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Theorem mgcmnt2 32968
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 32962 . . . . . 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 32966 . . 3 (𝜑 → (𝐹‘(𝐺𝑋)) 𝑋)
18 mgcmnt2.3 . . 3 (𝜑𝑋 𝑌)
194, 6prstr 18357 . . 3 ((𝑊 ∈ Proset ∧ ((𝐹‘(𝐺𝑋)) ∈ 𝐵𝑋𝐵𝑌𝐵) ∧ ((𝐹‘(𝐺𝑋)) 𝑋𝑋 𝑌)) → (𝐹‘(𝐺𝑋)) 𝑌)
201, 15, 13, 16, 17, 18, 19syl132anc 1387 . 2 (𝜑 → (𝐹‘(𝐺𝑋)) 𝑌)
21 breq2 5152 . . . 4 (𝑦 = 𝑌 → ((𝐹‘(𝐺𝑋)) 𝑦 ↔ (𝐹‘(𝐺𝑋)) 𝑌))
22 fveq2 6907 . . . . 5 (𝑦 = 𝑌 → (𝐺𝑦) = (𝐺𝑌))
2322breq2d 5160 . . . 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 6907 . . . . . . 7 (𝑥 = (𝐺𝑋) → (𝐹𝑥) = (𝐹‘(𝐺𝑋)))
2625breq1d 5158 . . . . . 6 (𝑥 = (𝐺𝑋) → ((𝐹𝑥) 𝑦 ↔ (𝐹‘(𝐺𝑋)) 𝑦))
27 breq1 5151 . . . . . 6 (𝑥 = (𝐺𝑋) → (𝑥 (𝐺𝑦) ↔ (𝐺𝑋) (𝐺𝑦)))
2826, 27bibi12d 345 . . . . 5 (𝑥 = (𝐺𝑋) → (((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)) ↔ ((𝐹‘(𝐺𝑋)) 𝑦 ↔ (𝐺𝑋) (𝐺𝑦))))
2928ralbidv 3176 . . . 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 1537  wcel 2106  wral 3059   class class class wbr 5148  wf 6559  cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305   Proset cproset 18350  MGalConncmgc 32954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-proset 18352  df-mgc 32956
This theorem is referenced by:  dfmgc2  32971  mgcf1olem2  32977
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