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Theorem infpr 8450
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 1081 . 2 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → 𝑅 Or 𝐴)
2 ifcl 4163 . . 3 ((𝐵𝐴𝐶𝐴) → if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝐴)
323adant1 1099 . 2 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝐴)
4 ifpr 4265 . . 3 ((𝐵𝐴𝐶𝐴) → if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ {𝐵, 𝐶})
543adant1 1099 . 2 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ {𝐵, 𝐶})
6 breq2 4689 . . . . . 6 (𝐵 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐵𝑅𝐵𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
76notbid 307 . . . . 5 (𝐵 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (¬ 𝐵𝑅𝐵 ↔ ¬ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
8 breq2 4689 . . . . . 6 (𝐶 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐵𝑅𝐶𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
98notbid 307 . . . . 5 (𝐶 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (¬ 𝐵𝑅𝐶 ↔ ¬ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
10 sonr 5085 . . . . . . 7 ((𝑅 Or 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
11103adant3 1101 . . . . . 6 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → ¬ 𝐵𝑅𝐵)
1211adantr 480 . . . . 5 (((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) ∧ 𝐵𝑅𝐶) → ¬ 𝐵𝑅𝐵)
13 simpr 476 . . . . 5 (((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) ∧ ¬ 𝐵𝑅𝐶) → ¬ 𝐵𝑅𝐶)
147, 9, 12, 13ifbothda 4156 . . . 4 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → ¬ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))
15 breq2 4689 . . . . . 6 (𝐵 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐶𝑅𝐵𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
1615notbid 307 . . . . 5 (𝐵 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (¬ 𝐶𝑅𝐵 ↔ ¬ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
17 breq2 4689 . . . . . 6 (𝐶 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐶𝑅𝐶𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
1817notbid 307 . . . . 5 (𝐶 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (¬ 𝐶𝑅𝐶 ↔ ¬ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
19 so2nr 5088 . . . . . . . 8 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))
20193impb 1279 . . . . . . 7 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))
21 imnan 437 . . . . . . 7 ((𝐵𝑅𝐶 → ¬ 𝐶𝑅𝐵) ↔ ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))
2220, 21sylibr 224 . . . . . 6 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → (𝐵𝑅𝐶 → ¬ 𝐶𝑅𝐵))
2322imp 444 . . . . 5 (((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) ∧ 𝐵𝑅𝐶) → ¬ 𝐶𝑅𝐵)
24 sonr 5085 . . . . . . 7 ((𝑅 Or 𝐴𝐶𝐴) → ¬ 𝐶𝑅𝐶)
25243adant2 1100 . . . . . 6 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → ¬ 𝐶𝑅𝐶)
2625adantr 480 . . . . 5 (((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) ∧ ¬ 𝐵𝑅𝐶) → ¬ 𝐶𝑅𝐶)
2716, 18, 23, 26ifbothda 4156 . . . 4 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → ¬ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))
28 breq1 4688 . . . . . . 7 (𝑦 = 𝐵 → (𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
2928notbid 307 . . . . . 6 (𝑦 = 𝐵 → (¬ 𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ ¬ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
30 breq1 4688 . . . . . . 7 (𝑦 = 𝐶 → (𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
3130notbid 307 . . . . . 6 (𝑦 = 𝐶 → (¬ 𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ ¬ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
3229, 31ralprg 4266 . . . . 5 ((𝐵𝐴𝐶𝐴) → (∀𝑦 ∈ {𝐵, 𝐶} ¬ 𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ (¬ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ∧ ¬ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))))
33323adant1 1099 . . . 4 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → (∀𝑦 ∈ {𝐵, 𝐶} ¬ 𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ (¬ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ∧ ¬ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))))
3414, 27, 33mpbir2and 977 . . 3 ((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → ∀𝑦 ∈ {𝐵, 𝐶} ¬ 𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))
3534r19.21bi 2961 . 2 (((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) ∧ 𝑦 ∈ {𝐵, 𝐶}) → ¬ 𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))
361, 3, 5, 35infmin 8441 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 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  ifcif 4119  {cpr 4212   class class class wbr 4685   Or wor 5063  infcinf 8388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-po 5064  df-so 5065  df-cnv 5151  df-iota 5889  df-riota 6651  df-sup 8389  df-inf 8390
This theorem is referenced by:  infsn  8451  liminf10ex  40324
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