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Theorem infpr 8955
Description: The infimum of a pair. (Contributed by AV, 4-Sep-2020.)
Assertion
Ref Expression
infpr ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → inf({𝐵, 𝐶}, 𝐴, 𝑅) = if(𝐵𝑅𝐶, 𝐵, 𝐶))

Proof of Theorem infpr
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 simp1 1128 . 2 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → 𝑅 Or 𝐴)
2 ifcl 4507 . . 3 ((𝐵𝐴𝐶𝐴) → if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝐴)
323adant1 1122 . 2 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝐴)
4 ifpr 4621 . . 3 ((𝐵𝐴𝐶𝐴) → if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ {𝐵, 𝐶})
543adant1 1122 . 2 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ {𝐵, 𝐶})
6 breq2 5061 . . . . . 6 (𝐵 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐵𝑅𝐵𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
76notbid 319 . . . . 5 (𝐵 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (¬ 𝐵𝑅𝐵 ↔ ¬ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
8 breq2 5061 . . . . . 6 (𝐶 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐵𝑅𝐶𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
98notbid 319 . . . . 5 (𝐶 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (¬ 𝐵𝑅𝐶 ↔ ¬ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
10 sonr 5489 . . . . . . 7 ((𝑅 Or 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
11103adant3 1124 . . . . . 6 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → ¬ 𝐵𝑅𝐵)
1211adantr 481 . . . . 5 (((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) ∧ 𝐵𝑅𝐶) → ¬ 𝐵𝑅𝐵)
13 simpr 485 . . . . 5 (((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) ∧ ¬ 𝐵𝑅𝐶) → ¬ 𝐵𝑅𝐶)
147, 9, 12, 13ifbothda 4500 . . . 4 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → ¬ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))
15 breq2 5061 . . . . . 6 (𝐵 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐶𝑅𝐵𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
1615notbid 319 . . . . 5 (𝐵 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (¬ 𝐶𝑅𝐵 ↔ ¬ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
17 breq2 5061 . . . . . 6 (𝐶 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐶𝑅𝐶𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
1817notbid 319 . . . . 5 (𝐶 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (¬ 𝐶𝑅𝐶 ↔ ¬ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
19 so2nr 5492 . . . . . . . 8 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))
20193impb 1107 . . . . . . 7 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))
21 imnan 400 . . . . . . 7 ((𝐵𝑅𝐶 → ¬ 𝐶𝑅𝐵) ↔ ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))
2220, 21sylibr 235 . . . . . 6 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → (𝐵𝑅𝐶 → ¬ 𝐶𝑅𝐵))
2322imp 407 . . . . 5 (((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) ∧ 𝐵𝑅𝐶) → ¬ 𝐶𝑅𝐵)
24 sonr 5489 . . . . . . 7 ((𝑅 Or 𝐴𝐶𝐴) → ¬ 𝐶𝑅𝐶)
25243adant2 1123 . . . . . 6 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → ¬ 𝐶𝑅𝐶)
2625adantr 481 . . . . 5 (((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) ∧ ¬ 𝐵𝑅𝐶) → ¬ 𝐶𝑅𝐶)
2716, 18, 23, 26ifbothda 4500 . . . 4 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → ¬ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))
28 breq1 5060 . . . . . . 7 (𝑦 = 𝐵 → (𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
2928notbid 319 . . . . . 6 (𝑦 = 𝐵 → (¬ 𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ ¬ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
30 breq1 5060 . . . . . . 7 (𝑦 = 𝐶 → (𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
3130notbid 319 . . . . . 6 (𝑦 = 𝐶 → (¬ 𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ ¬ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
3229, 31ralprg 4624 . . . . 5 ((𝐵𝐴𝐶𝐴) → (∀𝑦 ∈ {𝐵, 𝐶} ¬ 𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ (¬ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ∧ ¬ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))))
33323adant1 1122 . . . 4 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → (∀𝑦 ∈ {𝐵, 𝐶} ¬ 𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ (¬ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ∧ ¬ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))))
3414, 27, 33mpbir2and 709 . . 3 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → ∀𝑦 ∈ {𝐵, 𝐶} ¬ 𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))
3534r19.21bi 3205 . 2 (((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) ∧ 𝑦 ∈ {𝐵, 𝐶}) → ¬ 𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))
361, 3, 5, 35infmin 8946 1 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → inf({𝐵, 𝐶}, 𝐴, 𝑅) = if(𝐵𝑅𝐶, 𝐵, 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  ifcif 4463  {cpr 4559   class class class wbr 5057   Or wor 5466  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:  infsupprpr  8956  infsn  8957  liminf10ex  41931  prproropf1olem2  43543  prproropf1olem3  43544  prproropf1olem4  43545
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