MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  infltoreq Structured version   Visualization version   GIF version

Theorem infltoreq 8573
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 5835 . . . 4 (𝑅 Or 𝐴𝑅 Or 𝐴)
31, 2sylib 208 . . 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 8514 . . . 4 inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, 𝑅)
97, 8syl6eq 2810 . . 3 (𝜑𝑆 = sup(𝐵, 𝐴, 𝑅))
103, 4, 5, 6, 9supgtoreq 8541 . 2 (𝜑 → (𝐶𝑅𝑆𝐶 = 𝑆))
11 ne0i 4064 . . . . . . 7 (𝐶𝐵𝐵 ≠ ∅)
126, 11syl 17 . . . . . 6 (𝜑𝐵 ≠ ∅)
13 fiinfcl 8572 . . . . . 6 ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵𝐴)) → inf(𝐵, 𝐴, 𝑅) ∈ 𝐵)
141, 5, 12, 4, 13syl13anc 1479 . . . . 5 (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐵)
157, 14eqeltrd 2839 . . . 4 (𝜑𝑆𝐵)
16 brcnvg 5458 . . . . 5 ((𝐶𝐵𝑆𝐵) → (𝐶𝑅𝑆𝑆𝑅𝐶))
1716bicomd 213 . . . 4 ((𝐶𝐵𝑆𝐵) → (𝑆𝑅𝐶𝐶𝑅𝑆))
186, 15, 17syl2anc 696 . . 3 (𝜑 → (𝑆𝑅𝐶𝐶𝑅𝑆))
1918orbi1d 741 . 2 (𝜑 → ((𝑆𝑅𝐶𝐶 = 𝑆) ↔ (𝐶𝑅𝑆𝐶 = 𝑆)))
2010, 19mpbird 247 1 (𝜑 → (𝑆𝑅𝐶𝐶 = 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383   = wceq 1632  wcel 2139  wne 2932  wss 3715  c0 4058   class class class wbr 4804   Or wor 5186  ccnv 5265  Fincfn 8121  supcsup 8511  infcinf 8512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-om 7231  df-1o 7729  df-er 7911  df-en 8122  df-fin 8125  df-sup 8513  df-inf 8514
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator