prproropf1olem3 - Mathbox for Alexander van der Vekens

	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 46159

Description: Lemma 3 for prproropf1o 46161. (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 9467	. . . 4 ⊢ (𝑝 = {(1^st ‘𝑊), (2^nd ‘𝑊)} → inf(𝑝, 𝑉, 𝑅) = inf({(1^st ‘𝑊), (2^nd ‘𝑊)}, 𝑉, 𝑅))
3		supeq1 9436	. . . 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 4880	. . 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 46156	. . . 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 483	. . . . . . . 8 ⊢ ((𝑅 Or 𝑉 ∧ (((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉) ∧ (1^st ‘𝑊)𝑅(2^nd ‘𝑊))) → 𝑅 Or 𝑉)
8		simprll 777	. . . . . . . 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 778	. . . . . . . 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 9494	. . . . . . . 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 1371	. . . . . . 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 4533	. . . . . . . 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 727	. . . . . . 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 2772	. . . . . 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 9462	. . . . . . . 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 1371	. . . . . . 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 5618	. . . . . . . . 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 455	. . . . . . . 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 4538	. . . . . . 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 2772	. . . . . 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 4880	. . . . 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 1169	. . . 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 2794	. 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 46157	. 2 ⊢ ((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑂) → {(1^st ‘𝑊), (2^nd ‘𝑊)} ∈ 𝑃)
27		opex 5463	. . 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 7003	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 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 {crab 3432 Vcvv 3474 ∩ cin 3946 ifcif 4527 𝒫 cpw 4601 {cpr 4629 ⟨cop 4633 class class class wbr 5147 ↦ cmpt 5230 Or wor 5586 × cxp 5673 ‘cfv 6540 1^st c1st 7969 2^nd c2nd 7970 supcsup 9431 infcinf 9432 2c2 12263 ♯chash 14286

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-oadd 8466 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-dju 9892 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-hash 14287

This theorem is referenced by: prproropf1o 46161

W3C validator