Theorem infpr 9262

Description: The infimum of a pair. (Contributed by AV, 4-Sep-2020.)

Assertion

Ref	Expression
infpr	⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → inf({𝐵, 𝐶}, 𝐴, 𝑅) = if(𝐵𝑅𝐶, 𝐵, 𝐶))

Proof of Theorem infpr

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1135	. 2 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → 𝑅 Or 𝐴)
2		ifcl 4504	. . 3 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝐴)
3	2	3adant1 1129	. 2 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝐴)
4		ifpr 4627	. . 3 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ {𝐵, 𝐶})
5	4	3adant1 1129	. 2 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ {𝐵, 𝐶})
6		breq2 5078	. . . . . 6 ⊢ (𝐵 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐵𝑅𝐵 ↔ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
7	6	notbid 318	. . . . 5 ⊢ (𝐵 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (¬ 𝐵𝑅𝐵 ↔ ¬ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
8		breq2 5078	. . . . . 6 ⊢ (𝐶 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐵𝑅𝐶 ↔ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
9	8	notbid 318	. . . . 5 ⊢ (𝐶 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (¬ 𝐵𝑅𝐶 ↔ ¬ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
10		sonr 5526	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵)
11	10	3adant3 1131	. . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ¬ 𝐵𝑅𝐵)
12	11	adantr 481	. . . . 5 ⊢ (((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ 𝐵𝑅𝐶) → ¬ 𝐵𝑅𝐵)
13		simpr 485	. . . . 5 ⊢ (((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ ¬ 𝐵𝑅𝐶) → ¬ 𝐵𝑅𝐶)
14	7, 9, 12, 13	ifbothda 4497	. . . 4 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ¬ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))
15		breq2 5078	. . . . . 6 ⊢ (𝐵 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐶𝑅𝐵 ↔ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
16	15	notbid 318	. . . . 5 ⊢ (𝐵 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (¬ 𝐶𝑅𝐵 ↔ ¬ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
17		breq2 5078	. . . . . 6 ⊢ (𝐶 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐶𝑅𝐶 ↔ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
18	17	notbid 318	. . . . 5 ⊢ (𝐶 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (¬ 𝐶𝑅𝐶 ↔ ¬ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
19		so2nr 5529	. . . . . . . 8 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵))
20	19	3impb 1114	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵))
21		imnan 400	. . . . . . 7 ⊢ ((𝐵𝑅𝐶 → ¬ 𝐶𝑅𝐵) ↔ ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵))
22	20, 21	sylibr 233	. . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵𝑅𝐶 → ¬ 𝐶𝑅𝐵))
23	22	imp 407	. . . . 5 ⊢ (((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ 𝐵𝑅𝐶) → ¬ 𝐶𝑅𝐵)
24		sonr 5526	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ 𝐶 ∈ 𝐴) → ¬ 𝐶𝑅𝐶)
25	24	3adant2 1130	. . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ¬ 𝐶𝑅𝐶)
26	25	adantr 481	. . . . 5 ⊢ (((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ ¬ 𝐵𝑅𝐶) → ¬ 𝐶𝑅𝐶)
27	16, 18, 23, 26	ifbothda 4497	. . . 4 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ¬ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))
28		breq1 5077	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
29	28	notbid 318	. . . . . 6 ⊢ (𝑦 = 𝐵 → (¬ 𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ ¬ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
30		breq1 5077	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
31	30	notbid 318	. . . . . 6 ⊢ (𝑦 = 𝐶 → (¬ 𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ ¬ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
32	29, 31	ralprg 4630	. . . . 5 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (∀𝑦 ∈ {𝐵, 𝐶} ¬ 𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ (¬ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ∧ ¬ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))))
33	32	3adant1 1129	. . . 4 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (∀𝑦 ∈ {𝐵, 𝐶} ¬ 𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ (¬ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ∧ ¬ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))))
34	14, 27, 33	mpbir2and 710	. . 3 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ∀𝑦 ∈ {𝐵, 𝐶} ¬ 𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))
35	34	r19.21bi 3134	. 2 ⊢ (((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ 𝑦 ∈ {𝐵, 𝐶}) → ¬ 𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))
36	1, 3, 5, 35	infmin 9253	1 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → inf({𝐵, 𝐶}, 𝐴, 𝑅) = if(𝐵𝑅𝐶, 𝐵, 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ifcif 4459  {cpr 4563   class class class wbr 5074   Or wor 5502  infcinf 9200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-po 5503  df-so 5504  df-cnv 5597  df-iota 6391  df-riota 7232  df-sup 9201  df-inf 9202

This theorem is referenced by:  infsupprpr  9263  infsn  9264  liminf10ex  43315  prproropf1olem2  44956  prproropf1olem3  44957  prproropf1olem4  44958