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Theorem somin1 6091
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 4496 . . . . 5 (𝐴𝑅𝐵 → if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴)
21olcd 873 . . . 4 (𝐴𝑅𝐵 → (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴))
32adantl 483 . . 3 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) ∧ 𝐴𝑅𝐵) → (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴))
4 sotric 5577 . . . . . . 7 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝑅𝐴)))
5 orcom 869 . . . . . . . . 9 ((𝐴 = 𝐵𝐵𝑅𝐴) ↔ (𝐵𝑅𝐴𝐴 = 𝐵))
6 eqcom 2740 . . . . . . . . . 10 (𝐴 = 𝐵𝐵 = 𝐴)
76orbi2i 912 . . . . . . . . 9 ((𝐵𝑅𝐴𝐴 = 𝐵) ↔ (𝐵𝑅𝐴𝐵 = 𝐴))
85, 7bitri 275 . . . . . . . 8 ((𝐴 = 𝐵𝐵𝑅𝐴) ↔ (𝐵𝑅𝐴𝐵 = 𝐴))
98notbii 320 . . . . . . 7 (¬ (𝐴 = 𝐵𝐵𝑅𝐴) ↔ ¬ (𝐵𝑅𝐴𝐵 = 𝐴))
104, 9bitrdi 287 . . . . . 6 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝑅𝐵 ↔ ¬ (𝐵𝑅𝐴𝐵 = 𝐴)))
1110con2bid 355 . . . . 5 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → ((𝐵𝑅𝐴𝐵 = 𝐴) ↔ ¬ 𝐴𝑅𝐵))
1211biimpar 479 . . . 4 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) ∧ ¬ 𝐴𝑅𝐵) → (𝐵𝑅𝐴𝐵 = 𝐴))
13 iffalse 4499 . . . . . 6 𝐴𝑅𝐵 → if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐵)
14 breq1 5112 . . . . . . 7 (if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐵 → (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴𝐵𝑅𝐴))
15 eqeq1 2737 . . . . . . 7 (if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐵 → (if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴𝐵 = 𝐴))
1614, 15orbi12d 918 . . . . . 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 483 . . . 4 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) ∧ ¬ 𝐴𝑅𝐵) → ((if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴) ↔ (𝐵𝑅𝐴𝐵 = 𝐴)))
1912, 18mpbird 257 . . 3 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) ∧ ¬ 𝐴𝑅𝐵) → (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴))
203, 19pm2.61dan 812 . 2 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴))
21 poleloe 6089 . . 3 (𝐴𝑋 → (if(𝐴𝑅𝐵, 𝐴, 𝐵)(𝑅 ∪ I )𝐴 ↔ (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴)))
2221ad2antrl 727 . 2 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → (if(𝐴𝑅𝐵, 𝐴, 𝐵)(𝑅 ∪ I )𝐴 ↔ (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴)))
2320, 22mpbird 257 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 397  wo 846   = wceq 1542  wcel 2107  cun 3912  ifcif 4490   class class class wbr 5109   I cid 5534   Or wor 5548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-id 5535  df-po 5549  df-so 5550  df-xp 5643  df-rel 5644
This theorem is referenced by:  somin2  6093  soltmin  6094
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