Theorem somin1 6132

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 4492	. . . . 5 ⊢ (𝐴𝑅𝐵 → if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴)
2	1	olcd 887	. . . 4 ⊢ (𝐴𝑅𝐵 → (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴))
3	2	adantl 486	. . 3 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) ∧ 𝐴𝑅𝐵) → (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴))
4		sotric 5598	. . . . . . 7 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵𝑅𝐴)))
5		orcom 883	. . . . . . . . 9 ⊢ ((𝐴 = 𝐵 ∨ 𝐵𝑅𝐴) ↔ (𝐵𝑅𝐴 ∨ 𝐴 = 𝐵))
6		eqcom 2769	. . . . . . . . . 10 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
7	6	orbi2i 925	. . . . . . . . 9 ⊢ ((𝐵𝑅𝐴 ∨ 𝐴 = 𝐵) ↔ (𝐵𝑅𝐴 ∨ 𝐵 = 𝐴))
8	5, 7	bitri 278	. . . . . . . 8 ⊢ ((𝐴 = 𝐵 ∨ 𝐵𝑅𝐴) ↔ (𝐵𝑅𝐴 ∨ 𝐵 = 𝐴))
9	8	notbii 323	. . . . . . 7 ⊢ (¬ (𝐴 = 𝐵 ∨ 𝐵𝑅𝐴) ↔ ¬ (𝐵𝑅𝐴 ∨ 𝐵 = 𝐴))
10	4, 9	bitrdi 290	. . . . . 6 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝑅𝐵 ↔ ¬ (𝐵𝑅𝐴 ∨ 𝐵 = 𝐴)))
11	10	con2bid 357	. . . . 5 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐵𝑅𝐴 ∨ 𝐵 = 𝐴) ↔ ¬ 𝐴𝑅𝐵))
12	11	biimpar 482	. . . 4 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) ∧ ¬ 𝐴𝑅𝐵) → (𝐵𝑅𝐴 ∨ 𝐵 = 𝐴))
13		iffalse 4495	. . . . . 6 ⊢ (¬ 𝐴𝑅𝐵 → if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐵)
14		breq1 5111	. . . . . . 7 ⊢ (if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐵 → (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ↔ 𝐵𝑅𝐴))
15		eqeq1 2766	. . . . . . 7 ⊢ (if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐵 → (if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴 ↔ 𝐵 = 𝐴))
16	14, 15	orbi12d 931	. . . . . 6 ⊢ (if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐵 → ((if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴) ↔ (𝐵𝑅𝐴 ∨ 𝐵 = 𝐴)))
17	13, 16	syl 18	. . . . 5 ⊢ (¬ 𝐴𝑅𝐵 → ((if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴) ↔ (𝐵𝑅𝐴 ∨ 𝐵 = 𝐴)))
18	17	adantl 486	. . . 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 260	. . 3 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) ∧ ¬ 𝐴𝑅𝐵) → (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴))
20	3, 19	pm2.61dan 824	. 2 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴))
21		poleloe 6130	. . 3 ⊢ (𝐴 ∈ 𝑋 → (if(𝐴𝑅𝐵, 𝐴, 𝐵)(𝑅 ∪ I )𝐴 ↔ (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴)))
22	21	ad2antrl 740	. 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 260	1 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → if(𝐴𝑅𝐵, 𝐴, 𝐵)(𝑅 ∪ I )𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1569   ∈ wcel 2142   ∪ cun 3902  ifcif 4486   class class class wbr 5108   I cid 5554   Or wor 5567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-id 5555  df-po 5568  df-so 5569  df-xp 5666  df-rel 5667

This theorem is used by:  somin2  6134  soltmin  6135