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Theorem infltoreq 9503
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 9444 . . . 4 inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, 𝑅)
97, 8eqtrdi 2787 . . 3 (𝜑𝑆 = sup(𝐵, 𝐴, 𝑅))
103, 4, 5, 6, 9supgtoreq 9471 . 2 (𝜑 → (𝐶𝑅𝑆𝐶 = 𝑆))
116ne0d 4335 . . . . . 6 (𝜑𝐵 ≠ ∅)
12 fiinfcl 9502 . . . . . 6 ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵𝐴)) → inf(𝐵, 𝐴, 𝑅) ∈ 𝐵)
131, 5, 11, 4, 12syl13anc 1371 . . . . 5 (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐵)
147, 13eqeltrd 2832 . . . 4 (𝜑𝑆𝐵)
15 brcnvg 5879 . . . . 5 ((𝐶𝐵𝑆𝐵) → (𝐶𝑅𝑆𝑆𝑅𝐶))
1615bicomd 222 . . . 4 ((𝐶𝐵𝑆𝐵) → (𝑆𝑅𝐶𝐶𝑅𝑆))
176, 14, 16syl2anc 583 . . 3 (𝜑 → (𝑆𝑅𝐶𝐶𝑅𝑆))
1817orbi1d 914 . 2 (𝜑 → ((𝑆𝑅𝐶𝐶 = 𝑆) ↔ (𝐶𝑅𝑆𝐶 = 𝑆)))
1910, 18mpbird 257 1 (𝜑 → (𝑆𝑅𝐶𝐶 = 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wo 844   = wceq 1540  wcel 2105  wne 2939  wss 3948  c0 4322   class class class wbr 5148   Or wor 5587  ccnv 5675  Fincfn 8945  supcsup 9441  infcinf 9442
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  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 7368  df-om 7860  df-en 8946  df-fin 8949  df-sup 9443  df-inf 9444
This theorem is referenced by: (None)
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