Theorem somin1 6139

Description: Property of a minimum in a strict order. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Assertion

Ref	Expression
somin1	⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → if(𝐴𝑅𝐵, 𝐴, 𝐵)(𝑅 ∪ I )𝐴)

Proof of Theorem somin1

Step	Hyp	Ref	Expression
1		iftrue 4535	. . . . 5 ⊢ (𝐴𝑅𝐵 → if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴)
2	1	olcd 873	. . . 4 ⊢ (𝐴𝑅𝐵 → (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴))
3	2	adantl 481	. . 3 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) ∧ 𝐴𝑅𝐵) → (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴))
4		sotric 5618	. . . . . . 7 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵𝑅𝐴)))
5		orcom 869	. . . . . . . . 9 ⊢ ((𝐴 = 𝐵 ∨ 𝐵𝑅𝐴) ↔ (𝐵𝑅𝐴 ∨ 𝐴 = 𝐵))
6		eqcom 2735	. . . . . . . . . 10 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
7	6	orbi2i 911	. . . . . . . . 9 ⊢ ((𝐵𝑅𝐴 ∨ 𝐴 = 𝐵) ↔ (𝐵𝑅𝐴 ∨ 𝐵 = 𝐴))
8	5, 7	bitri 275	. . . . . . . 8 ⊢ ((𝐴 = 𝐵 ∨ 𝐵𝑅𝐴) ↔ (𝐵𝑅𝐴 ∨ 𝐵 = 𝐴))
9	8	notbii 320	. . . . . . 7 ⊢ (¬ (𝐴 = 𝐵 ∨ 𝐵𝑅𝐴) ↔ ¬ (𝐵𝑅𝐴 ∨ 𝐵 = 𝐴))
10	4, 9	bitrdi 287	. . . . . 6 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝑅𝐵 ↔ ¬ (𝐵𝑅𝐴 ∨ 𝐵 = 𝐴)))
11	10	con2bid 354	. . . . 5 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐵𝑅𝐴 ∨ 𝐵 = 𝐴) ↔ ¬ 𝐴𝑅𝐵))
12	11	biimpar 477	. . . 4 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) ∧ ¬ 𝐴𝑅𝐵) → (𝐵𝑅𝐴 ∨ 𝐵 = 𝐴))
13		iffalse 4538	. . . . . 6 ⊢ (¬ 𝐴𝑅𝐵 → if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐵)
14		breq1 5151	. . . . . . 7 ⊢ (if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐵 → (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ↔ 𝐵𝑅𝐴))
15		eqeq1 2732	. . . . . . 7 ⊢ (if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐵 → (if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴 ↔ 𝐵 = 𝐴))
16	14, 15	orbi12d 917	. . . . . 6 ⊢ (if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐵 → ((if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴) ↔ (𝐵𝑅𝐴 ∨ 𝐵 = 𝐴)))
17	13, 16	syl 17	. . . . 5 ⊢ (¬ 𝐴𝑅𝐵 → ((if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴) ↔ (𝐵𝑅𝐴 ∨ 𝐵 = 𝐴)))
18	17	adantl 481	. . . 4 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) ∧ ¬ 𝐴𝑅𝐵) → ((if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴) ↔ (𝐵𝑅𝐴 ∨ 𝐵 = 𝐴)))
19	12, 18	mpbird 257	. . 3 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) ∧ ¬ 𝐴𝑅𝐵) → (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴))
20	3, 19	pm2.61dan 812	. 2 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴))
21		poleloe 6137	. . 3 ⊢ (𝐴 ∈ 𝑋 → (if(𝐴𝑅𝐵, 𝐴, 𝐵)(𝑅 ∪ I )𝐴 ↔ (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴)))
22	21	ad2antrl 727	. 2 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (if(𝐴𝑅𝐵, 𝐴, 𝐵)(𝑅 ∪ I )𝐴 ↔ (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴)))
23	20, 22	mpbird 257	1 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → if(𝐴𝑅𝐵, 𝐴, 𝐵)(𝑅 ∪ I )𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 846   = wceq 1534   ∈ wcel 2099   ∪ cun 3945  ifcif 4529   class class class wbr 5148   I cid 5575   Or wor 5589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-id 5576  df-po 5590  df-so 5591  df-xp 5684  df-rel 5685

This theorem is referenced by:  somin2  6141  soltmin  6142