prproropf1olem3 - Mathbox for Alexander van der Vekens

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Theorem prproropf1olem3 47506

Description: Lemma 3 for prproropf1o 47508. (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 = (𝑝 ∈ 𝑃 ↦ ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩)

**Assertion**
Ref	Expression
prproropf1olem3	⊢ ((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑂) → (𝐹‘{(1^st ‘𝑊), (2^nd ‘𝑊)}) = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩)

Distinct variable groups: 𝑉,𝑝 𝑊,𝑝 𝑂,𝑝 𝑃,𝑝 𝑅,𝑝 Allowed substitution hint: 𝐹(𝑝)

**Proof of Theorem prproropf1olem3**
Step	Hyp	Ref	Expression
1		prproropf1o.f	. 2 ⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩)
2		infeq1 9428	. . . 4 ⊢ (𝑝 = {(1^st ‘𝑊), (2^nd ‘𝑊)} → inf(𝑝, 𝑉, 𝑅) = inf({(1^st ‘𝑊), (2^nd ‘𝑊)}, 𝑉, 𝑅))
3		supeq1 9396	. . . 4 ⊢ (𝑝 = {(1^st ‘𝑊), (2^nd ‘𝑊)} → sup(𝑝, 𝑉, 𝑅) = sup({(1^st ‘𝑊), (2^nd ‘𝑊)}, 𝑉, 𝑅))
4	2, 3	opeq12d 4845	. . 3 ⊢ (𝑝 = {(1^st ‘𝑊), (2^nd ‘𝑊)} → ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩ = ⟨inf({(1^st ‘𝑊), (2^nd ‘𝑊)}, 𝑉, 𝑅), sup({(1^st ‘𝑊), (2^nd ‘𝑊)}, 𝑉, 𝑅)⟩)
5		prproropf1o.o	. . . . 5 ⊢ 𝑂 = (𝑅 ∩ (𝑉 × 𝑉))
6	5	prproropf1olem0 47503	. . . 4 ⊢ (𝑊 ∈ 𝑂 ↔ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∧ ((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉) ∧ (1^st ‘𝑊)𝑅(2^nd ‘𝑊)))
7		simpl 482	. . . . . . . 8 ⊢ ((𝑅 Or 𝑉 ∧ (((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉) ∧ (1^st ‘𝑊)𝑅(2^nd ‘𝑊))) → 𝑅 Or 𝑉)
8		simprll 778	. . . . . . . 8 ⊢ ((𝑅 Or 𝑉 ∧ (((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉) ∧ (1^st ‘𝑊)𝑅(2^nd ‘𝑊))) → (1^st ‘𝑊) ∈ 𝑉)
9		simprlr 779	. . . . . . . 8 ⊢ ((𝑅 Or 𝑉 ∧ (((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉) ∧ (1^st ‘𝑊)𝑅(2^nd ‘𝑊))) → (2^nd ‘𝑊) ∈ 𝑉)
10		infpr 9456	. . . . . . . 8 ⊢ ((𝑅 Or 𝑉 ∧ (1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉) → inf({(1^st ‘𝑊), (2^nd ‘𝑊)}, 𝑉, 𝑅) = if((1^st ‘𝑊)𝑅(2^nd ‘𝑊), (1^st ‘𝑊), (2^nd ‘𝑊)))
11	7, 8, 9, 10	syl3anc 1373	. . . . . . 7 ⊢ ((𝑅 Or 𝑉 ∧ (((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉) ∧ (1^st ‘𝑊)𝑅(2^nd ‘𝑊))) → inf({(1^st ‘𝑊), (2^nd ‘𝑊)}, 𝑉, 𝑅) = if((1^st ‘𝑊)𝑅(2^nd ‘𝑊), (1^st ‘𝑊), (2^nd ‘𝑊)))
12		iftrue 4494	. . . . . . . 8 ⊢ ((1^st ‘𝑊)𝑅(2^nd ‘𝑊) → if((1^st ‘𝑊)𝑅(2^nd ‘𝑊), (1^st ‘𝑊), (2^nd ‘𝑊)) = (1^st ‘𝑊))
13	12	ad2antll 729	. . . . . . 7 ⊢ ((𝑅 Or 𝑉 ∧ (((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉) ∧ (1^st ‘𝑊)𝑅(2^nd ‘𝑊))) → if((1^st ‘𝑊)𝑅(2^nd ‘𝑊), (1^st ‘𝑊), (2^nd ‘𝑊)) = (1^st ‘𝑊))
14	11, 13	eqtrd 2764	. . . . . 6 ⊢ ((𝑅 Or 𝑉 ∧ (((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉) ∧ (1^st ‘𝑊)𝑅(2^nd ‘𝑊))) → inf({(1^st ‘𝑊), (2^nd ‘𝑊)}, 𝑉, 𝑅) = (1^st ‘𝑊))
15		suppr 9423	. . . . . . . 8 ⊢ ((𝑅 Or 𝑉 ∧ (1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉) → sup({(1^st ‘𝑊), (2^nd ‘𝑊)}, 𝑉, 𝑅) = if((2^nd ‘𝑊)𝑅(1^st ‘𝑊), (1^st ‘𝑊), (2^nd ‘𝑊)))
16	7, 8, 9, 15	syl3anc 1373	. . . . . . 7 ⊢ ((𝑅 Or 𝑉 ∧ (((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉) ∧ (1^st ‘𝑊)𝑅(2^nd ‘𝑊))) → sup({(1^st ‘𝑊), (2^nd ‘𝑊)}, 𝑉, 𝑅) = if((2^nd ‘𝑊)𝑅(1^st ‘𝑊), (1^st ‘𝑊), (2^nd ‘𝑊)))
17		soasym 5579	. . . . . . . . 9 ⊢ ((𝑅 Or 𝑉 ∧ ((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉)) → ((1^st ‘𝑊)𝑅(2^nd ‘𝑊) → ¬ (2^nd ‘𝑊)𝑅(1^st ‘𝑊)))
18	17	impr 454	. . . . . . . 8 ⊢ ((𝑅 Or 𝑉 ∧ (((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉) ∧ (1^st ‘𝑊)𝑅(2^nd ‘𝑊))) → ¬ (2^nd ‘𝑊)𝑅(1^st ‘𝑊))
19	18	iffalsed 4499	. . . . . . 7 ⊢ ((𝑅 Or 𝑉 ∧ (((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉) ∧ (1^st ‘𝑊)𝑅(2^nd ‘𝑊))) → if((2^nd ‘𝑊)𝑅(1^st ‘𝑊), (1^st ‘𝑊), (2^nd ‘𝑊)) = (2^nd ‘𝑊))
20	16, 19	eqtrd 2764	. . . . . 6 ⊢ ((𝑅 Or 𝑉 ∧ (((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉) ∧ (1^st ‘𝑊)𝑅(2^nd ‘𝑊))) → sup({(1^st ‘𝑊), (2^nd ‘𝑊)}, 𝑉, 𝑅) = (2^nd ‘𝑊))
21	14, 20	opeq12d 4845	. . . . 5 ⊢ ((𝑅 Or 𝑉 ∧ (((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉) ∧ (1^st ‘𝑊)𝑅(2^nd ‘𝑊))) → ⟨inf({(1^st ‘𝑊), (2^nd ‘𝑊)}, 𝑉, 𝑅), sup({(1^st ‘𝑊), (2^nd ‘𝑊)}, 𝑉, 𝑅)⟩ = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩)
22	21	3adantr1 1170	. . . 4 ⊢ ((𝑅 Or 𝑉 ∧ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∧ ((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉) ∧ (1^st ‘𝑊)𝑅(2^nd ‘𝑊))) → ⟨inf({(1^st ‘𝑊), (2^nd ‘𝑊)}, 𝑉, 𝑅), sup({(1^st ‘𝑊), (2^nd ‘𝑊)}, 𝑉, 𝑅)⟩ = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩)
23	6, 22	sylan2b 594	. . 3 ⊢ ((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑂) → ⟨inf({(1^st ‘𝑊), (2^nd ‘𝑊)}, 𝑉, 𝑅), sup({(1^st ‘𝑊), (2^nd ‘𝑊)}, 𝑉, 𝑅)⟩ = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩)
24	4, 23	sylan9eqr 2786	. 2 ⊢ (((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑂) ∧ 𝑝 = {(1^st ‘𝑊), (2^nd ‘𝑊)}) → ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩ = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩)
25		prproropf1o.p	. . 3 ⊢ 𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}
26	5, 25	prproropf1olem1 47504	. 2 ⊢ ((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑂) → {(1^st ‘𝑊), (2^nd ‘𝑊)} ∈ 𝑃)
27		opex 5424	. . 3 ⊢ ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∈ V
28	27	a1i 11	. 2 ⊢ ((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑂) → ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∈ V)
29	1, 24, 26, 28	fvmptd2 6976	1 ⊢ ((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑂) → (𝐹‘{(1^st ‘𝑊), (2^nd ‘𝑊)}) = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 {crab 3405 Vcvv 3447 ∩ cin 3913 ifcif 4488 𝒫 cpw 4563 {cpr 4591 ⟨cop 4595 class class class wbr 5107 ↦ cmpt 5188 Or wor 5545 × cxp 5636 ‘cfv 6511 1^st c1st 7966 2^nd c2nd 7967 supcsup 9391 infcinf 9392 2c2 12241 ♯chash 14295

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-oadd 8438 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-sup 9393 df-inf 9394 df-dju 9854 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-n0 12443 df-z 12530 df-uz 12794 df-fz 13469 df-hash 14296

This theorem is referenced by: prproropf1o 47508

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