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Theorem mgcmnt1 30796
 Description: The lower adjoint 𝐹 of a Galois connection is monotonically increasing. (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)
mgcmnt1.1 (𝜑𝑋𝐴)
mgcmnt1.2 (𝜑𝑌𝐴)
mgcmnt1.3 (𝜑𝑋 𝑌)
Assertion
Ref Expression
mgcmnt1 (𝜑 → (𝐹𝑋) (𝐹𝑌))

Proof of Theorem mgcmnt1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mgcval.2 . . 3 (𝜑𝑉 ∈ Proset )
2 mgcmnt1.1 . . 3 (𝜑𝑋𝐴)
3 mgcmnt1.2 . . 3 (𝜑𝑌𝐴)
4 mgccole.1 . . . . . 6 (𝜑𝐹𝐻𝐺)
5 mgcoval.1 . . . . . . 7 𝐴 = (Base‘𝑉)
6 mgcoval.2 . . . . . . 7 𝐵 = (Base‘𝑊)
7 mgcoval.3 . . . . . . 7 = (le‘𝑉)
8 mgcoval.4 . . . . . . 7 = (le‘𝑊)
9 mgcval.1 . . . . . . 7 𝐻 = (𝑉MGalConn𝑊)
10 mgcval.3 . . . . . . 7 (𝜑𝑊 ∈ Proset )
115, 6, 7, 8, 9, 1, 10mgcval 30791 . . . . . 6 (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)))))
124, 11mpbid 235 . . . . 5 (𝜑 → ((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦))))
1312simplrd 769 . . . 4 (𝜑𝐺:𝐵𝐴)
1412simplld 767 . . . . 5 (𝜑𝐹:𝐴𝐵)
1514, 3ffvelrnd 6843 . . . 4 (𝜑 → (𝐹𝑌) ∈ 𝐵)
1613, 15ffvelrnd 6843 . . 3 (𝜑 → (𝐺‘(𝐹𝑌)) ∈ 𝐴)
17 mgcmnt1.3 . . 3 (𝜑𝑋 𝑌)
185, 6, 7, 8, 9, 1, 10, 4, 3mgccole1 30794 . . 3 (𝜑𝑌 (𝐺‘(𝐹𝑌)))
195, 7prstr 17609 . . 3 ((𝑉 ∈ Proset ∧ (𝑋𝐴𝑌𝐴 ∧ (𝐺‘(𝐹𝑌)) ∈ 𝐴) ∧ (𝑋 𝑌𝑌 (𝐺‘(𝐹𝑌)))) → 𝑋 (𝐺‘(𝐹𝑌)))
201, 2, 3, 16, 17, 18, 19syl132anc 1385 . 2 (𝜑𝑋 (𝐺‘(𝐹𝑌)))
2112simprd 499 . . . 4 (𝜑 → ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)))
22 fveq2 6658 . . . . . . . . 9 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
2322breq1d 5042 . . . . . . . 8 (𝑥 = 𝑋 → ((𝐹𝑥) 𝑦 ↔ (𝐹𝑋) 𝑦))
24 breq1 5035 . . . . . . . 8 (𝑥 = 𝑋 → (𝑥 (𝐺𝑦) ↔ 𝑋 (𝐺𝑦)))
2523, 24bibi12d 349 . . . . . . 7 (𝑥 = 𝑋 → (((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)) ↔ ((𝐹𝑋) 𝑦𝑋 (𝐺𝑦))))
2625adantl 485 . . . . . 6 ((𝜑𝑥 = 𝑋) → (((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)) ↔ ((𝐹𝑋) 𝑦𝑋 (𝐺𝑦))))
2726ralbidv 3126 . . . . 5 ((𝜑𝑥 = 𝑋) → (∀𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)) ↔ ∀𝑦𝐵 ((𝐹𝑋) 𝑦𝑋 (𝐺𝑦))))
282, 27rspcdv 3533 . . . 4 (𝜑 → (∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)) → ∀𝑦𝐵 ((𝐹𝑋) 𝑦𝑋 (𝐺𝑦))))
2921, 28mpd 15 . . 3 (𝜑 → ∀𝑦𝐵 ((𝐹𝑋) 𝑦𝑋 (𝐺𝑦)))
30 simpr 488 . . . . . 6 ((𝜑𝑦 = (𝐹𝑌)) → 𝑦 = (𝐹𝑌))
3130breq2d 5044 . . . . 5 ((𝜑𝑦 = (𝐹𝑌)) → ((𝐹𝑋) 𝑦 ↔ (𝐹𝑋) (𝐹𝑌)))
3230fveq2d 6662 . . . . . 6 ((𝜑𝑦 = (𝐹𝑌)) → (𝐺𝑦) = (𝐺‘(𝐹𝑌)))
3332breq2d 5044 . . . . 5 ((𝜑𝑦 = (𝐹𝑌)) → (𝑋 (𝐺𝑦) ↔ 𝑋 (𝐺‘(𝐹𝑌))))
3431, 33bibi12d 349 . . . 4 ((𝜑𝑦 = (𝐹𝑌)) → (((𝐹𝑋) 𝑦𝑋 (𝐺𝑦)) ↔ ((𝐹𝑋) (𝐹𝑌) ↔ 𝑋 (𝐺‘(𝐹𝑌)))))
3515, 34rspcdv 3533 . . 3 (𝜑 → (∀𝑦𝐵 ((𝐹𝑋) 𝑦𝑋 (𝐺𝑦)) → ((𝐹𝑋) (𝐹𝑌) ↔ 𝑋 (𝐺‘(𝐹𝑌)))))
3629, 35mpd 15 . 2 (𝜑 → ((𝐹𝑋) (𝐹𝑌) ↔ 𝑋 (𝐺‘(𝐹𝑌))))
3720, 36mpbird 260 1 (𝜑 → (𝐹𝑋) (𝐹𝑌))
 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:  dfmgc2  30800  mgcf1olem1  30805
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