oduglb - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > oduglb		Structured version Visualization version GIF version

Theorem oduglb 18361

Description: Greatest lower bounds in a dual order are least upper bounds in the original order. (Contributed by Stefan O'Rear, 29-Jan-2015.)

**Hypotheses**
Ref	Expression
oduglb.d	⊢ 𝐷 = (ODual‘𝑂)
oduglb.l	⊢ 𝑈 = (lub‘𝑂)

**Assertion**
Ref	Expression
oduglb	⊢ (𝑂 ∈ 𝑉 → 𝑈 = (glb‘𝐷))

Proof of Theorem oduglb Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oduglb.l	. 2 ⊢ 𝑈 = (lub‘𝑂)
2		vex 3478	. . . . . . . . . . 11 ⊢ 𝑏 ∈ V
3		vex 3478	. . . . . . . . . . 11 ⊢ 𝑐 ∈ V
4	2, 3	brcnv 5882	. . . . . . . . . 10 ⊢ (𝑏^◡(le‘𝑂)𝑐 ↔ 𝑐(le‘𝑂)𝑏)
5	4	ralbii 3093	. . . . . . . . 9 ⊢ (∀𝑐 ∈ 𝑎 𝑏^◡(le‘𝑂)𝑐 ↔ ∀𝑐 ∈ 𝑎 𝑐(le‘𝑂)𝑏)
6		vex 3478	. . . . . . . . . . . . 13 ⊢ 𝑑 ∈ V
7	6, 3	brcnv 5882	. . . . . . . . . . . 12 ⊢ (𝑑^◡(le‘𝑂)𝑐 ↔ 𝑐(le‘𝑂)𝑑)
8	7	ralbii 3093	. . . . . . . . . . 11 ⊢ (∀𝑐 ∈ 𝑎 𝑑^◡(le‘𝑂)𝑐 ↔ ∀𝑐 ∈ 𝑎 𝑐(le‘𝑂)𝑑)
9	6, 2	brcnv 5882	. . . . . . . . . . 11 ⊢ (𝑑^◡(le‘𝑂)𝑏 ↔ 𝑏(le‘𝑂)𝑑)
10	8, 9	imbi12i 350	. . . . . . . . . 10 ⊢ ((∀𝑐 ∈ 𝑎 𝑑^◡(le‘𝑂)𝑐 → 𝑑^◡(le‘𝑂)𝑏) ↔ (∀𝑐 ∈ 𝑎 𝑐(le‘𝑂)𝑑 → 𝑏(le‘𝑂)𝑑))
11	10	ralbii 3093	. . . . . . . . 9 ⊢ (∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑑^◡(le‘𝑂)𝑐 → 𝑑^◡(le‘𝑂)𝑏) ↔ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑐(le‘𝑂)𝑑 → 𝑏(le‘𝑂)𝑑))
12	5, 11	anbi12i 627	. . . . . . . 8 ⊢ ((∀𝑐 ∈ 𝑎 𝑏^◡(le‘𝑂)𝑐 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑑^◡(le‘𝑂)𝑐 → 𝑑^◡(le‘𝑂)𝑏)) ↔ (∀𝑐 ∈ 𝑎 𝑐(le‘𝑂)𝑏 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑐(le‘𝑂)𝑑 → 𝑏(le‘𝑂)𝑑)))
13	12	a1i 11	. . . . . . 7 ⊢ (𝑏 ∈ (Base‘𝑂) → ((∀𝑐 ∈ 𝑎 𝑏^◡(le‘𝑂)𝑐 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑑^◡(le‘𝑂)𝑐 → 𝑑^◡(le‘𝑂)𝑏)) ↔ (∀𝑐 ∈ 𝑎 𝑐(le‘𝑂)𝑏 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑐(le‘𝑂)𝑑 → 𝑏(le‘𝑂)𝑑))))
14	13	riotabiia 7385	. . . . . 6 ⊢ (℩𝑏 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑏^◡(le‘𝑂)𝑐 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑑^◡(le‘𝑂)𝑐 → 𝑑^◡(le‘𝑂)𝑏))) = (℩𝑏 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑐(le‘𝑂)𝑏 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑐(le‘𝑂)𝑑 → 𝑏(le‘𝑂)𝑑)))
15	14	mpteq2i 5253	. . . . 5 ⊢ (𝑎 ∈ 𝒫 (Base‘𝑂) ↦ (℩𝑏 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑏^◡(le‘𝑂)𝑐 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑑^◡(le‘𝑂)𝑐 → 𝑑^◡(le‘𝑂)𝑏)))) = (𝑎 ∈ 𝒫 (Base‘𝑂) ↦ (℩𝑏 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑐(le‘𝑂)𝑏 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑐(le‘𝑂)𝑑 → 𝑏(le‘𝑂)𝑑))))
16	12	reubii 3385	. . . . . 6 ⊢ (∃!𝑏 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑏^◡(le‘𝑂)𝑐 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑑^◡(le‘𝑂)𝑐 → 𝑑^◡(le‘𝑂)𝑏)) ↔ ∃!𝑏 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑐(le‘𝑂)𝑏 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑐(le‘𝑂)𝑑 → 𝑏(le‘𝑂)𝑑)))
17	16	abbii 2802	. . . . 5 ⊢ {𝑎 ∣ ∃!𝑏 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑏^◡(le‘𝑂)𝑐 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑑^◡(le‘𝑂)𝑐 → 𝑑^◡(le‘𝑂)𝑏))} = {𝑎 ∣ ∃!𝑏 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑐(le‘𝑂)𝑏 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑐(le‘𝑂)𝑑 → 𝑏(le‘𝑂)𝑑))}
18	15, 17	reseq12i 5979	. . . 4 ⊢ ((𝑎 ∈ 𝒫 (Base‘𝑂) ↦ (℩𝑏 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑏^◡(le‘𝑂)𝑐 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑑^◡(le‘𝑂)𝑐 → 𝑑^◡(le‘𝑂)𝑏)))) ↾ {𝑎 ∣ ∃!𝑏 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑏^◡(le‘𝑂)𝑐 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑑^◡(le‘𝑂)𝑐 → 𝑑^◡(le‘𝑂)𝑏))}) = ((𝑎 ∈ 𝒫 (Base‘𝑂) ↦ (℩𝑏 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑐(le‘𝑂)𝑏 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑐(le‘𝑂)𝑑 → 𝑏(le‘𝑂)𝑑)))) ↾ {𝑎 ∣ ∃!𝑏 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑐(le‘𝑂)𝑏 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑐(le‘𝑂)𝑑 → 𝑏(le‘𝑂)𝑑))})
19	18	eqcomi 2741	. . 3 ⊢ ((𝑎 ∈ 𝒫 (Base‘𝑂) ↦ (℩𝑏 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑐(le‘𝑂)𝑏 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑐(le‘𝑂)𝑑 → 𝑏(le‘𝑂)𝑑)))) ↾ {𝑎 ∣ ∃!𝑏 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑐(le‘𝑂)𝑏 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑐(le‘𝑂)𝑑 → 𝑏(le‘𝑂)𝑑))}) = ((𝑎 ∈ 𝒫 (Base‘𝑂) ↦ (℩𝑏 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑏^◡(le‘𝑂)𝑐 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑑^◡(le‘𝑂)𝑐 → 𝑑^◡(le‘𝑂)𝑏)))) ↾ {𝑎 ∣ ∃!𝑏 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑏^◡(le‘𝑂)𝑐 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑑^◡(le‘𝑂)𝑐 → 𝑑^◡(le‘𝑂)𝑏))})
20		eqid 2732	. . . 4 ⊢ (Base‘𝑂) = (Base‘𝑂)
21		eqid 2732	. . . 4 ⊢ (le‘𝑂) = (le‘𝑂)
22		eqid 2732	. . . 4 ⊢ (lub‘𝑂) = (lub‘𝑂)
23		biid 260	. . . 4 ⊢ ((∀𝑐 ∈ 𝑎 𝑐(le‘𝑂)𝑏 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑐(le‘𝑂)𝑑 → 𝑏(le‘𝑂)𝑑)) ↔ (∀𝑐 ∈ 𝑎 𝑐(le‘𝑂)𝑏 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑐(le‘𝑂)𝑑 → 𝑏(le‘𝑂)𝑑)))
24		id 22	. . . 4 ⊢ (𝑂 ∈ 𝑉 → 𝑂 ∈ 𝑉)
25	20, 21, 22, 23, 24	lubfval 18302	. . 3 ⊢ (𝑂 ∈ 𝑉 → (lub‘𝑂) = ((𝑎 ∈ 𝒫 (Base‘𝑂) ↦ (℩𝑏 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑐(le‘𝑂)𝑏 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑐(le‘𝑂)𝑑 → 𝑏(le‘𝑂)𝑑)))) ↾ {𝑎 ∣ ∃!𝑏 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑐(le‘𝑂)𝑏 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑐(le‘𝑂)𝑑 → 𝑏(le‘𝑂)𝑑))}))
26		oduglb.d	. . . . 5 ⊢ 𝐷 = (ODual‘𝑂)
27	26	fvexi 6905	. . . 4 ⊢ 𝐷 ∈ V
28	26, 20	odubas 18243	. . . . 5 ⊢ (Base‘𝑂) = (Base‘𝐷)
29	26, 21	oduleval 18241	. . . . 5 ⊢ ^◡(le‘𝑂) = (le‘𝐷)
30		eqid 2732	. . . . 5 ⊢ (glb‘𝐷) = (glb‘𝐷)
31		biid 260	. . . . 5 ⊢ ((∀𝑐 ∈ 𝑎 𝑏^◡(le‘𝑂)𝑐 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑑^◡(le‘𝑂)𝑐 → 𝑑^◡(le‘𝑂)𝑏)) ↔ (∀𝑐 ∈ 𝑎 𝑏^◡(le‘𝑂)𝑐 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑑^◡(le‘𝑂)𝑐 → 𝑑^◡(le‘𝑂)𝑏)))
32		id 22	. . . . 5 ⊢ (𝐷 ∈ V → 𝐷 ∈ V)
33	28, 29, 30, 31, 32	glbfval 18315	. . . 4 ⊢ (𝐷 ∈ V → (glb‘𝐷) = ((𝑎 ∈ 𝒫 (Base‘𝑂) ↦ (℩𝑏 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑏^◡(le‘𝑂)𝑐 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑑^◡(le‘𝑂)𝑐 → 𝑑^◡(le‘𝑂)𝑏)))) ↾ {𝑎 ∣ ∃!𝑏 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑏^◡(le‘𝑂)𝑐 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑑^◡(le‘𝑂)𝑐 → 𝑑^◡(le‘𝑂)𝑏))}))
34	27, 33	mp1i 13	. . 3 ⊢ (𝑂 ∈ 𝑉 → (glb‘𝐷) = ((𝑎 ∈ 𝒫 (Base‘𝑂) ↦ (℩𝑏 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑏^◡(le‘𝑂)𝑐 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑑^◡(le‘𝑂)𝑐 → 𝑑^◡(le‘𝑂)𝑏)))) ↾ {𝑎 ∣ ∃!𝑏 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑏^◡(le‘𝑂)𝑐 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑑^◡(le‘𝑂)𝑐 → 𝑑^◡(le‘𝑂)𝑏))}))
35	19, 25, 34	3eqtr4a 2798	. 2 ⊢ (𝑂 ∈ 𝑉 → (lub‘𝑂) = (glb‘𝐷))
36	1, 35	eqtrid 2784	1 ⊢ (𝑂 ∈ 𝑉 → 𝑈 = (glb‘𝐷))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {cab 2709 ∀wral 3061 ∃!wreu 3374 Vcvv 3474 𝒫 cpw 4602 class class class wbr 5148 ↦ cmpt 5231 ^◡ccnv 5675 ↾ cres 5678 ‘cfv 6543 ℩crio 7363 Basecbs 17143 lecple 17203 ODualcodu 18238 lubclub 18261 glbcglb 18262

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-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186

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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-dec 12677 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ple 17216 df-odu 18239 df-lub 18298 df-glb 18299

This theorem is referenced by: odumeet 18362 oduclatb 18459

W3C validator