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Theorem infpr 8616
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 1166 . 2 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → 𝑅 Or 𝐴)
2 ifcl 4287 . . 3 ((𝐵𝐴𝐶𝐴) → if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝐴)
323adant1 1160 . 2 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝐴)
4 ifpr 4389 . . 3 ((𝐵𝐴𝐶𝐴) → if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ {𝐵, 𝐶})
543adant1 1160 . 2 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ {𝐵, 𝐶})
6 breq2 4813 . . . . . 6 (𝐵 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐵𝑅𝐵𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
76notbid 309 . . . . 5 (𝐵 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (¬ 𝐵𝑅𝐵 ↔ ¬ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
8 breq2 4813 . . . . . 6 (𝐶 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐵𝑅𝐶𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
98notbid 309 . . . . 5 (𝐶 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (¬ 𝐵𝑅𝐶 ↔ ¬ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
10 sonr 5219 . . . . . . 7 ((𝑅 Or 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
11103adant3 1162 . . . . . 6 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → ¬ 𝐵𝑅𝐵)
1211adantr 472 . . . . 5 (((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) ∧ 𝐵𝑅𝐶) → ¬ 𝐵𝑅𝐵)
13 simpr 477 . . . . 5 (((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) ∧ ¬ 𝐵𝑅𝐶) → ¬ 𝐵𝑅𝐶)
147, 9, 12, 13ifbothda 4280 . . . 4 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → ¬ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))
15 breq2 4813 . . . . . 6 (𝐵 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐶𝑅𝐵𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
1615notbid 309 . . . . 5 (𝐵 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (¬ 𝐶𝑅𝐵 ↔ ¬ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
17 breq2 4813 . . . . . 6 (𝐶 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐶𝑅𝐶𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
1817notbid 309 . . . . 5 (𝐶 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (¬ 𝐶𝑅𝐶 ↔ ¬ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
19 so2nr 5222 . . . . . . . 8 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))
20193impb 1143 . . . . . . 7 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))
21 imnan 388 . . . . . . 7 ((𝐵𝑅𝐶 → ¬ 𝐶𝑅𝐵) ↔ ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))
2220, 21sylibr 225 . . . . . 6 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → (𝐵𝑅𝐶 → ¬ 𝐶𝑅𝐵))
2322imp 395 . . . . 5 (((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) ∧ 𝐵𝑅𝐶) → ¬ 𝐶𝑅𝐵)
24 sonr 5219 . . . . . . 7 ((𝑅 Or 𝐴𝐶𝐴) → ¬ 𝐶𝑅𝐶)
25243adant2 1161 . . . . . 6 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → ¬ 𝐶𝑅𝐶)
2625adantr 472 . . . . 5 (((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) ∧ ¬ 𝐵𝑅𝐶) → ¬ 𝐶𝑅𝐶)
2716, 18, 23, 26ifbothda 4280 . . . 4 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → ¬ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))
28 breq1 4812 . . . . . . 7 (𝑦 = 𝐵 → (𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
2928notbid 309 . . . . . 6 (𝑦 = 𝐵 → (¬ 𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ ¬ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
30 breq1 4812 . . . . . . 7 (𝑦 = 𝐶 → (𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
3130notbid 309 . . . . . 6 (𝑦 = 𝐶 → (¬ 𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ ¬ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
3229, 31ralprg 4390 . . . . 5 ((𝐵𝐴𝐶𝐴) → (∀𝑦 ∈ {𝐵, 𝐶} ¬ 𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ (¬ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ∧ ¬ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))))
33323adant1 1160 . . . 4 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → (∀𝑦 ∈ {𝐵, 𝐶} ¬ 𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ (¬ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ∧ ¬ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))))
3414, 27, 33mpbir2and 704 . . 3 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → ∀𝑦 ∈ {𝐵, 𝐶} ¬ 𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))
3534r19.21bi 3079 . 2 (((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) ∧ 𝑦 ∈ {𝐵, 𝐶}) → ¬ 𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))
361, 3, 5, 35infmin 8607 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 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055  ifcif 4243  {cpr 4336   class class class wbr 4809   Or wor 5197  infcinf 8554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-po 5198  df-so 5199  df-cnv 5285  df-iota 6031  df-riota 6803  df-sup 8555  df-inf 8556
This theorem is referenced by:  infsn  8617  liminf10ex  40576
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