Theorem tosglb 33040

Description: Same theorem as toslub 33038, 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 2737	. . . . 5 ⊢ (le‘𝐾) = (le‘𝐾)
6	1, 2, 3, 4, 5	tosglblem 33039	. . . 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 7334	. . 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 2737	. . . 4 ⊢ (glb‘𝐾) = (glb‘𝐾)
9		biid 261	. . . 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 18291	. . 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 18341	. . . . . . 7 ⊢ (𝐾 ∈ Toset → (𝐾 ∈ Toset ↔ ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝐾))))
12	11	ibi 267	. . . . . 6 ⊢ (𝐾 ∈ Toset → ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝐾)))
13	12	simpld 494	. . . . 5 ⊢ (𝐾 ∈ Toset → < Or 𝐵)
14		cnvso 6244	. . . . 5 ⊢ ( < Or 𝐵 ↔ ◡ < Or 𝐵)
15	13, 14	sylib 218	. . . 4 ⊢ (𝐾 ∈ Toset → ◡ < Or 𝐵)
16		id 22	. . . . 5 ⊢ (◡ < Or 𝐵 → ◡ < Or 𝐵)
17	16	supval2 9359	. . . 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 2782	. 2 ⊢ (𝜑 → ((glb‘𝐾)‘𝐴) = sup(𝐴, 𝐵, ◡ < ))
20		df-inf 9347	. . . 4 ⊢ inf(𝐴, 𝐵, < ) = sup(𝐴, 𝐵, ◡ < )
21	20	eqcomi 2746	. . 3 ⊢ sup(𝐴, 𝐵, ◡ < ) = inf(𝐴, 𝐵, < )
22	21	a1i 11	. 2 ⊢ (𝜑 → sup(𝐴, 𝐵, ◡ < ) = inf(𝐴, 𝐵, < ))
23	19, 22	eqtrd 2772	1 ⊢ (𝜑 → ((glb‘𝐾)‘𝐴) = inf(𝐴, 𝐵, < ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ⊆ wss 3890   class class class wbr 5086   I cid 5516   Or wor 5529  ^◡ccnv 5621   ↾ cres 5624  ‘cfv 6490  ℩crio 7314  supcsup 9344  infcinf 9345  Basecbs 17137  lecple 17185  ltcplt 18232  glbcglb 18234  Tosetctos 18338

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-po 5530  df-so 5531  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7315  df-sup 9346  df-inf 9347  df-proset 18218  df-poset 18237  df-plt 18252  df-glb 18269  df-toset 18339

This theorem is referenced by:  xrsp0  33077