ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ltmprr GIF version

Theorem ltmprr 7632
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 7628 . . . . 5 (𝐶P → ∃𝑦P (𝐶 ·P 𝑦) = 1P)
213ad2ant3 1020 . . . 4 ((𝐴P𝐵P𝐶P) → ∃𝑦P (𝐶 ·P 𝑦) = 1P)
32adantr 276 . . 3 (((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) → ∃𝑦P (𝐶 ·P 𝑦) = 1P)
4 ltexpri 7603 . . . . 5 ((𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵) → ∃𝑥P ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))
54ad2antlr 489 . . . 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 533 . . . . . . 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 1009 . . . . . 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 535 . . . . . . 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 529 . . . . . . 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 7562 . . . . . . 7 ((𝑦P𝑥P) → (𝑦 ·P 𝑥) ∈ P)
118, 9, 10syl2anc 411 . . . . . 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 7587 . . . . . 6 ((𝐴P ∧ (𝑦 ·P 𝑥) ∈ P) → 𝐴<P (𝐴 +P (𝑦 ·P 𝑥)))
137, 11, 12syl2anc 411 . . . . 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 531 . . . . . . 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 5885 . . . . . 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 1011 . . . . . . . . 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 7562 . . . . . . . . 9 ((𝐶P𝐴P) → (𝐶 ·P 𝐴) ∈ P)
1816, 7, 17syl2anc 411 . . . . . . . 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 7578 . . . . . . . 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 1238 . . . . . . 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 7571 . . . . . . . . 9 ((𝑦P𝐶P𝐴P) → ((𝑦 ·P 𝐶) ·P 𝐴) = (𝑦 ·P (𝐶 ·P 𝐴)))
228, 16, 7, 21syl3anc 1238 . . . . . . . 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 5884 . . . . . . 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 7570 . . . . . . . . . . . 12 ((𝑦P𝐶P) → (𝑦 ·P 𝐶) = (𝐶 ·P 𝑦))
258, 16, 24syl2anc 411 . . . . . . . . . . 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 536 . . . . . . . . . . 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 2210 . . . . . . . . . 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 5884 . . . . . . . . 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 7544 . . . . . . . . . . . 12 1PP
30 mulcomprg 7570 . . . . . . . . . . . 12 ((𝐴P ∧ 1PP) → (𝐴 ·P 1P) = (1P ·P 𝐴))
3129, 30mpan2 425 . . . . . . . . . . 11 (𝐴P → (𝐴 ·P 1P) = (1P ·P 𝐴))
32 1idpr 7582 . . . . . . . . . . 11 (𝐴P → (𝐴 ·P 1P) = 𝐴)
3331, 32eqtr3d 2212 . . . . . . . . . 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 2210 . . . . . . . 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 5884 . . . . . . 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 2216 . . . . . 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 5884 . . . . . . 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 1010 . . . . . . . 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 7571 . . . . . . . 8 ((𝑦P𝐶P𝐵P) → ((𝑦 ·P 𝐶) ·P 𝐵) = (𝑦 ·P (𝐶 ·P 𝐵)))
418, 16, 39, 40syl3anc 1238 . . . . . . 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 7570 . . . . . . . . . 10 ((𝐵P ∧ 1PP) → (𝐵 ·P 1P) = (1P ·P 𝐵))
4329, 42mpan2 425 . . . . . . . . 9 (𝐵P → (𝐵 ·P 1P) = (1P ·P 𝐵))
44 1idpr 7582 . . . . . . . . 9 (𝐵P → (𝐵 ·P 1P) = 𝐵)
4543, 44eqtr3d 2212 . . . . . . . 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 2218 . . . . . 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 2218 . . . . 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 4026 . . . 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 2599 . . 3 ((((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦P ∧ (𝐶 ·P 𝑦) = 1P)) → 𝐴<P 𝐵)
513, 50rexlimddv 2599 . 2 (((𝐴P𝐵P𝐶P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) → 𝐴<P 𝐵)
5251ex 115 1 ((𝐴P𝐵P𝐶P) → ((𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵) → 𝐴<P 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978   = wceq 1353  wcel 2148  wrex 2456   class class class wbr 4000  (class class class)co 5869  Pcnp 7281  1Pc1p 7282   +P cpp 7283   ·P cmp 7284  <P cltp 7285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-2o 6412  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343  df-enq0 7414  df-nq0 7415  df-0nq0 7416  df-plq0 7417  df-mq0 7418  df-inp 7456  df-i1p 7457  df-iplp 7458  df-imp 7459  df-iltp 7460
This theorem is referenced by:  mulextsr1lem  7770
  Copyright terms: Public domain W3C validator