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Theorem mgcmnt1 32982
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 32977 . . . . . 6 (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)))))
124, 11mpbid 232 . . . . 5 (𝜑 → ((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦))))
1312simplrd 770 . . . 4 (𝜑𝐺:𝐵𝐴)
1412simplld 768 . . . . 5 (𝜑𝐹:𝐴𝐵)
1514, 3ffvelcdmd 7105 . . . 4 (𝜑 → (𝐹𝑌) ∈ 𝐵)
1613, 15ffvelcdmd 7105 . . 3 (𝜑 → (𝐺‘(𝐹𝑌)) ∈ 𝐴)
17 mgcmnt1.3 . . 3 (𝜑𝑋 𝑌)
185, 6, 7, 8, 9, 1, 10, 4, 3mgccole1 32980 . . 3 (𝜑𝑌 (𝐺‘(𝐹𝑌)))
195, 7prstr 18345 . . 3 ((𝑉 ∈ Proset ∧ (𝑋𝐴𝑌𝐴 ∧ (𝐺‘(𝐹𝑌)) ∈ 𝐴) ∧ (𝑋 𝑌𝑌 (𝐺‘(𝐹𝑌)))) → 𝑋 (𝐺‘(𝐹𝑌)))
201, 2, 3, 16, 17, 18, 19syl132anc 1390 . 2 (𝜑𝑋 (𝐺‘(𝐹𝑌)))
2112simprd 495 . . . 4 (𝜑 → ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)))
22 fveq2 6906 . . . . . . . . 9 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
2322breq1d 5153 . . . . . . . 8 (𝑥 = 𝑋 → ((𝐹𝑥) 𝑦 ↔ (𝐹𝑋) 𝑦))
24 breq1 5146 . . . . . . . 8 (𝑥 = 𝑋 → (𝑥 (𝐺𝑦) ↔ 𝑋 (𝐺𝑦)))
2523, 24bibi12d 345 . . . . . . 7 (𝑥 = 𝑋 → (((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)) ↔ ((𝐹𝑋) 𝑦𝑋 (𝐺𝑦))))
2625adantl 481 . . . . . 6 ((𝜑𝑥 = 𝑋) → (((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)) ↔ ((𝐹𝑋) 𝑦𝑋 (𝐺𝑦))))
2726ralbidv 3178 . . . . 5 ((𝜑𝑥 = 𝑋) → (∀𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)) ↔ ∀𝑦𝐵 ((𝐹𝑋) 𝑦𝑋 (𝐺𝑦))))
282, 27rspcdv 3614 . . . 4 (𝜑 → (∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)) → ∀𝑦𝐵 ((𝐹𝑋) 𝑦𝑋 (𝐺𝑦))))
2921, 28mpd 15 . . 3 (𝜑 → ∀𝑦𝐵 ((𝐹𝑋) 𝑦𝑋 (𝐺𝑦)))
30 simpr 484 . . . . . 6 ((𝜑𝑦 = (𝐹𝑌)) → 𝑦 = (𝐹𝑌))
3130breq2d 5155 . . . . 5 ((𝜑𝑦 = (𝐹𝑌)) → ((𝐹𝑋) 𝑦 ↔ (𝐹𝑋) (𝐹𝑌)))
3230fveq2d 6910 . . . . . 6 ((𝜑𝑦 = (𝐹𝑌)) → (𝐺𝑦) = (𝐺‘(𝐹𝑌)))
3332breq2d 5155 . . . . 5 ((𝜑𝑦 = (𝐹𝑌)) → (𝑋 (𝐺𝑦) ↔ 𝑋 (𝐺‘(𝐹𝑌))))
3431, 33bibi12d 345 . . . 4 ((𝜑𝑦 = (𝐹𝑌)) → (((𝐹𝑋) 𝑦𝑋 (𝐺𝑦)) ↔ ((𝐹𝑋) (𝐹𝑌) ↔ 𝑋 (𝐺‘(𝐹𝑌)))))
3515, 34rspcdv 3614 . . 3 (𝜑 → (∀𝑦𝐵 ((𝐹𝑋) 𝑦𝑋 (𝐺𝑦)) → ((𝐹𝑋) (𝐹𝑌) ↔ 𝑋 (𝐺‘(𝐹𝑌)))))
3629, 35mpd 15 . 2 (𝜑 → ((𝐹𝑋) (𝐹𝑌) ↔ 𝑋 (𝐺‘(𝐹𝑌))))
3720, 36mpbird 257 1 (𝜑 → (𝐹𝑋) (𝐹𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061   class class class wbr 5143  wf 6557  cfv 6561  (class class class)co 7431  Basecbs 17247  lecple 17304   Proset cproset 18338  MGalConncmgc 32969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-proset 18340  df-mgc 32971
This theorem is referenced by:  dfmgc2  32986  mgcf1olem1  32991
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