prproropf1olem3 - Mathbox for Alexander van der Vekens

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Theorem prproropf1olem3 47510

Description: Lemma 3 for prproropf1o 47512. (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 9435	. . . 4 ⊢ (𝑝 = {(1^st ‘𝑊), (2^nd ‘𝑊)} → inf(𝑝, 𝑉, 𝑅) = inf({(1^st ‘𝑊), (2^nd ‘𝑊)}, 𝑉, 𝑅))
3		supeq1 9403	. . . 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 4848	. . 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 47507	. . . 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 9463	. . . . . . . 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 4497	. . . . . . . 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 2765	. . . . . 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 9430	. . . . . . . 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 5582	. . . . . . . . 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 4502	. . . . . . 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 2765	. . . . . 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 4848	. . . . 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 2787	. 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 47508	. 2 ⊢ ((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑂) → {(1^st ‘𝑊), (2^nd ‘𝑊)} ∈ 𝑃)
27		opex 5427	. . 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 6979	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 3408 Vcvv 3450 ∩ cin 3916 ifcif 4491 𝒫 cpw 4566 {cpr 4594 ⟨cop 4598 class class class wbr 5110 ↦ cmpt 5191 Or wor 5548 × cxp 5639 ‘cfv 6514 1^st c1st 7969 2^nd c2nd 7970 supcsup 9398 infcinf 9399 2c2 12248 ♯chash 14302

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 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-oadd 8441 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-sup 9400 df-inf 9401 df-dju 9861 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-n0 12450 df-z 12537 df-uz 12801 df-fz 13476 df-hash 14303

This theorem is referenced by: prproropf1o 47512

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