Theorem somincom 6153

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

Assertion

Ref	Expression
somincom	⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → if(𝐴𝑅𝐵, 𝐴, 𝐵) = if(𝐵𝑅𝐴, 𝐵, 𝐴))

Proof of Theorem somincom

Step	Hyp	Ref	Expression
1		so2nr 5619	. . . . 5 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ¬ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴))
2		nan 829	. . . . 5 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ¬ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴)) ↔ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) ∧ 𝐴𝑅𝐵) → ¬ 𝐵𝑅𝐴))
3	1, 2	mpbi 230	. . . 4 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) ∧ 𝐴𝑅𝐵) → ¬ 𝐵𝑅𝐴)
4	3	iffalsed 4535	. . 3 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) ∧ 𝐴𝑅𝐵) → if(𝐵𝑅𝐴, 𝐵, 𝐴) = 𝐴)
5	4	eqcomd 2742	. 2 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) ∧ 𝐴𝑅𝐵) → 𝐴 = if(𝐵𝑅𝐴, 𝐵, 𝐴))
6		sotric 5621	. . . . 5 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵𝑅𝐴)))
7	6	con2bid 354	. . . 4 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴 = 𝐵 ∨ 𝐵𝑅𝐴) ↔ ¬ 𝐴𝑅𝐵))
8		ifeq2 4529	. . . . . 6 ⊢ (𝐴 = 𝐵 → if(𝐵𝑅𝐴, 𝐵, 𝐴) = if(𝐵𝑅𝐴, 𝐵, 𝐵))
9		ifid 4565	. . . . . 6 ⊢ if(𝐵𝑅𝐴, 𝐵, 𝐵) = 𝐵
10	8, 9	eqtr2di 2793	. . . . 5 ⊢ (𝐴 = 𝐵 → 𝐵 = if(𝐵𝑅𝐴, 𝐵, 𝐴))
11		iftrue 4530	. . . . . 6 ⊢ (𝐵𝑅𝐴 → if(𝐵𝑅𝐴, 𝐵, 𝐴) = 𝐵)
12	11	eqcomd 2742	. . . . 5 ⊢ (𝐵𝑅𝐴 → 𝐵 = if(𝐵𝑅𝐴, 𝐵, 𝐴))
13	10, 12	jaoi 857	. . . 4 ⊢ ((𝐴 = 𝐵 ∨ 𝐵𝑅𝐴) → 𝐵 = if(𝐵𝑅𝐴, 𝐵, 𝐴))
14	7, 13	biimtrrdi 254	. . 3 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (¬ 𝐴𝑅𝐵 → 𝐵 = if(𝐵𝑅𝐴, 𝐵, 𝐴)))
15	14	imp 406	. 2 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) ∧ ¬ 𝐴𝑅𝐵) → 𝐵 = if(𝐵𝑅𝐴, 𝐵, 𝐴))
16	5, 15	ifeqda 4561	1 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → if(𝐴𝑅𝐵, 𝐴, 𝐵) = if(𝐵𝑅𝐴, 𝐵, 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1539   ∈ wcel 2107  ifcif 4524   class class class wbr 5142   Or wor 5590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-po 5591  df-so 5592

This theorem is referenced by:  somin2  6154