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Theorem ltmprr 6740
 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 6736 . . . . 5 (𝐶P → ∃𝑦P (𝐶 ·P 𝑦) = 1P)
213ad2ant3 927 . . . 4 ((𝐴P𝐵P𝐶P) → ∃𝑦P (𝐶 ·P 𝑦) = 1P)
32adantr 261 . . 3 (((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) → ∃𝑦P (𝐶 ·P 𝑦) = 1P)
4 ltexpri 6711 . . . . 5 ((𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵) → ∃𝑥P ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))
54ad2antlr 458 . . . 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 485 . . . . . . 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 916 . . . . . 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 487 . . . . . . 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 483 . . . . . . 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 6670 . . . . . . 7 ((𝑦P𝑥P) → (𝑦 ·P 𝑥) ∈ P)
118, 9, 10syl2anc 391 . . . . . 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 6695 . . . . . 6 ((𝐴P ∧ (𝑦 ·P 𝑥) ∈ P) → 𝐴<P (𝐴 +P (𝑦 ·P 𝑥)))
137, 11, 12syl2anc 391 . . . . 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 484 . . . . . . 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 5528 . . . . . 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 918 . . . . . . . . 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 6670 . . . . . . . . 9 ((𝐶P𝐴P) → (𝐶 ·P 𝐴) ∈ P)
1816, 7, 17syl2anc 391 . . . . . . . 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 6686 . . . . . . . 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 1135 . . . . . . 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 6679 . . . . . . . . 9 ((𝑦P𝐶P𝐴P) → ((𝑦 ·P 𝐶) ·P 𝐴) = (𝑦 ·P (𝐶 ·P 𝐴)))
228, 16, 7, 21syl3anc 1135 . . . . . . . 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 5527 . . . . . . 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 6678 . . . . . . . . . . . 12 ((𝑦P𝐶P) → (𝑦 ·P 𝐶) = (𝐶 ·P 𝑦))
258, 16, 24syl2anc 391 . . . . . . . . . . 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 488 . . . . . . . . . . 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 2072 . . . . . . . . . 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 5527 . . . . . . . . 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 6652 . . . . . . . . . . . 12 1PP
30 mulcomprg 6678 . . . . . . . . . . . 12 ((𝐴P ∧ 1PP) → (𝐴 ·P 1P) = (1P ·P 𝐴))
3129, 30mpan2 401 . . . . . . . . . . 11 (𝐴P → (𝐴 ·P 1P) = (1P ·P 𝐴))
32 1idpr 6690 . . . . . . . . . . 11 (𝐴P → (𝐴 ·P 1P) = 𝐴)
3331, 32eqtr3d 2074 . . . . . . . . . 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 2072 . . . . . . . 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 5527 . . . . . . 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 2078 . . . . . 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 5527 . . . . . . 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 917 . . . . . . . 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 6679 . . . . . . . 8 ((𝑦P𝐶P𝐵P) → ((𝑦 ·P 𝐶) ·P 𝐵) = (𝑦 ·P (𝐶 ·P 𝐵)))
418, 16, 39, 40syl3anc 1135 . . . . . . 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 6678 . . . . . . . . . 10 ((𝐵P ∧ 1PP) → (𝐵 ·P 1P) = (1P ·P 𝐵))
4329, 42mpan2 401 . . . . . . . . 9 (𝐵P → (𝐵 ·P 1P) = (1P ·P 𝐵))
44 1idpr 6690 . . . . . . . . 9 (𝐵P → (𝐵 ·P 1P) = 𝐵)
4543, 44eqtr3d 2074 . . . . . . . 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 2080 . . . . . 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 2080 . . . . 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 3788 . . . 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 2437 . . 3 ((((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦P ∧ (𝐶 ·P 𝑦) = 1P)) → 𝐴<P 𝐵)
513, 50rexlimddv 2437 . 2 (((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) → 𝐴<P 𝐵)
5251ex 108 1 ((𝐴P𝐵P𝐶P) → ((𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵) → 𝐴<P 𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 885   = wceq 1243   ∈ wcel 1393  ∃wrex 2307   class class class wbr 3764  (class class class)co 5512  Pcnp 6389  1Pc1p 6390   +P cpp 6391   ·P cmp 6392
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