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

Theorem mgccole2 30795
 Description: Inequality for the closure operator (𝐹 ∘ 𝐺) 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 30791 . . . . . 6 (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)))))
102, 9mpbid 235 . . . . 5 (𝜑 → ((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦))))
1110simplrd 769 . . . 4 (𝜑𝐺:𝐵𝐴)
12 mgccole2.1 . . . 4 (𝜑𝑌𝐵)
1311, 12ffvelrnd 6843 . . 3 (𝜑 → (𝐺𝑌) ∈ 𝐴)
143, 5prsref 17608 . . 3 ((𝑉 ∈ Proset ∧ (𝐺𝑌) ∈ 𝐴) → (𝐺𝑌) (𝐺𝑌))
151, 13, 14syl2anc 587 . 2 (𝜑 → (𝐺𝑌) (𝐺𝑌))
1610simprd 499 . . . 4 (𝜑 → ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)))
17 fveq2 6658 . . . . . . . . 9 (𝑥 = (𝐺𝑌) → (𝐹𝑥) = (𝐹‘(𝐺𝑌)))
1817breq1d 5042 . . . . . . . 8 (𝑥 = (𝐺𝑌) → ((𝐹𝑥) 𝑦 ↔ (𝐹‘(𝐺𝑌)) 𝑦))
19 breq1 5035 . . . . . . . 8 (𝑥 = (𝐺𝑌) → (𝑥 (𝐺𝑦) ↔ (𝐺𝑌) (𝐺𝑦)))
2018, 19bibi12d 349 . . . . . . 7 (𝑥 = (𝐺𝑌) → (((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)) ↔ ((𝐹‘(𝐺𝑌)) 𝑦 ↔ (𝐺𝑌) (𝐺𝑦))))
2120adantl 485 . . . . . 6 ((𝜑𝑥 = (𝐺𝑌)) → (((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)) ↔ ((𝐹‘(𝐺𝑌)) 𝑦 ↔ (𝐺𝑌) (𝐺𝑦))))
2221ralbidv 3126 . . . . 5 ((𝜑𝑥 = (𝐺𝑌)) → (∀𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)) ↔ ∀𝑦𝐵 ((𝐹‘(𝐺𝑌)) 𝑦 ↔ (𝐺𝑌) (𝐺𝑦))))
2313, 22rspcdv 3533 . . . 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 488 . . . . . 6 ((𝜑𝑦 = 𝑌) → 𝑦 = 𝑌)
2625breq2d 5044 . . . . 5 ((𝜑𝑦 = 𝑌) → ((𝐹‘(𝐺𝑌)) 𝑦 ↔ (𝐹‘(𝐺𝑌)) 𝑌))
27 fveq2 6658 . . . . . . 7 (𝑦 = 𝑌 → (𝐺𝑦) = (𝐺𝑌))
2827adantl 485 . . . . . 6 ((𝜑𝑦 = 𝑌) → (𝐺𝑦) = (𝐺𝑌))
2928breq2d 5044 . . . . 5 ((𝜑𝑦 = 𝑌) → ((𝐺𝑌) (𝐺𝑦) ↔ (𝐺𝑌) (𝐺𝑌)))
3026, 29bibi12d 349 . . . 4 ((𝜑𝑦 = 𝑌) → (((𝐹‘(𝐺𝑌)) 𝑦 ↔ (𝐺𝑌) (𝐺𝑦)) ↔ ((𝐹‘(𝐺𝑌)) 𝑌 ↔ (𝐺𝑌) (𝐺𝑌))))
3112, 30rspcdv 3533 . . 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 260 1 (𝜑 → (𝐹‘(𝐺𝑌)) 𝑌)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070   class class class wbr 5032  ⟶wf 6331  ‘cfv 6335  (class class class)co 7150  Basecbs 16541  lecple 16630   Proset cproset 17602  MGalConncmgc 30783 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-map 8418  df-proset 17604  df-mgc 30785 This theorem is referenced by:  mgcmnt2  30797  mgcmntco  30798  dfmgc2  30800  mgcf1olem1  30805  mgcf1olem2  30806
 Copyright terms: Public domain W3C validator