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Theorem ltmprr 6738
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 6734 . . . . 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 6709 . . . . 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 6668 . . . . . . 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 6693 . . . . . 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 6668 . . . . . . . . 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 6684 . . . . . . . 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 6677 . . . . . . . . 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 6676 . . . . . . . . . . . 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 6650 . . . . . . . . . . . 12 1PP
30 mulcomprg 6676 . . . . . . . . . . . 12 ((𝐴P ∧ 1PP) → (𝐴 ·P 1P) = (1P ·P 𝐴))
3129, 30mpan2 401 . . . . . . . . . . 11 (𝐴P → (𝐴 ·P 1P) = (1P ·P 𝐴))
32 1idpr 6688 . . . . . . . . . . 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 6677 . . . . . . . 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 6676 . . . . . . . . . 10 ((𝐵P ∧ 1PP) → (𝐵 ·P 1P) = (1P ·P 𝐵))
4329, 42mpan2 401 . . . . . . . . 9 (𝐵P → (𝐵 ·P 1P) = (1P ·P 𝐵))
44 1idpr 6688 . . . . . . . . 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 6387  1Pc1p 6388   +P cpp 6389   ·P cmp 6390  <P cltp 6391
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6400  df-pli 6401  df-mi 6402  df-lti 6403  df-plpq 6440  df-mpq 6441  df-enq 6443  df-nqqs 6444  df-plqqs 6445  df-mqqs 6446  df-1nqqs 6447  df-rq 6448  df-ltnqqs 6449  df-enq0 6520  df-nq0 6521  df-0nq0 6522  df-plq0 6523  df-mq0 6524  df-inp 6562  df-i1p 6563  df-iplp 6564  df-imp 6565  df-iltp 6566
This theorem is referenced by:  mulextsr1lem  6862
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