Theorem tosglb 31155

Description: Same theorem as toslub 31153, for infinimum. (Contributed by Thierry Arnoux, 17-Feb-2018.) (Revised by AV, 28-Sep-2020.)

Hypotheses

Ref	Expression
tosglb.b	⊢ 𝐵 = (Base‘𝐾)
tosglb.l	⊢ < = (lt‘𝐾)
tosglb.1	⊢ (𝜑 → 𝐾 ∈ Toset)
tosglb.2	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Assertion

Ref	Expression
tosglb	⊢ (𝜑 → ((glb‘𝐾)‘𝐴) = inf(𝐴, 𝐵, < ))

Proof of Theorem tosglb

Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tosglb.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
2		tosglb.l	. . . . 5 ⊢ < = (lt‘𝐾)
3		tosglb.1	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Toset)
4		tosglb.2	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
5		eqid 2738	. . . . 5 ⊢ (le‘𝐾) = (le‘𝐾)
6	1, 2, 3, 4, 5	tosglblem 31154	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((∀𝑏 ∈ 𝐴 𝑎(le‘𝐾)𝑏 ∧ ∀𝑐 ∈ 𝐵 (∀𝑏 ∈ 𝐴 𝑐(le‘𝐾)𝑏 → 𝑐(le‘𝐾)𝑎)) ↔ (∀𝑏 ∈ 𝐴 ¬ 𝑎◡ < 𝑏 ∧ ∀𝑏 ∈ 𝐵 (𝑏◡ < 𝑎 → ∃𝑑 ∈ 𝐴 𝑏◡ < 𝑑))))
7	6	riotabidva 7232	. . 3 ⊢ (𝜑 → (℩𝑎 ∈ 𝐵 (∀𝑏 ∈ 𝐴 𝑎(le‘𝐾)𝑏 ∧ ∀𝑐 ∈ 𝐵 (∀𝑏 ∈ 𝐴 𝑐(le‘𝐾)𝑏 → 𝑐(le‘𝐾)𝑎))) = (℩𝑎 ∈ 𝐵 (∀𝑏 ∈ 𝐴 ¬ 𝑎◡ < 𝑏 ∧ ∀𝑏 ∈ 𝐵 (𝑏◡ < 𝑎 → ∃𝑑 ∈ 𝐴 𝑏◡ < 𝑑))))
8		eqid 2738	. . . 4 ⊢ (glb‘𝐾) = (glb‘𝐾)
9		biid 260	. . . 4 ⊢ ((∀𝑏 ∈ 𝐴 𝑎(le‘𝐾)𝑏 ∧ ∀𝑐 ∈ 𝐵 (∀𝑏 ∈ 𝐴 𝑐(le‘𝐾)𝑏 → 𝑐(le‘𝐾)𝑎)) ↔ (∀𝑏 ∈ 𝐴 𝑎(le‘𝐾)𝑏 ∧ ∀𝑐 ∈ 𝐵 (∀𝑏 ∈ 𝐴 𝑐(le‘𝐾)𝑏 → 𝑐(le‘𝐾)𝑎)))
10	1, 5, 8, 9, 3, 4	glbval 18002	. . 3 ⊢ (𝜑 → ((glb‘𝐾)‘𝐴) = (℩𝑎 ∈ 𝐵 (∀𝑏 ∈ 𝐴 𝑎(le‘𝐾)𝑏 ∧ ∀𝑐 ∈ 𝐵 (∀𝑏 ∈ 𝐴 𝑐(le‘𝐾)𝑏 → 𝑐(le‘𝐾)𝑎))))
11	1, 5, 2	tosso 18052	. . . . . . 7 ⊢ (𝐾 ∈ Toset → (𝐾 ∈ Toset ↔ ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝐾))))
12	11	ibi 266	. . . . . 6 ⊢ (𝐾 ∈ Toset → ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝐾)))
13	12	simpld 494	. . . . 5 ⊢ (𝐾 ∈ Toset → < Or 𝐵)
14		cnvso 6180	. . . . 5 ⊢ ( < Or 𝐵 ↔ ◡ < Or 𝐵)
15	13, 14	sylib 217	. . . 4 ⊢ (𝐾 ∈ Toset → ◡ < Or 𝐵)
16		id 22	. . . . 5 ⊢ (◡ < Or 𝐵 → ◡ < Or 𝐵)
17	16	supval2 9144	. . . 4 ⊢ (◡ < Or 𝐵 → sup(𝐴, 𝐵, ◡ < ) = (℩𝑎 ∈ 𝐵 (∀𝑏 ∈ 𝐴 ¬ 𝑎◡ < 𝑏 ∧ ∀𝑏 ∈ 𝐵 (𝑏◡ < 𝑎 → ∃𝑑 ∈ 𝐴 𝑏◡ < 𝑑))))
18	3, 15, 17	3syl 18	. . 3 ⊢ (𝜑 → sup(𝐴, 𝐵, ◡ < ) = (℩𝑎 ∈ 𝐵 (∀𝑏 ∈ 𝐴 ¬ 𝑎◡ < 𝑏 ∧ ∀𝑏 ∈ 𝐵 (𝑏◡ < 𝑎 → ∃𝑑 ∈ 𝐴 𝑏◡ < 𝑑))))
19	7, 10, 18	3eqtr4d 2788	. 2 ⊢ (𝜑 → ((glb‘𝐾)‘𝐴) = sup(𝐴, 𝐵, ◡ < ))
20		df-inf 9132	. . . 4 ⊢ inf(𝐴, 𝐵, < ) = sup(𝐴, 𝐵, ◡ < )
21	20	eqcomi 2747	. . 3 ⊢ sup(𝐴, 𝐵, ◡ < ) = inf(𝐴, 𝐵, < )
22	21	a1i 11	. 2 ⊢ (𝜑 → sup(𝐴, 𝐵, ◡ < ) = inf(𝐴, 𝐵, < ))
23	19, 22	eqtrd 2778	1 ⊢ (𝜑 → ((glb‘𝐾)‘𝐴) = inf(𝐴, 𝐵, < ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064   ⊆ wss 3883   class class class wbr 5070   I cid 5479   Or wor 5493  ^◡ccnv 5579   ↾ cres 5582  ‘cfv 6418  ℩crio 7211  supcsup 9129  infcinf 9130  Basecbs 16840  lecple 16895  ltcplt 17941  glbcglb 17943  Tosetctos 18049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-po 5494  df-so 5495  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-sup 9131  df-inf 9132  df-proset 17928  df-poset 17946  df-plt 17963  df-glb 17980  df-toset 18050

This theorem is referenced by:  xrsp0  31192