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Theorem posglbd 17754

Description: Properties which determine the greatest lower bound in a poset. (Contributed by Stefan O'Rear, 31-Jan-2015.)

**Hypotheses**
Ref	Expression
posglbd.l	⊢ ≤ = (le‘𝐾)
posglbd.b	⊢ (𝜑 → 𝐵 = (Base‘𝐾))
posglbd.g	⊢ (𝜑 → 𝐺 = (glb‘𝐾))
posglbd.k	⊢ (𝜑 → 𝐾 ∈ Poset)
posglbd.s	⊢ (𝜑 → 𝑆 ⊆ 𝐵)
posglbd.t	⊢ (𝜑 → 𝑇 ∈ 𝐵)
posglbd.lb	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑇 ≤ 𝑥)
posglbd.gt	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 𝑦 ≤ 𝑥) → 𝑦 ≤ 𝑇)

**Assertion**
Ref	Expression
posglbd	⊢ (𝜑 → (𝐺‘𝑆) = 𝑇)

Distinct variable groups: 𝑥, ≤ ,𝑦 𝑥,𝐵,𝑦 𝑥,𝐾,𝑦 𝑥,𝑆,𝑦 𝑥,𝐺,𝑦 𝑥,𝑇,𝑦 𝜑,𝑥,𝑦

**Proof of Theorem posglbd**
Step	Hyp	Ref	Expression
1		eqid 2821	. . 3 ⊢ (ODual‘𝐾) = (ODual‘𝐾)
2		posglbd.l	. . 3 ⊢ ≤ = (le‘𝐾)
3	1, 2	oduleval 17735	. 2 ⊢ ^◡ ≤ = (le‘(ODual‘𝐾))
4		posglbd.b	. . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
5		eqid 2821	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
6	1, 5	odubas 17737	. . 3 ⊢ (Base‘𝐾) = (Base‘(ODual‘𝐾))
7	4, 6	syl6eq 2872	. 2 ⊢ (𝜑 → 𝐵 = (Base‘(ODual‘𝐾)))
8		posglbd.g	. . 3 ⊢ (𝜑 → 𝐺 = (glb‘𝐾))
9		posglbd.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Poset)
10		eqid 2821	. . . . 5 ⊢ (glb‘𝐾) = (glb‘𝐾)
11	1, 10	odulub 17745	. . . 4 ⊢ (𝐾 ∈ Poset → (glb‘𝐾) = (lub‘(ODual‘𝐾)))
12	9, 11	syl 17	. . 3 ⊢ (𝜑 → (glb‘𝐾) = (lub‘(ODual‘𝐾)))
13	8, 12	eqtrd 2856	. 2 ⊢ (𝜑 → 𝐺 = (lub‘(ODual‘𝐾)))
14	1	odupos 17739	. . 3 ⊢ (𝐾 ∈ Poset → (ODual‘𝐾) ∈ Poset)
15	9, 14	syl 17	. 2 ⊢ (𝜑 → (ODual‘𝐾) ∈ Poset)
16		posglbd.s	. 2 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
17		posglbd.t	. 2 ⊢ (𝜑 → 𝑇 ∈ 𝐵)
18		posglbd.lb	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑇 ≤ 𝑥)
19		vex 3497	. . . . 5 ⊢ 𝑥 ∈ V
20		brcnvg 5744	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑇 ∈ 𝐵) → (𝑥^◡ ≤ 𝑇 ↔ 𝑇 ≤ 𝑥))
21	19, 17, 20	sylancr 589	. . . 4 ⊢ (𝜑 → (𝑥^◡ ≤ 𝑇 ↔ 𝑇 ≤ 𝑥))
22	21	adantr 483	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥^◡ ≤ 𝑇 ↔ 𝑇 ≤ 𝑥))
23	18, 22	mpbird 259	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥^◡ ≤ 𝑇)
24		vex 3497	. . . . . 6 ⊢ 𝑦 ∈ V
25	19, 24	brcnv 5747	. . . . 5 ⊢ (𝑥^◡ ≤ 𝑦 ↔ 𝑦 ≤ 𝑥)
26	25	ralbii 3165	. . . 4 ⊢ (∀𝑥 ∈ 𝑆 𝑥^◡ ≤ 𝑦 ↔ ∀𝑥 ∈ 𝑆 𝑦 ≤ 𝑥)
27		posglbd.gt	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 𝑦 ≤ 𝑥) → 𝑦 ≤ 𝑇)
28	26, 27	syl3an3b 1401	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 𝑥^◡ ≤ 𝑦) → 𝑦 ≤ 𝑇)
29		brcnvg 5744	. . . . 5 ⊢ ((𝑇 ∈ 𝐵 ∧ 𝑦 ∈ V) → (𝑇^◡ ≤ 𝑦 ↔ 𝑦 ≤ 𝑇))
30	17, 24, 29	sylancl 588	. . . 4 ⊢ (𝜑 → (𝑇^◡ ≤ 𝑦 ↔ 𝑦 ≤ 𝑇))
31	30	3ad2ant1 1129	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 𝑥^◡ ≤ 𝑦) → (𝑇^◡ ≤ 𝑦 ↔ 𝑦 ≤ 𝑇))
32	28, 31	mpbird 259	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 𝑥^◡ ≤ 𝑦) → 𝑇^◡ ≤ 𝑦)
33	3, 7, 13, 15, 16, 17, 23, 32	poslubdg 17753	1 ⊢ (𝜑 → (𝐺‘𝑆) = 𝑇)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 Vcvv 3494 ⊆ wss 3935 class class class wbr 5058 ^◡ccnv 5548 ‘cfv 6349 Basecbs 16477 lecple 16566 Posetcpo 17544 lubclub 17546 glbcglb 17547 ODualcodu 17732

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-dec 12093 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ple 16579 df-proset 17532 df-poset 17550 df-lub 17578 df-glb 17579 df-odu 17733

This theorem is referenced by: mrelatglb 17788 mrelatglb0 17789

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