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 44909

Description: Lemma 3 for prproropf1o 44911. (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 9196	. . . 4 ⊢ (𝑝 = {(1^st ‘𝑊), (2^nd ‘𝑊)} → inf(𝑝, 𝑉, 𝑅) = inf({(1^st ‘𝑊), (2^nd ‘𝑊)}, 𝑉, 𝑅))
3		supeq1 9165	. . . 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 4817	. . 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 44906	. . . 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 775	. . . . . . . 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 776	. . . . . . . 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 9223	. . . . . . . 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 4470	. . . . . . . 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 725	. . . . . . 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 2779	. . . . . 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 9191	. . . . . . . 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 5533	. . . . . . . . 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 4475	. . . . . . 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 2779	. . . . . 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 4817	. . . . 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 2801	. 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 44907	. 2 ⊢ ((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑂) → {(1^st ‘𝑊), (2^nd ‘𝑊)} ∈ 𝑃)
27		opex 5381	. . 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 6877	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 1541 ∈ wcel 2109 {crab 3069 Vcvv 3430 ∩ cin 3890 ifcif 4464 𝒫 cpw 4538 {cpr 4568 ⟨cop 4572 class class class wbr 5078 ↦ cmpt 5161 Or wor 5501 × cxp 5586 ‘cfv 6430 1^st c1st 7815 2^nd c2nd 7816 supcsup 9160 infcinf 9161 2c2 12011 ♯chash 14025

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-1st 7817 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-oadd 8285 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-sup 9162 df-inf 9163 df-dju 9643 df-card 9681 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-nn 11957 df-2 12019 df-n0 12217 df-z 12303 df-uz 12565 df-fz 13222 df-hash 14026

This theorem is referenced by: prproropf1o 44911

W3C validator