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Theorem mgcmnt2 32983
Description: The upper adjoint 𝐺 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)
mgcmnt2.1 (𝜑𝑋𝐵)
mgcmnt2.2 (𝜑𝑌𝐵)
mgcmnt2.3 (𝜑𝑋 𝑌)
Assertion
Ref Expression
mgcmnt2 (𝜑 → (𝐺𝑋) (𝐺𝑌))

Proof of Theorem mgcmnt2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mgcval.3 . . 3 (𝜑𝑊 ∈ 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.2 . . . . . . 7 (𝜑𝑉 ∈ Proset )
93, 4, 5, 6, 7, 8, 1mgcval 32977 . . . . . 6 (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)))))
102, 9mpbid 232 . . . . 5 (𝜑 → ((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦))))
1110simplld 768 . . . 4 (𝜑𝐹:𝐴𝐵)
1210simplrd 770 . . . . 5 (𝜑𝐺:𝐵𝐴)
13 mgcmnt2.1 . . . . 5 (𝜑𝑋𝐵)
1412, 13ffvelcdmd 7105 . . . 4 (𝜑 → (𝐺𝑋) ∈ 𝐴)
1511, 14ffvelcdmd 7105 . . 3 (𝜑 → (𝐹‘(𝐺𝑋)) ∈ 𝐵)
16 mgcmnt2.2 . . 3 (𝜑𝑌𝐵)
173, 4, 5, 6, 7, 8, 1, 2, 13mgccole2 32981 . . 3 (𝜑 → (𝐹‘(𝐺𝑋)) 𝑋)
18 mgcmnt2.3 . . 3 (𝜑𝑋 𝑌)
194, 6prstr 18345 . . 3 ((𝑊 ∈ Proset ∧ ((𝐹‘(𝐺𝑋)) ∈ 𝐵𝑋𝐵𝑌𝐵) ∧ ((𝐹‘(𝐺𝑋)) 𝑋𝑋 𝑌)) → (𝐹‘(𝐺𝑋)) 𝑌)
201, 15, 13, 16, 17, 18, 19syl132anc 1390 . 2 (𝜑 → (𝐹‘(𝐺𝑋)) 𝑌)
21 breq2 5147 . . . 4 (𝑦 = 𝑌 → ((𝐹‘(𝐺𝑋)) 𝑦 ↔ (𝐹‘(𝐺𝑋)) 𝑌))
22 fveq2 6906 . . . . 5 (𝑦 = 𝑌 → (𝐺𝑦) = (𝐺𝑌))
2322breq2d 5155 . . . 4 (𝑦 = 𝑌 → ((𝐺𝑋) (𝐺𝑦) ↔ (𝐺𝑋) (𝐺𝑌)))
2421, 23bibi12d 345 . . 3 (𝑦 = 𝑌 → (((𝐹‘(𝐺𝑋)) 𝑦 ↔ (𝐺𝑋) (𝐺𝑦)) ↔ ((𝐹‘(𝐺𝑋)) 𝑌 ↔ (𝐺𝑋) (𝐺𝑌))))
25 fveq2 6906 . . . . . . 7 (𝑥 = (𝐺𝑋) → (𝐹𝑥) = (𝐹‘(𝐺𝑋)))
2625breq1d 5153 . . . . . 6 (𝑥 = (𝐺𝑋) → ((𝐹𝑥) 𝑦 ↔ (𝐹‘(𝐺𝑋)) 𝑦))
27 breq1 5146 . . . . . 6 (𝑥 = (𝐺𝑋) → (𝑥 (𝐺𝑦) ↔ (𝐺𝑋) (𝐺𝑦)))
2826, 27bibi12d 345 . . . . 5 (𝑥 = (𝐺𝑋) → (((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)) ↔ ((𝐹‘(𝐺𝑋)) 𝑦 ↔ (𝐺𝑋) (𝐺𝑦))))
2928ralbidv 3178 . . . 4 (𝑥 = (𝐺𝑋) → (∀𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)) ↔ ∀𝑦𝐵 ((𝐹‘(𝐺𝑋)) 𝑦 ↔ (𝐺𝑋) (𝐺𝑦))))
3010simprd 495 . . . 4 (𝜑 → ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)))
3129, 30, 14rspcdva 3623 . . 3 (𝜑 → ∀𝑦𝐵 ((𝐹‘(𝐺𝑋)) 𝑦 ↔ (𝐺𝑋) (𝐺𝑦)))
3224, 31, 16rspcdva 3623 . 2 (𝜑 → ((𝐹‘(𝐺𝑋)) 𝑌 ↔ (𝐺𝑋) (𝐺𝑌)))
3320, 32mpbid 232 1 (𝜑 → (𝐺𝑋) (𝐺𝑌))
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  mgcf1olem2  32992
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