Theorem infpr 9119

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 1138	. 2 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → 𝑅 Or 𝐴)
2		ifcl 4484	. . 3 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝐴)
3	2	3adant1 1132	. 2 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝐴)
4		ifpr 4607	. . 3 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ {𝐵, 𝐶})
5	4	3adant1 1132	. 2 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ {𝐵, 𝐶})
6		breq2 5057	. . . . . 6 ⊢ (𝐵 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐵𝑅𝐵 ↔ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
7	6	notbid 321	. . . . 5 ⊢ (𝐵 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (¬ 𝐵𝑅𝐵 ↔ ¬ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
8		breq2 5057	. . . . . 6 ⊢ (𝐶 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐵𝑅𝐶 ↔ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
9	8	notbid 321	. . . . 5 ⊢ (𝐶 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (¬ 𝐵𝑅𝐶 ↔ ¬ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
10		sonr 5491	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵)
11	10	3adant3 1134	. . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ¬ 𝐵𝑅𝐵)
12	11	adantr 484	. . . . 5 ⊢ (((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ 𝐵𝑅𝐶) → ¬ 𝐵𝑅𝐵)
13		simpr 488	. . . . 5 ⊢ (((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ ¬ 𝐵𝑅𝐶) → ¬ 𝐵𝑅𝐶)
14	7, 9, 12, 13	ifbothda 4477	. . . 4 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ¬ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))
15		breq2 5057	. . . . . 6 ⊢ (𝐵 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐶𝑅𝐵 ↔ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
16	15	notbid 321	. . . . 5 ⊢ (𝐵 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (¬ 𝐶𝑅𝐵 ↔ ¬ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
17		breq2 5057	. . . . . 6 ⊢ (𝐶 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐶𝑅𝐶 ↔ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
18	17	notbid 321	. . . . 5 ⊢ (𝐶 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (¬ 𝐶𝑅𝐶 ↔ ¬ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
19		so2nr 5494	. . . . . . . 8 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵))
20	19	3impb 1117	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵))
21		imnan 403	. . . . . . 7 ⊢ ((𝐵𝑅𝐶 → ¬ 𝐶𝑅𝐵) ↔ ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵))
22	20, 21	sylibr 237	. . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵𝑅𝐶 → ¬ 𝐶𝑅𝐵))
23	22	imp 410	. . . . 5 ⊢ (((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ 𝐵𝑅𝐶) → ¬ 𝐶𝑅𝐵)
24		sonr 5491	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ 𝐶 ∈ 𝐴) → ¬ 𝐶𝑅𝐶)
25	24	3adant2 1133	. . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ¬ 𝐶𝑅𝐶)
26	25	adantr 484	. . . . 5 ⊢ (((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ ¬ 𝐵𝑅𝐶) → ¬ 𝐶𝑅𝐶)
27	16, 18, 23, 26	ifbothda 4477	. . . 4 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ¬ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))
28		breq1 5056	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
29	28	notbid 321	. . . . . 6 ⊢ (𝑦 = 𝐵 → (¬ 𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ ¬ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
30		breq1 5056	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
31	30	notbid 321	. . . . . 6 ⊢ (𝑦 = 𝐶 → (¬ 𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ ¬ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
32	29, 31	ralprg 4610	. . . . 5 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (∀𝑦 ∈ {𝐵, 𝐶} ¬ 𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ (¬ 𝐵𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ∧ ¬ 𝐶𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))))
33	32	3adant1 1132	. . . 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 713	. . 3 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ∀𝑦 ∈ {𝐵, 𝐶} ¬ 𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))
35	34	r19.21bi 3130	. 2 ⊢ (((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ 𝑦 ∈ {𝐵, 𝐶}) → ¬ 𝑦𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))
36	1, 3, 5, 35	infmin 9110	1 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → inf({𝐵, 𝐶}, 𝐴, 𝑅) = if(𝐵𝑅𝐶, 𝐵, 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  ∀wral 3061  ifcif 4439  {cpr 4543   class class class wbr 5053   Or wor 5467  infcinf 9057

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-po 5468  df-so 5469  df-cnv 5559  df-iota 6338  df-riota 7170  df-sup 9058  df-inf 9059

This theorem is referenced by:  infsupprpr  9120  infsn  9121  liminf10ex  42990  prproropf1olem2  44629  prproropf1olem3  44630  prproropf1olem4  44631