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Theorem somin1 6133
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 4533 . . . . 5 (𝐴𝑅𝐵 → if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴)
21olcd 870 . . . 4 (𝐴𝑅𝐵 → (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴))
32adantl 480 . . 3 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) ∧ 𝐴𝑅𝐵) → (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴))
4 sotric 5615 . . . . . . 7 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝑅𝐴)))
5 orcom 866 . . . . . . . . 9 ((𝐴 = 𝐵𝐵𝑅𝐴) ↔ (𝐵𝑅𝐴𝐴 = 𝐵))
6 eqcom 2737 . . . . . . . . . 10 (𝐴 = 𝐵𝐵 = 𝐴)
76orbi2i 909 . . . . . . . . 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 353 . . . . 5 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → ((𝐵𝑅𝐴𝐵 = 𝐴) ↔ ¬ 𝐴𝑅𝐵))
1211biimpar 476 . . . 4 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) ∧ ¬ 𝐴𝑅𝐵) → (𝐵𝑅𝐴𝐵 = 𝐴))
13 iffalse 4536 . . . . . 6 𝐴𝑅𝐵 → if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐵)
14 breq1 5150 . . . . . . 7 (if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐵 → (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴𝐵𝑅𝐴))
15 eqeq1 2734 . . . . . . 7 (if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐵 → (if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴𝐵 = 𝐴))
1614, 15orbi12d 915 . . . . . 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 480 . . . 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 809 . 2 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴))
21 poleloe 6131 . . 3 (𝐴𝑋 → (if(𝐴𝑅𝐵, 𝐴, 𝐵)(𝑅 ∪ I )𝐴 ↔ (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴)))
2221ad2antrl 724 . 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 394  wo 843   = wceq 1539  wcel 2104  cun 3945  ifcif 4527   class class class wbr 5147   I cid 5572   Or wor 5586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-id 5573  df-po 5587  df-so 5588  df-xp 5681  df-rel 5682
This theorem is referenced by:  somin2  6135  soltmin  6136
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