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Theorem ltmprr 7414
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 7410 . . . . 5 (𝐶P → ∃𝑦P (𝐶 ·P 𝑦) = 1P)
213ad2ant3 987 . . . 4 ((𝐴P𝐵P𝐶P) → ∃𝑦P (𝐶 ·P 𝑦) = 1P)
32adantr 272 . . 3 (((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) → ∃𝑦P (𝐶 ·P 𝑦) = 1P)
4 ltexpri 7385 . . . . 5 ((𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵) → ∃𝑥P ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))
54ad2antlr 478 . . . 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 505 . . . . . . 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 976 . . . . . 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 507 . . . . . . 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 503 . . . . . . 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 7344 . . . . . . 7 ((𝑦P𝑥P) → (𝑦 ·P 𝑥) ∈ P)
118, 9, 10syl2anc 406 . . . . . 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 7369 . . . . . 6 ((𝐴P ∧ (𝑦 ·P 𝑥) ∈ P) → 𝐴<P (𝐴 +P (𝑦 ·P 𝑥)))
137, 11, 12syl2anc 406 . . . . 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 504 . . . . . . 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 5756 . . . . . 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 978 . . . . . . . . 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 7344 . . . . . . . . 9 ((𝐶P𝐴P) → (𝐶 ·P 𝐴) ∈ P)
1816, 7, 17syl2anc 406 . . . . . . . 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 7360 . . . . . . . 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 1199 . . . . . . 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 7353 . . . . . . . . 9 ((𝑦P𝐶P𝐴P) → ((𝑦 ·P 𝐶) ·P 𝐴) = (𝑦 ·P (𝐶 ·P 𝐴)))
228, 16, 7, 21syl3anc 1199 . . . . . . . 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 5755 . . . . . . 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 7352 . . . . . . . . . . . 12 ((𝑦P𝐶P) → (𝑦 ·P 𝐶) = (𝐶 ·P 𝑦))
258, 16, 24syl2anc 406 . . . . . . . . . . 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 508 . . . . . . . . . . 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 2148 . . . . . . . . . 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 5755 . . . . . . . . 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 7326 . . . . . . . . . . . 12 1PP
30 mulcomprg 7352 . . . . . . . . . . . 12 ((𝐴P ∧ 1PP) → (𝐴 ·P 1P) = (1P ·P 𝐴))
3129, 30mpan2 419 . . . . . . . . . . 11 (𝐴P → (𝐴 ·P 1P) = (1P ·P 𝐴))
32 1idpr 7364 . . . . . . . . . . 11 (𝐴P → (𝐴 ·P 1P) = 𝐴)
3331, 32eqtr3d 2150 . . . . . . . . . 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 2148 . . . . . . . 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 5755 . . . . . . 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 2154 . . . . . 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 5755 . . . . . . 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 977 . . . . . . . 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 7353 . . . . . . . 8 ((𝑦P𝐶P𝐵P) → ((𝑦 ·P 𝐶) ·P 𝐵) = (𝑦 ·P (𝐶 ·P 𝐵)))
418, 16, 39, 40syl3anc 1199 . . . . . . 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 7352 . . . . . . . . . 10 ((𝐵P ∧ 1PP) → (𝐵 ·P 1P) = (1P ·P 𝐵))
4329, 42mpan2 419 . . . . . . . . 9 (𝐵P → (𝐵 ·P 1P) = (1P ·P 𝐵))
44 1idpr 7364 . . . . . . . . 9 (𝐵P → (𝐵 ·P 1P) = 𝐵)
4543, 44eqtr3d 2150 . . . . . . . 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 2156 . . . . . 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 2156 . . . . 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 3922 . . . 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 2529 . . 3 ((((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦P ∧ (𝐶 ·P 𝑦) = 1P)) → 𝐴<P 𝐵)
513, 50rexlimddv 2529 . 2 (((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) → 𝐴<P 𝐵)
5251ex 114 1 ((𝐴P𝐵P𝐶P) → ((𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵) → 𝐴<P 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 945   = wceq 1314  wcel 1463  wrex 2392   class class class wbr 3897  (class class class)co 5740  Pcnp 7063  1Pc1p 7064   +P cpp 7065   ·P cmp 7066  <P cltp 7067
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-eprel 4179  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-1o 6279  df-2o 6280  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-pli 7077  df-mi 7078  df-lti 7079  df-plpq 7116  df-mpq 7117  df-enq 7119  df-nqqs 7120  df-plqqs 7121  df-mqqs 7122  df-1nqqs 7123  df-rq 7124  df-ltnqqs 7125  df-enq0 7196  df-nq0 7197  df-0nq0 7198  df-plq0 7199  df-mq0 7200  df-inp 7238  df-i1p 7239  df-iplp 7240  df-imp 7241  df-iltp 7242
This theorem is referenced by:  mulextsr1lem  7552
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