Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  prproropf1olem3 Structured version   Visualization version   GIF version

Theorem prproropf1olem3 44190
 Description: Lemma 3 for prproropf1o 44192. (Contributed by AV, 13-Mar-2023.)
Hypotheses
Ref Expression
prproropf1o.o 𝑂 = (𝑅 ∩ (𝑉 × 𝑉))
prproropf1o.p 𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}
prproropf1o.f 𝐹 = (𝑝𝑃 ↦ ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩)
Assertion
Ref Expression
prproropf1olem3 ((𝑅 Or 𝑉𝑊𝑂) → (𝐹‘{(1st𝑊), (2nd𝑊)}) = ⟨(1st𝑊), (2nd𝑊)⟩)
Distinct variable groups:   𝑉,𝑝   𝑊,𝑝   𝑂,𝑝   𝑃,𝑝   𝑅,𝑝
Allowed substitution hint:   𝐹(𝑝)

Proof of Theorem prproropf1olem3
StepHypRef Expression
1 prproropf1o.f . 2 𝐹 = (𝑝𝑃 ↦ ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩)
2 infeq1 8942 . . . 4 (𝑝 = {(1st𝑊), (2nd𝑊)} → inf(𝑝, 𝑉, 𝑅) = inf({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅))
3 supeq1 8911 . . . 4 (𝑝 = {(1st𝑊), (2nd𝑊)} → sup(𝑝, 𝑉, 𝑅) = sup({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅))
42, 3opeq12d 4777 . . 3 (𝑝 = {(1st𝑊), (2nd𝑊)} → ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩ = ⟨inf({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅), sup({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅)⟩)
5 prproropf1o.o . . . . 5 𝑂 = (𝑅 ∩ (𝑉 × 𝑉))
65prproropf1olem0 44187 . . . 4 (𝑊𝑂 ↔ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊)))
7 simpl 486 . . . . . . . 8 ((𝑅 Or 𝑉 ∧ (((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → 𝑅 Or 𝑉)
8 simprll 778 . . . . . . . 8 ((𝑅 Or 𝑉 ∧ (((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → (1st𝑊) ∈ 𝑉)
9 simprlr 779 . . . . . . . 8 ((𝑅 Or 𝑉 ∧ (((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → (2nd𝑊) ∈ 𝑉)
10 infpr 8969 . . . . . . . 8 ((𝑅 Or 𝑉 ∧ (1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) → inf({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅) = if((1st𝑊)𝑅(2nd𝑊), (1st𝑊), (2nd𝑊)))
117, 8, 9, 10syl3anc 1368 . . . . . . 7 ((𝑅 Or 𝑉 ∧ (((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → inf({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅) = if((1st𝑊)𝑅(2nd𝑊), (1st𝑊), (2nd𝑊)))
12 iftrue 4434 . . . . . . . 8 ((1st𝑊)𝑅(2nd𝑊) → if((1st𝑊)𝑅(2nd𝑊), (1st𝑊), (2nd𝑊)) = (1st𝑊))
1312ad2antll 728 . . . . . . 7 ((𝑅 Or 𝑉 ∧ (((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → if((1st𝑊)𝑅(2nd𝑊), (1st𝑊), (2nd𝑊)) = (1st𝑊))
1411, 13eqtrd 2833 . . . . . 6 ((𝑅 Or 𝑉 ∧ (((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → inf({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅) = (1st𝑊))
15 suppr 8937 . . . . . . . 8 ((𝑅 Or 𝑉 ∧ (1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) → sup({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅) = if((2nd𝑊)𝑅(1st𝑊), (1st𝑊), (2nd𝑊)))
167, 8, 9, 15syl3anc 1368 . . . . . . 7 ((𝑅 Or 𝑉 ∧ (((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → sup({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅) = if((2nd𝑊)𝑅(1st𝑊), (1st𝑊), (2nd𝑊)))
17 soasym 5472 . . . . . . . . 9 ((𝑅 Or 𝑉 ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉)) → ((1st𝑊)𝑅(2nd𝑊) → ¬ (2nd𝑊)𝑅(1st𝑊)))
1817impr 458 . . . . . . . 8 ((𝑅 Or 𝑉 ∧ (((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → ¬ (2nd𝑊)𝑅(1st𝑊))
1918iffalsed 4439 . . . . . . 7 ((𝑅 Or 𝑉 ∧ (((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → if((2nd𝑊)𝑅(1st𝑊), (1st𝑊), (2nd𝑊)) = (2nd𝑊))
2016, 19eqtrd 2833 . . . . . 6 ((𝑅 Or 𝑉 ∧ (((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → sup({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅) = (2nd𝑊))
2114, 20opeq12d 4777 . . . . 5 ((𝑅 Or 𝑉 ∧ (((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → ⟨inf({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅), sup({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅)⟩ = ⟨(1st𝑊), (2nd𝑊)⟩)
22213adantr1 1166 . . . 4 ((𝑅 Or 𝑉 ∧ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → ⟨inf({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅), sup({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅)⟩ = ⟨(1st𝑊), (2nd𝑊)⟩)
236, 22sylan2b 596 . . 3 ((𝑅 Or 𝑉𝑊𝑂) → ⟨inf({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅), sup({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅)⟩ = ⟨(1st𝑊), (2nd𝑊)⟩)
244, 23sylan9eqr 2855 . 2 (((𝑅 Or 𝑉𝑊𝑂) ∧ 𝑝 = {(1st𝑊), (2nd𝑊)}) → ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩ = ⟨(1st𝑊), (2nd𝑊)⟩)
25 prproropf1o.p . . 3 𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}
265, 25prproropf1olem1 44188 . 2 ((𝑅 Or 𝑉𝑊𝑂) → {(1st𝑊), (2nd𝑊)} ∈ 𝑃)
27 opex 5325 . . 3 ⟨(1st𝑊), (2nd𝑊)⟩ ∈ V
2827a1i 11 . 2 ((𝑅 Or 𝑉𝑊𝑂) → ⟨(1st𝑊), (2nd𝑊)⟩ ∈ V)
291, 24, 26, 28fvmptd2 6763 1 ((𝑅 Or 𝑉𝑊𝑂) → (𝐹‘{(1st𝑊), (2nd𝑊)}) = ⟨(1st𝑊), (2nd𝑊)⟩)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  {crab 3110  Vcvv 3442   ∩ cin 3882  ifcif 4428  𝒫 cpw 4500  {cpr 4530  ⟨cop 4534   class class class wbr 5034   ↦ cmpt 5114   Or wor 5441   × cxp 5521  ‘cfv 6332  1st c1st 7682  2nd c2nd 7683  supcsup 8906  infcinf 8907  2c2 11698  ♯chash 13706 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-cnex 10600  ax-resscn 10601  ax-1cn 10602  ax-icn 10603  ax-addcl 10604  ax-addrcl 10605  ax-mulcl 10606  ax-mulrcl 10607  ax-mulcom 10608  ax-addass 10609  ax-mulass 10610  ax-distr 10611  ax-i2m1 10612  ax-1ne0 10613  ax-1rid 10614  ax-rnegex 10615  ax-rrecex 10616  ax-cnre 10617  ax-pre-lttri 10618  ax-pre-lttrn 10619  ax-pre-ltadd 10620  ax-pre-mulgt0 10621 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-int 4843  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7574  df-1st 7684  df-2nd 7685  df-wrecs 7948  df-recs 8009  df-rdg 8047  df-1o 8103  df-oadd 8107  df-er 8290  df-en 8511  df-dom 8512  df-sdom 8513  df-fin 8514  df-sup 8908  df-inf 8909  df-dju 9332  df-card 9370  df-pnf 10684  df-mnf 10685  df-xr 10686  df-ltxr 10687  df-le 10688  df-sub 10879  df-neg 10880  df-nn 11644  df-2 11706  df-n0 11904  df-z 11990  df-uz 12252  df-fz 12906  df-hash 13707 This theorem is referenced by:  prproropf1o  44192
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