Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mgccole2 Structured version   Visualization version   GIF version

Theorem mgccole2 32595
Description: Inequality for the closure operator (𝐹𝐺) of the Galois connection 𝐻. (Contributed by Thierry Arnoux, 26-Apr-2024.)
Hypotheses
Ref Expression
mgcoval.1 𝐴 = (Base‘𝑉)
mgcoval.2 𝐵 = (Base‘𝑊)
mgcoval.3 = (le‘𝑉)
mgcoval.4 = (le‘𝑊)
mgcval.1 𝐻 = (𝑉MGalConn𝑊)
mgcval.2 (𝜑𝑉 ∈ Proset )
mgcval.3 (𝜑𝑊 ∈ Proset )
mgccole.1 (𝜑𝐹𝐻𝐺)
mgccole2.1 (𝜑𝑌𝐵)
Assertion
Ref Expression
mgccole2 (𝜑 → (𝐹‘(𝐺𝑌)) 𝑌)

Proof of Theorem mgccole2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mgcval.2 . . 3 (𝜑𝑉 ∈ Proset )
2 mgccole.1 . . . . . 6 (𝜑𝐹𝐻𝐺)
3 mgcoval.1 . . . . . . 7 𝐴 = (Base‘𝑉)
4 mgcoval.2 . . . . . . 7 𝐵 = (Base‘𝑊)
5 mgcoval.3 . . . . . . 7 = (le‘𝑉)
6 mgcoval.4 . . . . . . 7 = (le‘𝑊)
7 mgcval.1 . . . . . . 7 𝐻 = (𝑉MGalConn𝑊)
8 mgcval.3 . . . . . . 7 (𝜑𝑊 ∈ Proset )
93, 4, 5, 6, 7, 1, 8mgcval 32591 . . . . . 6 (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)))))
102, 9mpbid 231 . . . . 5 (𝜑 → ((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦))))
1110simplrd 767 . . . 4 (𝜑𝐺:𝐵𝐴)
12 mgccole2.1 . . . 4 (𝜑𝑌𝐵)
1311, 12ffvelcdmd 7087 . . 3 (𝜑 → (𝐺𝑌) ∈ 𝐴)
143, 5prsref 18262 . . 3 ((𝑉 ∈ Proset ∧ (𝐺𝑌) ∈ 𝐴) → (𝐺𝑌) (𝐺𝑌))
151, 13, 14syl2anc 583 . 2 (𝜑 → (𝐺𝑌) (𝐺𝑌))
1610simprd 495 . . . 4 (𝜑 → ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)))
17 fveq2 6891 . . . . . . . . 9 (𝑥 = (𝐺𝑌) → (𝐹𝑥) = (𝐹‘(𝐺𝑌)))
1817breq1d 5158 . . . . . . . 8 (𝑥 = (𝐺𝑌) → ((𝐹𝑥) 𝑦 ↔ (𝐹‘(𝐺𝑌)) 𝑦))
19 breq1 5151 . . . . . . . 8 (𝑥 = (𝐺𝑌) → (𝑥 (𝐺𝑦) ↔ (𝐺𝑌) (𝐺𝑦)))
2018, 19bibi12d 345 . . . . . . 7 (𝑥 = (𝐺𝑌) → (((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)) ↔ ((𝐹‘(𝐺𝑌)) 𝑦 ↔ (𝐺𝑌) (𝐺𝑦))))
2120adantl 481 . . . . . 6 ((𝜑𝑥 = (𝐺𝑌)) → (((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)) ↔ ((𝐹‘(𝐺𝑌)) 𝑦 ↔ (𝐺𝑌) (𝐺𝑦))))
2221ralbidv 3176 . . . . 5 ((𝜑𝑥 = (𝐺𝑌)) → (∀𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)) ↔ ∀𝑦𝐵 ((𝐹‘(𝐺𝑌)) 𝑦 ↔ (𝐺𝑌) (𝐺𝑦))))
2313, 22rspcdv 3604 . . . 4 (𝜑 → (∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)) → ∀𝑦𝐵 ((𝐹‘(𝐺𝑌)) 𝑦 ↔ (𝐺𝑌) (𝐺𝑦))))
2416, 23mpd 15 . . 3 (𝜑 → ∀𝑦𝐵 ((𝐹‘(𝐺𝑌)) 𝑦 ↔ (𝐺𝑌) (𝐺𝑦)))
25 simpr 484 . . . . . 6 ((𝜑𝑦 = 𝑌) → 𝑦 = 𝑌)
2625breq2d 5160 . . . . 5 ((𝜑𝑦 = 𝑌) → ((𝐹‘(𝐺𝑌)) 𝑦 ↔ (𝐹‘(𝐺𝑌)) 𝑌))
27 fveq2 6891 . . . . . . 7 (𝑦 = 𝑌 → (𝐺𝑦) = (𝐺𝑌))
2827adantl 481 . . . . . 6 ((𝜑𝑦 = 𝑌) → (𝐺𝑦) = (𝐺𝑌))
2928breq2d 5160 . . . . 5 ((𝜑𝑦 = 𝑌) → ((𝐺𝑌) (𝐺𝑦) ↔ (𝐺𝑌) (𝐺𝑌)))
3026, 29bibi12d 345 . . . 4 ((𝜑𝑦 = 𝑌) → (((𝐹‘(𝐺𝑌)) 𝑦 ↔ (𝐺𝑌) (𝐺𝑦)) ↔ ((𝐹‘(𝐺𝑌)) 𝑌 ↔ (𝐺𝑌) (𝐺𝑌))))
3112, 30rspcdv 3604 . . 3 (𝜑 → (∀𝑦𝐵 ((𝐹‘(𝐺𝑌)) 𝑦 ↔ (𝐺𝑌) (𝐺𝑦)) → ((𝐹‘(𝐺𝑌)) 𝑌 ↔ (𝐺𝑌) (𝐺𝑌))))
3224, 31mpd 15 . 2 (𝜑 → ((𝐹‘(𝐺𝑌)) 𝑌 ↔ (𝐺𝑌) (𝐺𝑌)))
3315, 32mpbird 257 1 (𝜑 → (𝐹‘(𝐺𝑌)) 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1540  wcel 2105  wral 3060   class class class wbr 5148  wf 6539  cfv 6543  (class class class)co 7412  Basecbs 17151  lecple 17211   Proset cproset 18256  MGalConncmgc 32583
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8828  df-proset 18258  df-mgc 32585
This theorem is referenced by:  mgcmnt2  32597  mgcmntco  32598  dfmgc2  32600  mgcf1olem1  32605  mgcf1olem2  32606
  Copyright terms: Public domain W3C validator