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Theorem odulub 18360

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

**Hypotheses**
Ref	Expression
oduglb.d	⊢ 𝐷 = (ODual‘𝑂)
odulub.l	⊢ 𝐿 = (glb‘𝑂)

**Assertion**
Ref	Expression
odulub	⊢ (𝑂 ∈ 𝑉 → 𝐿 = (lub‘𝐷))

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

Step	Hyp	Ref	Expression
1		odulub.l	. 2 ⊢ 𝐿 = (glb‘𝑂)
2		vex 3479	. . . . . . . . . . 11 ⊢ 𝑐 ∈ V
3		vex 3479	. . . . . . . . . . 11 ⊢ 𝑏 ∈ V
4	2, 3	brcnv 5883	. . . . . . . . . 10 ⊢ (𝑐^◡(le‘𝑂)𝑏 ↔ 𝑏(le‘𝑂)𝑐)
5	4	ralbii 3094	. . . . . . . . 9 ⊢ (∀𝑐 ∈ 𝑎 𝑐^◡(le‘𝑂)𝑏 ↔ ∀𝑐 ∈ 𝑎 𝑏(le‘𝑂)𝑐)
6		vex 3479	. . . . . . . . . . . . 13 ⊢ 𝑑 ∈ V
7	2, 6	brcnv 5883	. . . . . . . . . . . 12 ⊢ (𝑐^◡(le‘𝑂)𝑑 ↔ 𝑑(le‘𝑂)𝑐)
8	7	ralbii 3094	. . . . . . . . . . 11 ⊢ (∀𝑐 ∈ 𝑎 𝑐^◡(le‘𝑂)𝑑 ↔ ∀𝑐 ∈ 𝑎 𝑑(le‘𝑂)𝑐)
9	3, 6	brcnv 5883	. . . . . . . . . . 11 ⊢ (𝑏^◡(le‘𝑂)𝑑 ↔ 𝑑(le‘𝑂)𝑏)
10	8, 9	imbi12i 351	. . . . . . . . . 10 ⊢ ((∀𝑐 ∈ 𝑎 𝑐^◡(le‘𝑂)𝑑 → 𝑏^◡(le‘𝑂)𝑑) ↔ (∀𝑐 ∈ 𝑎 𝑑(le‘𝑂)𝑐 → 𝑑(le‘𝑂)𝑏))
11	10	ralbii 3094	. . . . . . . . 9 ⊢ (∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑐^◡(le‘𝑂)𝑑 → 𝑏^◡(le‘𝑂)𝑑) ↔ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑑(le‘𝑂)𝑐 → 𝑑(le‘𝑂)𝑏))
12	5, 11	anbi12i 628	. . . . . . . 8 ⊢ ((∀𝑐 ∈ 𝑎 𝑐^◡(le‘𝑂)𝑏 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑐^◡(le‘𝑂)𝑑 → 𝑏^◡(le‘𝑂)𝑑)) ↔ (∀𝑐 ∈ 𝑎 𝑏(le‘𝑂)𝑐 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑑(le‘𝑂)𝑐 → 𝑑(le‘𝑂)𝑏)))
13	12	a1i 11	. . . . . . 7 ⊢ (𝑏 ∈ (Base‘𝑂) → ((∀𝑐 ∈ 𝑎 𝑐^◡(le‘𝑂)𝑏 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑐^◡(le‘𝑂)𝑑 → 𝑏^◡(le‘𝑂)𝑑)) ↔ (∀𝑐 ∈ 𝑎 𝑏(le‘𝑂)𝑐 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑑(le‘𝑂)𝑐 → 𝑑(le‘𝑂)𝑏))))
14	13	riotabiia 7386	. . . . . 6 ⊢ (℩𝑏 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑐^◡(le‘𝑂)𝑏 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑐^◡(le‘𝑂)𝑑 → 𝑏^◡(le‘𝑂)𝑑))) = (℩𝑏 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑏(le‘𝑂)𝑐 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑑(le‘𝑂)𝑐 → 𝑑(le‘𝑂)𝑏)))
15	14	mpteq2i 5254	. . . . 5 ⊢ (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 ∈ 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))))
16	12	reubii 3386	. . . . . 6 ⊢ (∃!𝑏 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑐^◡(le‘𝑂)𝑏 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑐^◡(le‘𝑂)𝑑 → 𝑏^◡(le‘𝑂)𝑑)) ↔ ∃!𝑏 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑏(le‘𝑂)𝑐 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑑(le‘𝑂)𝑐 → 𝑑(le‘𝑂)𝑏)))
17	16	abbii 2803	. . . . 5 ⊢ {𝑎 ∣ ∃!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 𝑏(le‘𝑂)𝑐 ∧ ∀𝑑 ∈ (Base‘𝑂)(∀𝑐 ∈ 𝑎 𝑑(le‘𝑂)𝑐 → 𝑑(le‘𝑂)𝑏))}
18	15, 17	reseq12i 5980	. . . 4 ⊢ ((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))})
19	18	eqcomi 2742	. . 3 ⊢ ((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))})
20		eqid 2733	. . . 4 ⊢ (Base‘𝑂) = (Base‘𝑂)
21		eqid 2733	. . . 4 ⊢ (le‘𝑂) = (le‘𝑂)
22		eqid 2733	. . . 4 ⊢ (glb‘𝑂) = (glb‘𝑂)
23		biid 261	. . . 4 ⊢ ((∀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)))
24		id 22	. . . 4 ⊢ (𝑂 ∈ 𝑉 → 𝑂 ∈ 𝑉)
25	20, 21, 22, 23, 24	glbfval 18316	. . 3 ⊢ (𝑂 ∈ 𝑉 → (glb‘𝑂) = ((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))}))
26		oduglb.d	. . . . 5 ⊢ 𝐷 = (ODual‘𝑂)
27	26	fvexi 6906	. . . 4 ⊢ 𝐷 ∈ V
28	26, 20	odubas 18244	. . . . 5 ⊢ (Base‘𝑂) = (Base‘𝐷)
29	26, 21	oduleval 18242	. . . . 5 ⊢ ^◡(le‘𝑂) = (le‘𝐷)
30		eqid 2733	. . . . 5 ⊢ (lub‘𝐷) = (lub‘𝐷)
31		biid 261	. . . . 5 ⊢ ((∀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)))
32		id 22	. . . . 5 ⊢ (𝐷 ∈ V → 𝐷 ∈ V)
33	28, 29, 30, 31, 32	lubfval 18303	. . . 4 ⊢ (𝐷 ∈ V → (lub‘𝐷) = ((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))}))
34	27, 33	mp1i 13	. . 3 ⊢ (𝑂 ∈ 𝑉 → (lub‘𝐷) = ((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))}))
35	19, 25, 34	3eqtr4a 2799	. 2 ⊢ (𝑂 ∈ 𝑉 → (glb‘𝑂) = (lub‘𝐷))
36	1, 35	eqtrid 2785	1 ⊢ (𝑂 ∈ 𝑉 → 𝐿 = (lub‘𝐷))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {cab 2710 ∀wral 3062 ∃!wreu 3375 Vcvv 3475 𝒫 cpw 4603 class class class wbr 5149 ↦ cmpt 5232 ^◡ccnv 5676 ↾ cres 5679 ‘cfv 6544 ℩crio 7364 Basecbs 17144 lecple 17204 ODualcodu 18239 lubclub 18262 glbcglb 18263

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-dec 12678 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ple 17217 df-odu 18240 df-lub 18299 df-glb 18300

This theorem is referenced by: odujoin 18361 posglbdg 18368 oduclatb 18460 glbprlem 47646

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