Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  infsupprpr Structured version   Visualization version   GIF version

Theorem infsupprpr 8955
 Description: The infimum of a proper pair is less than the supremum of this pair. (Contributed by AV, 13-Mar-2023.)
Assertion
Ref Expression
infsupprpr ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)) → inf({𝐵, 𝐶}, 𝐴, 𝑅)𝑅sup({𝐵, 𝐶}, 𝐴, 𝑅))

Proof of Theorem infsupprpr
StepHypRef Expression
1 solin 5463 . . . 4 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵))
213adantr3 1168 . . 3 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)) → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵))
3 iftrue 4431 . . . . . . 7 (𝐵𝑅𝐶 → if(𝐵𝑅𝐶, 𝐵, 𝐶) = 𝐵)
43adantr 484 . . . . . 6 ((𝐵𝑅𝐶 ∧ (𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶))) → if(𝐵𝑅𝐶, 𝐵, 𝐶) = 𝐵)
5 sotric 5466 . . . . . . . . 9 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶 ↔ ¬ (𝐵 = 𝐶𝐶𝑅𝐵)))
653adantr3 1168 . . . . . . . 8 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)) → (𝐵𝑅𝐶 ↔ ¬ (𝐵 = 𝐶𝐶𝑅𝐵)))
76biimpac 482 . . . . . . 7 ((𝐵𝑅𝐶 ∧ (𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶))) → ¬ (𝐵 = 𝐶𝐶𝑅𝐵))
8 ioran 981 . . . . . . . 8 (¬ (𝐵 = 𝐶𝐶𝑅𝐵) ↔ (¬ 𝐵 = 𝐶 ∧ ¬ 𝐶𝑅𝐵))
9 simprl 770 . . . . . . . . . 10 ((¬ 𝐶𝑅𝐵 ∧ (𝐵𝑅𝐶 ∧ (𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)))) → 𝐵𝑅𝐶)
10 iffalse 4434 . . . . . . . . . . 11 𝐶𝑅𝐵 → if(𝐶𝑅𝐵, 𝐵, 𝐶) = 𝐶)
1110adantr 484 . . . . . . . . . 10 ((¬ 𝐶𝑅𝐵 ∧ (𝐵𝑅𝐶 ∧ (𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)))) → if(𝐶𝑅𝐵, 𝐵, 𝐶) = 𝐶)
129, 11breqtrrd 5059 . . . . . . . . 9 ((¬ 𝐶𝑅𝐵 ∧ (𝐵𝑅𝐶 ∧ (𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)))) → 𝐵𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶))
1312ex 416 . . . . . . . 8 𝐶𝑅𝐵 → ((𝐵𝑅𝐶 ∧ (𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶))) → 𝐵𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶)))
148, 13simplbiim 508 . . . . . . 7 (¬ (𝐵 = 𝐶𝐶𝑅𝐵) → ((𝐵𝑅𝐶 ∧ (𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶))) → 𝐵𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶)))
157, 14mpcom 38 . . . . . 6 ((𝐵𝑅𝐶 ∧ (𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶))) → 𝐵𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶))
164, 15eqbrtrd 5053 . . . . 5 ((𝐵𝑅𝐶 ∧ (𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶))) → if(𝐵𝑅𝐶, 𝐵, 𝐶)𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶))
1716ex 416 . . . 4 (𝐵𝑅𝐶 → ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶)))
18 eqneqall 2998 . . . . . . 7 (𝐵 = 𝐶 → (𝐵𝐶 → if(𝐵𝑅𝐶, 𝐵, 𝐶)𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶)))
19182a1d 26 . . . . . 6 (𝐵 = 𝐶 → (𝐵𝐴 → (𝐶𝐴 → (𝐵𝐶 → if(𝐵𝑅𝐶, 𝐵, 𝐶)𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶)))))
20193impd 1345 . . . . 5 (𝐵 = 𝐶 → ((𝐵𝐴𝐶𝐴𝐵𝐶) → if(𝐵𝑅𝐶, 𝐵, 𝐶)𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶)))
2120adantld 494 . . . 4 (𝐵 = 𝐶 → ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶)))
22 pm3.22 463 . . . . . . . . 9 ((𝐵𝐴𝐶𝐴) → (𝐶𝐴𝐵𝐴))
23223adant3 1129 . . . . . . . 8 ((𝐵𝐴𝐶𝐴𝐵𝐶) → (𝐶𝐴𝐵𝐴))
24 sotric 5466 . . . . . . . . 9 ((𝑅 Or 𝐴 ∧ (𝐶𝐴𝐵𝐴)) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵𝐵𝑅𝐶)))
2524biimpd 232 . . . . . . . 8 ((𝑅 Or 𝐴 ∧ (𝐶𝐴𝐵𝐴)) → (𝐶𝑅𝐵 → ¬ (𝐶 = 𝐵𝐵𝑅𝐶)))
2623, 25sylan2 595 . . . . . . 7 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)) → (𝐶𝑅𝐵 → ¬ (𝐶 = 𝐵𝐵𝑅𝐶)))
2726impcom 411 . . . . . 6 ((𝐶𝑅𝐵 ∧ (𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶))) → ¬ (𝐶 = 𝐵𝐵𝑅𝐶))
28 ioran 981 . . . . . . 7 (¬ (𝐶 = 𝐵𝐵𝑅𝐶) ↔ (¬ 𝐶 = 𝐵 ∧ ¬ 𝐵𝑅𝐶))
29 simpr 488 . . . . . . . . . 10 ((¬ 𝐵𝑅𝐶𝐶𝑅𝐵) → 𝐶𝑅𝐵)
30 iffalse 4434 . . . . . . . . . . 11 𝐵𝑅𝐶 → if(𝐵𝑅𝐶, 𝐵, 𝐶) = 𝐶)
31 iftrue 4431 . . . . . . . . . . 11 (𝐶𝑅𝐵 → if(𝐶𝑅𝐵, 𝐵, 𝐶) = 𝐵)
3230, 31breqan12d 5047 . . . . . . . . . 10 ((¬ 𝐵𝑅𝐶𝐶𝑅𝐵) → (if(𝐵𝑅𝐶, 𝐵, 𝐶)𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶) ↔ 𝐶𝑅𝐵))
3329, 32mpbird 260 . . . . . . . . 9 ((¬ 𝐵𝑅𝐶𝐶𝑅𝐵) → if(𝐵𝑅𝐶, 𝐵, 𝐶)𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶))
3433a1d 25 . . . . . . . 8 ((¬ 𝐵𝑅𝐶𝐶𝑅𝐵) → ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶)))
3534expimpd 457 . . . . . . 7 𝐵𝑅𝐶 → ((𝐶𝑅𝐵 ∧ (𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶))) → if(𝐵𝑅𝐶, 𝐵, 𝐶)𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶)))
3628, 35simplbiim 508 . . . . . 6 (¬ (𝐶 = 𝐵𝐵𝑅𝐶) → ((𝐶𝑅𝐵 ∧ (𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶))) → if(𝐵𝑅𝐶, 𝐵, 𝐶)𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶)))
3727, 36mpcom 38 . . . . 5 ((𝐶𝑅𝐵 ∧ (𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶))) → if(𝐵𝑅𝐶, 𝐵, 𝐶)𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶))
3837ex 416 . . . 4 (𝐶𝑅𝐵 → ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶)))
3917, 21, 383jaoi 1424 . . 3 ((𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵) → ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶)))
402, 39mpcom 38 . 2 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶))
41 infpr 8954 . . . 4 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → inf({𝐵, 𝐶}, 𝐴, 𝑅) = if(𝐵𝑅𝐶, 𝐵, 𝐶))
42 suppr 8922 . . . 4 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → sup({𝐵, 𝐶}, 𝐴, 𝑅) = if(𝐶𝑅𝐵, 𝐵, 𝐶))
4341, 42breq12d 5044 . . 3 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → (inf({𝐵, 𝐶}, 𝐴, 𝑅)𝑅sup({𝐵, 𝐶}, 𝐴, 𝑅) ↔ if(𝐵𝑅𝐶, 𝐵, 𝐶)𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶)))
44433adant3r3 1181 . 2 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)) → (inf({𝐵, 𝐶}, 𝐴, 𝑅)𝑅sup({𝐵, 𝐶}, 𝐴, 𝑅) ↔ if(𝐵𝑅𝐶, 𝐵, 𝐶)𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶)))
4540, 44mpbird 260 1 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)) → inf({𝐵, 𝐶}, 𝐴, 𝑅)𝑅sup({𝐵, 𝐶}, 𝐴, 𝑅))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∨ w3o 1083   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ifcif 4425  {cpr 4527   class class class wbr 5031   Or wor 5438  supcsup 8891  infcinf 8892 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-po 5439  df-so 5440  df-cnv 5528  df-iota 6284  df-riota 7094  df-sup 8893  df-inf 8894 This theorem is referenced by:  prproropf1olem2  44064
 Copyright terms: Public domain W3C validator