prproropf1olem3 - Mathbox for Alexander van der Vekens

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Theorem prproropf1olem3 46768

Description: Lemma 3 for prproropf1o 46770. (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 9491	. . . 4 ⊢ (𝑝 = {(1^st ‘𝑊), (2^nd ‘𝑊)} → inf(𝑝, 𝑉, 𝑅) = inf({(1^st ‘𝑊), (2^nd ‘𝑊)}, 𝑉, 𝑅))
3		supeq1 9460	. . . 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 4877	. . 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 46765	. . . 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 9518	. . . . . . . 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 1369	. . . . . . 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 4530	. . . . . . . 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 728	. . . . . . 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 2767	. . . . . 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 9486	. . . . . . . 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 1369	. . . . . . 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 5615	. . . . . . . . 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 4535	. . . . . . 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 2767	. . . . . 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 4877	. . . . 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 1167	. . . 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 593	. . 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 2789	. 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 46766	. 2 ⊢ ((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑂) → {(1^st ‘𝑊), (2^nd ‘𝑊)} ∈ 𝑃)
27		opex 5460	. . 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 7007	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 1085 = wceq 1534 ∈ wcel 2099 {crab 3427 Vcvv 3469 ∩ cin 3943 ifcif 4524 𝒫 cpw 4598 {cpr 4626 ⟨cop 4630 class class class wbr 5142 ↦ cmpt 5225 Or wor 5583 × cxp 5670 ‘cfv 6542 1^st c1st 7985 2^nd c2nd 7986 supcsup 9455 infcinf 9456 2c2 12289 ♯chash 14313

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-oadd 8484 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-sup 9457 df-inf 9458 df-dju 9916 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-n0 12495 df-z 12581 df-uz 12845 df-fz 13509 df-hash 14314

This theorem is referenced by: prproropf1o 46770

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