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Theorem infsupprpr 9495
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 5603 . . . 4 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵))
213adantr3 1168 . . 3 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)) → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵))
3 iftrue 4526 . . . . . . 7 (𝐵𝑅𝐶 → if(𝐵𝑅𝐶, 𝐵, 𝐶) = 𝐵)
43adantr 480 . . . . . 6 ((𝐵𝑅𝐶 ∧ (𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶))) → if(𝐵𝑅𝐶, 𝐵, 𝐶) = 𝐵)
5 sotric 5606 . . . . . . . . 9 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶 ↔ ¬ (𝐵 = 𝐶𝐶𝑅𝐵)))
653adantr3 1168 . . . . . . . 8 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)) → (𝐵𝑅𝐶 ↔ ¬ (𝐵 = 𝐶𝐶𝑅𝐵)))
76biimpac 478 . . . . . . 7 ((𝐵𝑅𝐶 ∧ (𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶))) → ¬ (𝐵 = 𝐶𝐶𝑅𝐵))
8 ioran 980 . . . . . . . 8 (¬ (𝐵 = 𝐶𝐶𝑅𝐵) ↔ (¬ 𝐵 = 𝐶 ∧ ¬ 𝐶𝑅𝐵))
9 simprl 768 . . . . . . . . . 10 ((¬ 𝐶𝑅𝐵 ∧ (𝐵𝑅𝐶 ∧ (𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)))) → 𝐵𝑅𝐶)
10 iffalse 4529 . . . . . . . . . . 11 𝐶𝑅𝐵 → if(𝐶𝑅𝐵, 𝐵, 𝐶) = 𝐶)
1110adantr 480 . . . . . . . . . 10 ((¬ 𝐶𝑅𝐵 ∧ (𝐵𝑅𝐶 ∧ (𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)))) → if(𝐶𝑅𝐵, 𝐵, 𝐶) = 𝐶)
129, 11breqtrrd 5166 . . . . . . . . 9 ((¬ 𝐶𝑅𝐵 ∧ (𝐵𝑅𝐶 ∧ (𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)))) → 𝐵𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶))
1312ex 412 . . . . . . . 8 𝐶𝑅𝐵 → ((𝐵𝑅𝐶 ∧ (𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶))) → 𝐵𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶)))
148, 13simplbiim 504 . . . . . . 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 5160 . . . . 5 ((𝐵𝑅𝐶 ∧ (𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶))) → if(𝐵𝑅𝐶, 𝐵, 𝐶)𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶))
1716ex 412 . . . 4 (𝐵𝑅𝐶 → ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶)))
18 eqneqall 2943 . . . . . . 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 490 . . . 4 (𝐵 = 𝐶 → ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶)))
22 pm3.22 459 . . . . . . . . 9 ((𝐵𝐴𝐶𝐴) → (𝐶𝐴𝐵𝐴))
23223adant3 1129 . . . . . . . 8 ((𝐵𝐴𝐶𝐴𝐵𝐶) → (𝐶𝐴𝐵𝐴))
24 sotric 5606 . . . . . . . . 9 ((𝑅 Or 𝐴 ∧ (𝐶𝐴𝐵𝐴)) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵𝐵𝑅𝐶)))
2524biimpd 228 . . . . . . . 8 ((𝑅 Or 𝐴 ∧ (𝐶𝐴𝐵𝐴)) → (𝐶𝑅𝐵 → ¬ (𝐶 = 𝐵𝐵𝑅𝐶)))
2623, 25sylan2 592 . . . . . . 7 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)) → (𝐶𝑅𝐵 → ¬ (𝐶 = 𝐵𝐵𝑅𝐶)))
2726impcom 407 . . . . . 6 ((𝐶𝑅𝐵 ∧ (𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶))) → ¬ (𝐶 = 𝐵𝐵𝑅𝐶))
28 ioran 980 . . . . . . 7 (¬ (𝐶 = 𝐵𝐵𝑅𝐶) ↔ (¬ 𝐶 = 𝐵 ∧ ¬ 𝐵𝑅𝐶))
29 simpr 484 . . . . . . . . . 10 ((¬ 𝐵𝑅𝐶𝐶𝑅𝐵) → 𝐶𝑅𝐵)
30 iffalse 4529 . . . . . . . . . . 11 𝐵𝑅𝐶 → if(𝐵𝑅𝐶, 𝐵, 𝐶) = 𝐶)
31 iftrue 4526 . . . . . . . . . . 11 (𝐶𝑅𝐵 → if(𝐶𝑅𝐵, 𝐵, 𝐶) = 𝐵)
3230, 31breqan12d 5154 . . . . . . . . . 10 ((¬ 𝐵𝑅𝐶𝐶𝑅𝐵) → (if(𝐵𝑅𝐶, 𝐵, 𝐶)𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶) ↔ 𝐶𝑅𝐵))
3329, 32mpbird 257 . . . . . . . . 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 453 . . . . . . 7 𝐵𝑅𝐶 → ((𝐶𝑅𝐵 ∧ (𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶))) → if(𝐵𝑅𝐶, 𝐵, 𝐶)𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶)))
3628, 35simplbiim 504 . . . . . 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 412 . . . 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 9494 . . . 4 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → inf({𝐵, 𝐶}, 𝐴, 𝑅) = if(𝐵𝑅𝐶, 𝐵, 𝐶))
42 suppr 9462 . . . 4 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → sup({𝐵, 𝐶}, 𝐴, 𝑅) = if(𝐶𝑅𝐵, 𝐵, 𝐶))
4341, 42breq12d 5151 . . 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 257 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 205  wa 395  wo 844  w3o 1083  w3a 1084   = wceq 1533  wcel 2098  wne 2932  ifcif 4520  {cpr 4622   class class class wbr 5138   Or wor 5577  supcsup 9431  infcinf 9432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-po 5578  df-so 5579  df-cnv 5674  df-iota 6485  df-riota 7357  df-sup 9433  df-inf 9434
This theorem is referenced by:  prproropf1olem2  46657
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