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Theorem prproropf1olem3 47492
Description: Lemma 3 for prproropf1o 47494. (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 9516 . . . 4 (𝑝 = {(1st𝑊), (2nd𝑊)} → inf(𝑝, 𝑉, 𝑅) = inf({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅))
3 supeq1 9485 . . . 4 (𝑝 = {(1st𝑊), (2nd𝑊)} → sup(𝑝, 𝑉, 𝑅) = sup({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅))
42, 3opeq12d 4881 . . 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 47489 . . . 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 9543 . . . . . . . 8 ((𝑅 Or 𝑉 ∧ (1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) → inf({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅) = if((1st𝑊)𝑅(2nd𝑊), (1st𝑊), (2nd𝑊)))
117, 8, 9, 10syl3anc 1373 . . . . . . 7 ((𝑅 Or 𝑉 ∧ (((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → inf({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅) = if((1st𝑊)𝑅(2nd𝑊), (1st𝑊), (2nd𝑊)))
12 iftrue 4531 . . . . . . . 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 2777 . . . . . 6 ((𝑅 Or 𝑉 ∧ (((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → inf({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅) = (1st𝑊))
15 suppr 9511 . . . . . . . 8 ((𝑅 Or 𝑉 ∧ (1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) → sup({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅) = if((2nd𝑊)𝑅(1st𝑊), (1st𝑊), (2nd𝑊)))
167, 8, 9, 15syl3anc 1373 . . . . . . 7 ((𝑅 Or 𝑉 ∧ (((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → sup({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅) = if((2nd𝑊)𝑅(1st𝑊), (1st𝑊), (2nd𝑊)))
17 soasym 5625 . . . . . . . . 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 4536 . . . . . . 7 ((𝑅 Or 𝑉 ∧ (((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → if((2nd𝑊)𝑅(1st𝑊), (1st𝑊), (2nd𝑊)) = (2nd𝑊))
2016, 19eqtrd 2777 . . . . . 6 ((𝑅 Or 𝑉 ∧ (((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → sup({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅) = (2nd𝑊))
2114, 20opeq12d 4881 . . . . 5 ((𝑅 Or 𝑉 ∧ (((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → ⟨inf({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅), sup({(1st𝑊), (2nd𝑊)}, 𝑉, 𝑅)⟩ = ⟨(1st𝑊), (2nd𝑊)⟩)
22213adantr1 1170 . . . 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 2799 . 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 47490 . 2 ((𝑅 Or 𝑉𝑊𝑂) → {(1st𝑊), (2nd𝑊)} ∈ 𝑃)
27 opex 5469 . . 3 ⟨(1st𝑊), (2nd𝑊)⟩ ∈ V
2827a1i 11 . 2 ((𝑅 Or 𝑉𝑊𝑂) → ⟨(1st𝑊), (2nd𝑊)⟩ ∈ V)
291, 24, 26, 28fvmptd2 7024 1 ((𝑅 Or 𝑉𝑊𝑂) → (𝐹‘{(1st𝑊), (2nd𝑊)}) = ⟨(1st𝑊), (2nd𝑊)⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  {crab 3436  Vcvv 3480  cin 3950  ifcif 4525  𝒫 cpw 4600  {cpr 4628  cop 4632   class class class wbr 5143  cmpt 5225   Or wor 5591   × cxp 5683  cfv 6561  1st c1st 8012  2nd c2nd 8013  supcsup 9480  infcinf 9481  2c2 12321  chash 14369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-hash 14370
This theorem is referenced by:  prproropf1o  47494
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