Users' Mathboxes 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 47429
Description: Lemma 3 for prproropf1o 47431. (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 9513 . . . 4 (𝑝 = {(1st𝑊), (2nd𝑊)} → inf(𝑝, 𝑉, 𝑅) = inf({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅))
3 supeq1 9482 . . . 4 (𝑝 = {(1st𝑊), (2nd𝑊)} → sup(𝑝, 𝑉, 𝑅) = sup({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅))
42, 3opeq12d 4885 . . 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 47426 . . . 4 (𝑊𝑂 ↔ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊)))
7 simpl 482 . . . . . . . 8 ((𝑅 Or 𝑉 ∧ (((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → 𝑅 Or 𝑉)
8 simprll 779 . . . . . . . 8 ((𝑅 Or 𝑉 ∧ (((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → (1st𝑊) ∈ 𝑉)
9 simprlr 780 . . . . . . . 8 ((𝑅 Or 𝑉 ∧ (((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → (2nd𝑊) ∈ 𝑉)
10 infpr 9540 . . . . . . . 8 ((𝑅 Or 𝑉 ∧ (1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) → inf({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅) = if((1st𝑊)𝑅(2nd𝑊), (1st𝑊), (2nd𝑊)))
117, 8, 9, 10syl3anc 1370 . . . . . . 7 ((𝑅 Or 𝑉 ∧ (((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → inf({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅) = if((1st𝑊)𝑅(2nd𝑊), (1st𝑊), (2nd𝑊)))
12 iftrue 4536 . . . . . . . 8 ((1st𝑊)𝑅(2nd𝑊) → if((1st𝑊)𝑅(2nd𝑊), (1st𝑊), (2nd𝑊)) = (1st𝑊))
1312ad2antll 729 . . . . . . 7 ((𝑅 Or 𝑉 ∧ (((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → if((1st𝑊)𝑅(2nd𝑊), (1st𝑊), (2nd𝑊)) = (1st𝑊))
1411, 13eqtrd 2774 . . . . . 6 ((𝑅 Or 𝑉 ∧ (((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → inf({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅) = (1st𝑊))
15 suppr 9508 . . . . . . . 8 ((𝑅 Or 𝑉 ∧ (1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) → sup({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅) = if((2nd𝑊)𝑅(1st𝑊), (1st𝑊), (2nd𝑊)))
167, 8, 9, 15syl3anc 1370 . . . . . . 7 ((𝑅 Or 𝑉 ∧ (((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → sup({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅) = if((2nd𝑊)𝑅(1st𝑊), (1st𝑊), (2nd𝑊)))
17 soasym 5628 . . . . . . . . 9 ((𝑅 Or 𝑉 ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉)) → ((1st𝑊)𝑅(2nd𝑊) → ¬ (2nd𝑊)𝑅(1st𝑊)))
1817impr 454 . . . . . . . 8 ((𝑅 Or 𝑉 ∧ (((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → ¬ (2nd𝑊)𝑅(1st𝑊))
1918iffalsed 4541 . . . . . . 7 ((𝑅 Or 𝑉 ∧ (((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → if((2nd𝑊)𝑅(1st𝑊), (1st𝑊), (2nd𝑊)) = (2nd𝑊))
2016, 19eqtrd 2774 . . . . . 6 ((𝑅 Or 𝑉 ∧ (((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → sup({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅) = (2nd𝑊))
2114, 20opeq12d 4885 . . . . 5 ((𝑅 Or 𝑉 ∧ (((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → ⟨inf({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅), sup({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅)⟩ = ⟨(1st𝑊), (2nd𝑊)⟩)
22213adantr1 1168 . . . 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 594 . . 3 ((𝑅 Or 𝑉𝑊𝑂) → ⟨inf({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅), sup({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅)⟩ = ⟨(1st𝑊), (2nd𝑊)⟩)
244, 23sylan9eqr 2796 . 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 47427 . 2 ((𝑅 Or 𝑉𝑊𝑂) → {(1st𝑊), (2nd𝑊)} ∈ 𝑃)
27 opex 5474 . . 3 ⟨(1st𝑊), (2nd𝑊)⟩ ∈ V
2827a1i 11 . 2 ((𝑅 Or 𝑉𝑊𝑂) → ⟨(1st𝑊), (2nd𝑊)⟩ ∈ V)
291, 24, 26, 28fvmptd2 7023 1 ((𝑅 Or 𝑉𝑊𝑂) → (𝐹‘{(1st𝑊), (2nd𝑊)}) = ⟨(1st𝑊), (2nd𝑊)⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1086   = wceq 1536  wcel 2105  {crab 3432  Vcvv 3477  cin 3961  ifcif 4530  𝒫 cpw 4604  {cpr 4632  cop 4636   class class class wbr 5147  cmpt 5230   Or wor 5595   × cxp 5686  cfv 6562  1st c1st 8010  2nd c2nd 8011  supcsup 9477  infcinf 9478  2c2 12318  chash 14365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-oadd 8508  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-inf 9480  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-n0 12524  df-z 12611  df-uz 12876  df-fz 13544  df-hash 14366
This theorem is referenced by:  prproropf1o  47431
  Copyright terms: Public domain W3C validator