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Theorem infsupprpr 9120
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 5493 . . . 4 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵))
213adantr3 1173 . . 3 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)) → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵))
3 iftrue 4445 . . . . . . 7 (𝐵𝑅𝐶 → if(𝐵𝑅𝐶, 𝐵, 𝐶) = 𝐵)
43adantr 484 . . . . . 6 ((𝐵𝑅𝐶 ∧ (𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶))) → if(𝐵𝑅𝐶, 𝐵, 𝐶) = 𝐵)
5 sotric 5496 . . . . . . . . 9 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶 ↔ ¬ (𝐵 = 𝐶𝐶𝑅𝐵)))
653adantr3 1173 . . . . . . . 8 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)) → (𝐵𝑅𝐶 ↔ ¬ (𝐵 = 𝐶𝐶𝑅𝐵)))
76biimpac 482 . . . . . . 7 ((𝐵𝑅𝐶 ∧ (𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶))) → ¬ (𝐵 = 𝐶𝐶𝑅𝐵))
8 ioran 984 . . . . . . . 8 (¬ (𝐵 = 𝐶𝐶𝑅𝐵) ↔ (¬ 𝐵 = 𝐶 ∧ ¬ 𝐶𝑅𝐵))
9 simprl 771 . . . . . . . . . 10 ((¬ 𝐶𝑅𝐵 ∧ (𝐵𝑅𝐶 ∧ (𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)))) → 𝐵𝑅𝐶)
10 iffalse 4448 . . . . . . . . . . 11 𝐶𝑅𝐵 → if(𝐶𝑅𝐵, 𝐵, 𝐶) = 𝐶)
1110adantr 484 . . . . . . . . . 10 ((¬ 𝐶𝑅𝐵 ∧ (𝐵𝑅𝐶 ∧ (𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)))) → if(𝐶𝑅𝐵, 𝐵, 𝐶) = 𝐶)
129, 11breqtrrd 5081 . . . . . . . . 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 5075 . . . . 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 2951 . . . . . . 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 1350 . . . . 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 1134 . . . . . . . 8 ((𝐵𝐴𝐶𝐴𝐵𝐶) → (𝐶𝐴𝐵𝐴))
24 sotric 5496 . . . . . . . . 9 ((𝑅 Or 𝐴 ∧ (𝐶𝐴𝐵𝐴)) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵𝐵𝑅𝐶)))
2524biimpd 232 . . . . . . . 8 ((𝑅 Or 𝐴 ∧ (𝐶𝐴𝐵𝐴)) → (𝐶𝑅𝐵 → ¬ (𝐶 = 𝐵𝐵𝑅𝐶)))
2623, 25sylan2 596 . . . . . . 7 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)) → (𝐶𝑅𝐵 → ¬ (𝐶 = 𝐵𝐵𝑅𝐶)))
2726impcom 411 . . . . . 6 ((𝐶𝑅𝐵 ∧ (𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶))) → ¬ (𝐶 = 𝐵𝐵𝑅𝐶))
28 ioran 984 . . . . . . 7 (¬ (𝐶 = 𝐵𝐵𝑅𝐶) ↔ (¬ 𝐶 = 𝐵 ∧ ¬ 𝐵𝑅𝐶))
29 simpr 488 . . . . . . . . . 10 ((¬ 𝐵𝑅𝐶𝐶𝑅𝐵) → 𝐶𝑅𝐵)
30 iffalse 4448 . . . . . . . . . . 11 𝐵𝑅𝐶 → if(𝐵𝑅𝐶, 𝐵, 𝐶) = 𝐶)
31 iftrue 4445 . . . . . . . . . . 11 (𝐶𝑅𝐵 → if(𝐶𝑅𝐵, 𝐵, 𝐶) = 𝐵)
3230, 31breqan12d 5069 . . . . . . . . . 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 1429 . . 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 9119 . . . 4 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → inf({𝐵, 𝐶}, 𝐴, 𝑅) = if(𝐵𝑅𝐶, 𝐵, 𝐶))
42 suppr 9087 . . . 4 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → sup({𝐵, 𝐶}, 𝐴, 𝑅) = if(𝐶𝑅𝐵, 𝐵, 𝐶))
4341, 42breq12d 5066 . . 3 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → (inf({𝐵, 𝐶}, 𝐴, 𝑅)𝑅sup({𝐵, 𝐶}, 𝐴, 𝑅) ↔ if(𝐵𝑅𝐶, 𝐵, 𝐶)𝑅if(𝐶𝑅𝐵, 𝐵, 𝐶)))
44433adant3r3 1186 . 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 847  w3o 1088  w3a 1089   = wceq 1543  wcel 2110  wne 2940  ifcif 4439  {cpr 4543   class class class wbr 5053   Or wor 5467  supcsup 9056  infcinf 9057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-po 5468  df-so 5469  df-cnv 5559  df-iota 6338  df-riota 7170  df-sup 9058  df-inf 9059
This theorem is referenced by:  prproropf1olem2  44629
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