝐵P → (𝐵 ·P 1P) = (1P ·P 𝐵))
44 1idpr 6844 . . . . . . . . 9 (𝐵P → (𝐵 ·P 1P) = 𝐵)
4543, 44eqtr3d 2116 . . . . . . . 8 (𝐵P → (1P ·P 𝐵) = 𝐵)
4639, 45syl 14 . . . . . . 7 (((((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → (1P ·P 𝐵) = 𝐵)
4738, 41, 463eqtr3d 2122 . . . . . 6 (((((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → (𝑦 ·P (𝐶 ·P 𝐵)) = 𝐵)
4815, 37, 473eqtr3d 2122 . . . . 5 (((((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → (𝐴 +P (𝑦 ·P 𝑥)) = 𝐵)
4913, 48breqtrd 3817 . . . 4 (((((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → 𝐴<P 𝐵)
505, 49rexlimddv 2482 . . 3 ((((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦P ∧ (𝐶 ·P 𝑦) = 1P)) → 𝐴<P 𝐵)
513, 50rexlimddv 2482 . 2 (((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) → 𝐴<P 𝐵)
5251ex 113 1 ((𝐴P𝐵P𝐶P) → ((𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵) → 𝐴<P 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 920   = wceq 1285  wcel 1434  wrex 2350   class class class wbr 3793  (class class class)co 5543  Pcnp 6543  1Pc1p 6544   +P cpp 6545   ·P cmp 6546  <P cltp 6547
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-2o 6066  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-lti 6559  df-plpq 6596  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-plqqs 6601  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605  df-enq0 6676  df-nq0 6677  df-0nq0 6678  df-plq0 6679  df-mq0 6680  df-inp 6718  df-i1p 6719  df-iplp 6720  df-imp 6721  df-iltp 6722
This theorem is referenced by:  mulextsr1lem  7018
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