Theorem soltmin 6078

Description: Being less than a minimum, for a general total order. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Assertion

Ref	Expression
soltmin	⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ (𝐴𝑅𝐵 ∧ 𝐴𝑅𝐶)))

Proof of Theorem soltmin

Step	Hyp	Ref	Expression
1		sopo 5538	. . . . . 6 ⊢ (𝑅 Or 𝑋 → 𝑅 Po 𝑋)
2	1	ad2antrr 726	. . . . 5 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝑅 Po 𝑋)
3		simplr1 1216	. . . . . 6 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐴 ∈ 𝑋)
4		simplr2 1217	. . . . . . 7 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐵 ∈ 𝑋)
5		simplr3 1218	. . . . . . 7 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐶 ∈ 𝑋)
6	4, 5	ifcld 4517	. . . . . 6 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋)
7	3, 6, 4	3jca 1128	. . . . 5 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → (𝐴 ∈ 𝑋 ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋))
8		simpr 484	. . . . 5 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))
9		simpll 766	. . . . . 6 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝑅 Or 𝑋)
10		somin1 6075	. . . . . 6 ⊢ ((𝑅 Or 𝑋 ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐵)
11	9, 4, 5, 10	syl12anc 836	. . . . 5 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐵)
12		poltletr 6074	. . . . . 6 ⊢ ((𝑅 Po 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐵) → 𝐴𝑅𝐵))
13	12	imp 406	. . . . 5 ⊢ (((𝑅 Po 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) ∧ (𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐵)) → 𝐴𝑅𝐵)
14	2, 7, 8, 11, 13	syl22anc 838	. . . 4 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐴𝑅𝐵)
15	3, 6, 5	3jca 1128	. . . . 5 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → (𝐴 ∈ 𝑋 ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋))
16		somin2 6077	. . . . . 6 ⊢ ((𝑅 Or 𝑋 ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐶)
17	9, 4, 5, 16	syl12anc 836	. . . . 5 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐶)
18		poltletr 6074	. . . . . 6 ⊢ ((𝑅 Po 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐶) → 𝐴𝑅𝐶))
19	18	imp 406	. . . . 5 ⊢ (((𝑅 Po 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐶)) → 𝐴𝑅𝐶)
20	2, 15, 8, 17, 19	syl22anc 838	. . . 4 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐴𝑅𝐶)
21	14, 20	jca 511	. . 3 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → (𝐴𝑅𝐵 ∧ 𝐴𝑅𝐶))
22	21	ex 412	. 2 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐴𝑅𝐵 ∧ 𝐴𝑅𝐶)))
23		breq2 5090	. . 3 ⊢ (𝐵 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐴𝑅𝐵 ↔ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
24		breq2 5090	. . 3 ⊢ (𝐶 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐴𝑅𝐶 ↔ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
25	23, 24	ifboth 4510	. 2 ⊢ ((𝐴𝑅𝐵 ∧ 𝐴𝑅𝐶) → 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))
26	22, 25	impbid1 225	1 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ (𝐴𝑅𝐵 ∧ 𝐴𝑅𝐶)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2111   ∪ cun 3895  ifcif 4470   class class class wbr 5086   I cid 5505   Po wpo 5517   Or wor 5518

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-br 5087  df-opab 5149  df-id 5506  df-po 5519  df-so 5520  df-xp 5617  df-rel 5618

This theorem is referenced by:  wemaplem2  9428