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Theorem infltoreq 9525
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 6287 . . . 4 (𝑅 Or 𝐴𝑅 Or 𝐴)
31, 2sylib 217 . . 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 9466 . . . 4 inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, 𝑅)
97, 8eqtrdi 2781 . . 3 (𝜑𝑆 = sup(𝐵, 𝐴, 𝑅))
103, 4, 5, 6, 9supgtoreq 9493 . 2 (𝜑 → (𝐶𝑅𝑆𝐶 = 𝑆))
116ne0d 4331 . . . . . 6 (𝜑𝐵 ≠ ∅)
12 fiinfcl 9524 . . . . . 6 ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵𝐴)) → inf(𝐵, 𝐴, 𝑅) ∈ 𝐵)
131, 5, 11, 4, 12syl13anc 1369 . . . . 5 (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐵)
147, 13eqeltrd 2825 . . . 4 (𝜑𝑆𝐵)
15 brcnvg 5876 . . . . 5 ((𝐶𝐵𝑆𝐵) → (𝐶𝑅𝑆𝑆𝑅𝐶))
1615bicomd 222 . . . 4 ((𝐶𝐵𝑆𝐵) → (𝑆𝑅𝐶𝐶𝑅𝑆))
176, 14, 16syl2anc 582 . . 3 (𝜑 → (𝑆𝑅𝐶𝐶𝑅𝑆))
1817orbi1d 914 . 2 (𝜑 → ((𝑆𝑅𝐶𝐶 = 𝑆) ↔ (𝐶𝑅𝑆𝐶 = 𝑆)))
1910, 18mpbird 256 1 (𝜑 → (𝑆𝑅𝐶𝐶 = 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wo 845   = wceq 1533  wcel 2098  wne 2930  wss 3939  c0 4318   class class class wbr 5143   Or wor 5583  ccnv 5671  Fincfn 8962  supcsup 9463  infcinf 9464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7372  df-om 7869  df-en 8963  df-fin 8966  df-sup 9465  df-inf 9466
This theorem is referenced by: (None)
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