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Theorem somin1 6134
Description: Property of a minimum in a strict order. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Assertion
Ref Expression
somin1 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → if(𝐴𝑅𝐵, 𝐴, 𝐵)(𝑅 ∪ I )𝐴)

Proof of Theorem somin1
StepHypRef Expression
1 iftrue 4534 . . . . 5 (𝐴𝑅𝐵 → if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴)
21olcd 872 . . . 4 (𝐴𝑅𝐵 → (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴))
32adantl 482 . . 3 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) ∧ 𝐴𝑅𝐵) → (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴))
4 sotric 5616 . . . . . . 7 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝑅𝐴)))
5 orcom 868 . . . . . . . . 9 ((𝐴 = 𝐵𝐵𝑅𝐴) ↔ (𝐵𝑅𝐴𝐴 = 𝐵))
6 eqcom 2739 . . . . . . . . . 10 (𝐴 = 𝐵𝐵 = 𝐴)
76orbi2i 911 . . . . . . . . 9 ((𝐵𝑅𝐴𝐴 = 𝐵) ↔ (𝐵𝑅𝐴𝐵 = 𝐴))
85, 7bitri 274 . . . . . . . 8 ((𝐴 = 𝐵𝐵𝑅𝐴) ↔ (𝐵𝑅𝐴𝐵 = 𝐴))
98notbii 319 . . . . . . 7 (¬ (𝐴 = 𝐵𝐵𝑅𝐴) ↔ ¬ (𝐵𝑅𝐴𝐵 = 𝐴))
104, 9bitrdi 286 . . . . . 6 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝑅𝐵 ↔ ¬ (𝐵𝑅𝐴𝐵 = 𝐴)))
1110con2bid 354 . . . . 5 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → ((𝐵𝑅𝐴𝐵 = 𝐴) ↔ ¬ 𝐴𝑅𝐵))
1211biimpar 478 . . . 4 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) ∧ ¬ 𝐴𝑅𝐵) → (𝐵𝑅𝐴𝐵 = 𝐴))
13 iffalse 4537 . . . . . 6 𝐴𝑅𝐵 → if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐵)
14 breq1 5151 . . . . . . 7 (if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐵 → (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴𝐵𝑅𝐴))
15 eqeq1 2736 . . . . . . 7 (if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐵 → (if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴𝐵 = 𝐴))
1614, 15orbi12d 917 . . . . . 6 (if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐵 → ((if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴) ↔ (𝐵𝑅𝐴𝐵 = 𝐴)))
1713, 16syl 17 . . . . 5 𝐴𝑅𝐵 → ((if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴) ↔ (𝐵𝑅𝐴𝐵 = 𝐴)))
1817adantl 482 . . . 4 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) ∧ ¬ 𝐴𝑅𝐵) → ((if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴) ↔ (𝐵𝑅𝐴𝐵 = 𝐴)))
1912, 18mpbird 256 . . 3 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) ∧ ¬ 𝐴𝑅𝐵) → (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴))
203, 19pm2.61dan 811 . 2 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴))
21 poleloe 6132 . . 3 (𝐴𝑋 → (if(𝐴𝑅𝐵, 𝐴, 𝐵)(𝑅 ∪ I )𝐴 ↔ (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴)))
2221ad2antrl 726 . 2 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → (if(𝐴𝑅𝐵, 𝐴, 𝐵)(𝑅 ∪ I )𝐴 ↔ (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴)))
2320, 22mpbird 256 1 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → if(𝐴𝑅𝐵, 𝐴, 𝐵)(𝑅 ∪ I )𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845   = wceq 1541  wcel 2106  cun 3946  ifcif 4528   class class class wbr 5148   I cid 5573   Or wor 5587
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-id 5574  df-po 5588  df-so 5589  df-xp 5682  df-rel 5683
This theorem is referenced by:  somin2  6136  soltmin  6137
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