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Theorem infpr 8953
 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 1133 . 2 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → 𝑅 Or 𝐴)
2 ifcl 4469 . . 3 ((𝐵𝐴𝐶𝐴) → if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝐴)
323adant1 1127 . 2 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝐴)
4 ifpr 4589 . . 3 ((𝐵𝐴𝐶𝐴) → if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ {𝐵, 𝐶})
543adant1 1127 . 2 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ {𝐵, 𝐶})
6 breq2 5034 . . . . . 6 (𝐵 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐵𝑅𝐵𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
76notbid 321 . . . . 5 (𝐵 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (¬ 𝐵𝑅𝐵 ↔ ¬ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
8 breq2 5034 . . . . . 6 (𝐶 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐵𝑅𝐶𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
98notbid 321 . . . . 5 (𝐶 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (¬ 𝐵𝑅𝐶 ↔ ¬ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
10 sonr 5460 . . . . . . 7 ((𝑅 Or 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
11103adant3 1129 . . . . . 6 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → ¬ 𝐵𝑅𝐵)
1211adantr 484 . . . . 5 (((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) ∧ 𝐵𝑅𝐶) → ¬ 𝐵𝑅𝐵)
13 simpr 488 . . . . 5 (((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) ∧ ¬ 𝐵𝑅𝐶) → ¬ 𝐵𝑅𝐶)
147, 9, 12, 13ifbothda 4462 . . . 4 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → ¬ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))
15 breq2 5034 . . . . . 6 (𝐵 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐶𝑅𝐵𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
1615notbid 321 . . . . 5 (𝐵 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (¬ 𝐶𝑅𝐵 ↔ ¬ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
17 breq2 5034 . . . . . 6 (𝐶 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐶𝑅𝐶𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
1817notbid 321 . . . . 5 (𝐶 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (¬ 𝐶𝑅𝐶 ↔ ¬ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
19 so2nr 5463 . . . . . . . 8 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))
20193impb 1112 . . . . . . 7 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))
21 imnan 403 . . . . . . 7 ((𝐵𝑅𝐶 → ¬ 𝐶𝑅𝐵) ↔ ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))
2220, 21sylibr 237 . . . . . 6 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → (𝐵𝑅𝐶 → ¬ 𝐶𝑅𝐵))
2322imp 410 . . . . 5 (((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) ∧ 𝐵𝑅𝐶) → ¬ 𝐶𝑅𝐵)
24 sonr 5460 . . . . . . 7 ((𝑅 Or 𝐴𝐶𝐴) → ¬ 𝐶𝑅𝐶)
25243adant2 1128 . . . . . 6 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → ¬ 𝐶𝑅𝐶)
2625adantr 484 . . . . 5 (((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) ∧ ¬ 𝐵𝑅𝐶) → ¬ 𝐶𝑅𝐶)
2716, 18, 23, 26ifbothda 4462 . . . 4 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → ¬ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))
28 breq1 5033 . . . . . . 7 (𝑦 = 𝐵 → (𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
2928notbid 321 . . . . . 6 (𝑦 = 𝐵 → (¬ 𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ ¬ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
30 breq1 5033 . . . . . . 7 (𝑦 = 𝐶 → (𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
3130notbid 321 . . . . . 6 (𝑦 = 𝐶 → (¬ 𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ ¬ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
3229, 31ralprg 4592 . . . . 5 ((𝐵𝐴𝐶𝐴) → (∀𝑦 ∈ {𝐵, 𝐶} ¬ 𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ (¬ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ∧ ¬ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))))
33323adant1 1127 . . . 4 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → (∀𝑦 ∈ {𝐵, 𝐶} ¬ 𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ (¬ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ∧ ¬ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))))
3414, 27, 33mpbir2and 712 . . 3 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → ∀𝑦 ∈ {𝐵, 𝐶} ¬ 𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))
3534r19.21bi 3173 . 2 (((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) ∧ 𝑦 ∈ {𝐵, 𝐶}) → ¬ 𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))
361, 3, 5, 35infmin 8944 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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ifcif 4425  {cpr 4527   class class class wbr 5030   Or wor 5437  infcinf 8891 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 5167  ax-nul 5174  ax-pr 5295 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 4801  df-br 5031  df-opab 5093  df-po 5438  df-so 5439  df-cnv 5527  df-iota 6283  df-riota 7093  df-sup 8892  df-inf 8893 This theorem is referenced by:  infsupprpr  8954  infsn  8955  liminf10ex  42431  prproropf1olem2  44036  prproropf1olem3  44037  prproropf1olem4  44038
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