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Theorem infltoreq 9118
Description: The infimum of a finite set is less than or equal to all the elements of the set. (Contributed by AV, 4-Sep-2020.)
Hypotheses
Ref Expression
infltoreq.1 (𝜑𝑅 Or 𝐴)
infltoreq.2 (𝜑𝐵𝐴)
infltoreq.3 (𝜑𝐵 ∈ Fin)
infltoreq.4 (𝜑𝐶𝐵)
infltoreq.5 (𝜑𝑆 = inf(𝐵, 𝐴, 𝑅))
Assertion
Ref Expression
infltoreq (𝜑 → (𝑆𝑅𝐶𝐶 = 𝑆))

Proof of Theorem infltoreq
StepHypRef Expression
1 infltoreq.1 . . . 4 (𝜑𝑅 Or 𝐴)
2 cnvso 6151 . . . 4 (𝑅 Or 𝐴𝑅 Or 𝐴)
31, 2sylib 221 . . 3 (𝜑𝑅 Or 𝐴)
4 infltoreq.2 . . 3 (𝜑𝐵𝐴)
5 infltoreq.3 . . 3 (𝜑𝐵 ∈ Fin)
6 infltoreq.4 . . 3 (𝜑𝐶𝐵)
7 infltoreq.5 . . . 4 (𝜑𝑆 = inf(𝐵, 𝐴, 𝑅))
8 df-inf 9059 . . . 4 inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, 𝑅)
97, 8eqtrdi 2794 . . 3 (𝜑𝑆 = sup(𝐵, 𝐴, 𝑅))
103, 4, 5, 6, 9supgtoreq 9086 . 2 (𝜑 → (𝐶𝑅𝑆𝐶 = 𝑆))
116ne0d 4250 . . . . . 6 (𝜑𝐵 ≠ ∅)
12 fiinfcl 9117 . . . . . 6 ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵𝐴)) → inf(𝐵, 𝐴, 𝑅) ∈ 𝐵)
131, 5, 11, 4, 12syl13anc 1374 . . . . 5 (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐵)
147, 13eqeltrd 2838 . . . 4 (𝜑𝑆𝐵)
15 brcnvg 5748 . . . . 5 ((𝐶𝐵𝑆𝐵) → (𝐶𝑅𝑆𝑆𝑅𝐶))
1615bicomd 226 . . . 4 ((𝐶𝐵𝑆𝐵) → (𝑆𝑅𝐶𝐶𝑅𝑆))
176, 14, 16syl2anc 587 . . 3 (𝜑 → (𝑆𝑅𝐶𝐶𝑅𝑆))
1817orbi1d 917 . 2 (𝜑 → ((𝑆𝑅𝐶𝐶 = 𝑆) ↔ (𝐶𝑅𝑆𝐶 = 𝑆)))
1910, 18mpbird 260 1 (𝜑 → (𝑆𝑅𝐶𝐶 = 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 847   = wceq 1543  wcel 2110  wne 2940  wss 3866  c0 4237   class class class wbr 5053   Or wor 5467  ccnv 5550  Fincfn 8626  supcsup 9056  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  ax-un 7523
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-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-om 7645  df-en 8627  df-fin 8630  df-sup 9058  df-inf 9059
This theorem is referenced by: (None)
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