Theorem tosglb 30659

Description: Same theorem as toslub 30657, 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 2823	. . . . 5 ⊢ (le‘𝐾) = (le‘𝐾)
6	1, 2, 3, 4, 5	tosglblem 30658	. . . 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 7135	. . 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 2823	. . . 4 ⊢ (glb‘𝐾) = (glb‘𝐾)
9		biid 263	. . . 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 17609	. . 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 17648	. . . . . . 7 ⊢ (𝐾 ∈ Toset → (𝐾 ∈ Toset ↔ ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝐾))))
12	11	ibi 269	. . . . . 6 ⊢ (𝐾 ∈ Toset → ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝐾)))
13	12	simpld 497	. . . . 5 ⊢ (𝐾 ∈ Toset → < Or 𝐵)
14		cnvso 6141	. . . . 5 ⊢ ( < Or 𝐵 ↔ ◡ < Or 𝐵)
15	13, 14	sylib 220	. . . 4 ⊢ (𝐾 ∈ Toset → ◡ < Or 𝐵)
16		id 22	. . . . 5 ⊢ (◡ < Or 𝐵 → ◡ < Or 𝐵)
17	16	supval2 8921	. . . 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 2868	. 2 ⊢ (𝜑 → ((glb‘𝐾)‘𝐴) = sup(𝐴, 𝐵, ◡ < ))
20		df-inf 8909	. . . 4 ⊢ inf(𝐴, 𝐵, < ) = sup(𝐴, 𝐵, ◡ < )
21	20	eqcomi 2832	. . 3 ⊢ sup(𝐴, 𝐵, ◡ < ) = inf(𝐴, 𝐵, < )
22	21	a1i 11	. 2 ⊢ (𝜑 → sup(𝐴, 𝐵, ◡ < ) = inf(𝐴, 𝐵, < ))
23	19, 22	eqtrd 2858	1 ⊢ (𝜑 → ((glb‘𝐾)‘𝐴) = inf(𝐴, 𝐵, < ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141   ⊆ wss 3938   class class class wbr 5068   I cid 5461   Or wor 5475  ^◡ccnv 5556   ↾ cres 5559  ‘cfv 6357  ℩crio 7115  supcsup 8906  infcinf 8907  Basecbs 16485  lecple 16574  ltcplt 17553  glbcglb 17555  Tosetctos 17645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-po 5476  df-so 5477  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-sup 8908  df-inf 8909  df-proset 17540  df-poset 17558  df-plt 17570  df-glb 17587  df-toset 17646

This theorem is referenced by:  xrsp0  30670