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

Theorem infsupprpr 8956
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 5491 . . . 4 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵))
213adantr3 1163 . . 3 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)) → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵))
3 iftrue 4469 . . . . . . 7 (𝐵𝑅𝐶 → if(𝐵𝑅𝐶, 𝐵, 𝐶) = 𝐵)
43adantr 481 . . . . . 6 ((𝐵𝑅𝐶 ∧ (𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶))) → if(𝐵𝑅𝐶, 𝐵, 𝐶) = 𝐵)
5 sotric 5494 . . . . . . . . 9 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶 ↔ ¬ (𝐵 = 𝐶𝐶𝑅𝐵)))
653adantr3 1163 . . . . . . . 8 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)) → (𝐵𝑅𝐶 ↔ ¬ (𝐵 = 𝐶𝐶𝑅𝐵)))
76biimpac 479 . . . . . . 7 ((𝐵𝑅𝐶 ∧ (𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶))) → ¬ (𝐵 = 𝐶𝐶𝑅𝐵))
8 ioran 977 . . . . . . . 8 (¬ (𝐵 = 𝐶𝐶𝑅𝐵) ↔ (¬ 𝐵 = 𝐶 ∧ ¬ 𝐶𝑅𝐵))
9 simprl 767 . . . . . . . . . 10 ((¬ 𝐶𝑅𝐵 ∧ (𝐵𝑅𝐶 ∧ (𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)))) → 𝐵𝑅𝐶)
10 iffalse 4472 . . . . . . . . . . 11 𝐶𝑅𝐵 → if(𝐶𝑅𝐵, 𝐵, 𝐶) = 𝐶)
1110adantr 481 . . . . . . . . . 10 ((¬ 𝐶𝑅𝐵 ∧ (𝐵𝑅𝐶 ∧ (𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)))) → if(𝐶𝑅𝐵, 𝐵, 𝐶) = 𝐶)
129, 11breqtrrd 5085 . . . . . . . . 9 ((¬ 𝐶𝑅𝐵 ∧ (𝐵𝑅𝐶 ∧ (𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)))) → 𝐵𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶))
1312ex 413 . . . . . . . 8 𝐶𝑅𝐵 → ((𝐵𝑅𝐶 ∧ (𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶))) → 𝐵𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶)))
148, 13simplbiim 505 . . . . . . 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 5079 . . . . 5 ((𝐵𝑅𝐶 ∧ (𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶))) → if(𝐵𝑅𝐶, 𝐵, 𝐶)𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶))
1716ex 413 . . . 4 (𝐵𝑅𝐶 → ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶)))
18 eqneqall 3024 . . . . . . 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 1340 . . . . 5 (𝐵 = 𝐶 → ((𝐵𝐴𝐶𝐴𝐵𝐶) → if(𝐵𝑅𝐶, 𝐵, 𝐶)𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶)))
2120adantld 491 . . . 4 (𝐵 = 𝐶 → ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶)))
22 pm3.22 460 . . . . . . . . 9 ((𝐵𝐴𝐶𝐴) → (𝐶𝐴𝐵𝐴))
23223adant3 1124 . . . . . . . 8 ((𝐵𝐴𝐶𝐴𝐵𝐶) → (𝐶𝐴𝐵𝐴))
24 sotric 5494 . . . . . . . . 9 ((𝑅 Or 𝐴 ∧ (𝐶𝐴𝐵𝐴)) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵𝐵𝑅𝐶)))
2524biimpd 230 . . . . . . . 8 ((𝑅 Or 𝐴 ∧ (𝐶𝐴𝐵𝐴)) → (𝐶𝑅𝐵 → ¬ (𝐶 = 𝐵𝐵𝑅𝐶)))
2623, 25sylan2 592 . . . . . . 7 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)) → (𝐶𝑅𝐵 → ¬ (𝐶 = 𝐵𝐵𝑅𝐶)))
2726impcom 408 . . . . . 6 ((𝐶𝑅𝐵 ∧ (𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶))) → ¬ (𝐶 = 𝐵𝐵𝑅𝐶))
28 ioran 977 . . . . . . 7 (¬ (𝐶 = 𝐵𝐵𝑅𝐶) ↔ (¬ 𝐶 = 𝐵 ∧ ¬ 𝐵𝑅𝐶))
29 simpr 485 . . . . . . . . . 10 ((¬ 𝐵𝑅𝐶𝐶𝑅𝐵) → 𝐶𝑅𝐵)
30 iffalse 4472 . . . . . . . . . . 11 𝐵𝑅𝐶 → if(𝐵𝑅𝐶, 𝐵, 𝐶) = 𝐶)
31 iftrue 4469 . . . . . . . . . . 11 (𝐶𝑅𝐵 → if(𝐶𝑅𝐵, 𝐵, 𝐶) = 𝐵)
3230, 31breqan12d 5073 . . . . . . . . . 10 ((¬ 𝐵𝑅𝐶𝐶𝑅𝐵) → (if(𝐵𝑅𝐶, 𝐵, 𝐶)𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶) ↔ 𝐶𝑅𝐵))
3329, 32mpbird 258 . . . . . . . . 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 454 . . . . . . 7 𝐵𝑅𝐶 → ((𝐶𝑅𝐵 ∧ (𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶))) → if(𝐵𝑅𝐶, 𝐵, 𝐶)𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶)))
3628, 35simplbiim 505 . . . . . 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 413 . . . 4 (𝐶𝑅𝐵 → ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶)))
3917, 21, 383jaoi 1419 . . 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 8955 . . . 4 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → inf({𝐵, 𝐶}, 𝐴, 𝑅) = if(𝐵𝑅𝐶, 𝐵, 𝐶))
42 suppr 8923 . . . 4 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → sup({𝐵, 𝐶}, 𝐴, 𝑅) = if(𝐶𝑅𝐵, 𝐵, 𝐶))
4341, 42breq12d 5070 . . 3 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → (inf({𝐵, 𝐶}, 𝐴, 𝑅)𝑅sup({𝐵, 𝐶}, 𝐴, 𝑅) ↔ if(𝐵𝑅𝐶, 𝐵, 𝐶)𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶)))
44433adant3r3 1176 . 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 258 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 207  wa 396  wo 841  w3o 1078  w3a 1079   = wceq 1528  wcel 2105  wne 3013  ifcif 4463  {cpr 4559   class class class wbr 5057   Or wor 5466  supcsup 8892  infcinf 8893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-po 5467  df-so 5468  df-cnv 5556  df-iota 6307  df-riota 7103  df-sup 8894  df-inf 8895
This theorem is referenced by:  prproropf1olem2  43543
  Copyright terms: Public domain W3C validator