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Theorem mgcmnt1d 32987
Description: Galois connection implies monotonicity of the left adjoint. (Contributed by Thierry Arnoux, 21-Jul-2024.)
Hypotheses
Ref Expression
mgcmntd.1 𝐻 = (𝑉MGalConn𝑊)
mgcmntd.2 (𝜑𝑉 ∈ Proset )
mgcmntd.3 (𝜑𝑊 ∈ Proset )
mgcmntd.4 (𝜑𝐹𝐻𝐺)
Assertion
Ref Expression
mgcmnt1d (𝜑𝐹 ∈ (𝑉Monot𝑊))

Proof of Theorem mgcmnt1d
Dummy variables 𝑢 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mgcmntd.2 . 2 (𝜑𝑉 ∈ Proset )
2 mgcmntd.3 . 2 (𝜑𝑊 ∈ Proset )
3 eqid 2737 . . 3 (Base‘𝑉) = (Base‘𝑉)
4 eqid 2737 . . 3 (Base‘𝑊) = (Base‘𝑊)
5 eqid 2737 . . 3 (le‘𝑉) = (le‘𝑉)
6 eqid 2737 . . 3 (le‘𝑊) = (le‘𝑊)
7 mgcmntd.1 . . 3 𝐻 = (𝑉MGalConn𝑊)
8 mgcmntd.4 . . 3 (𝜑𝐹𝐻𝐺)
93, 4, 5, 6, 7, 1, 2, 8mgcf1 32978 . 2 (𝜑𝐹:(Base‘𝑉)⟶(Base‘𝑊))
103, 4, 5, 6, 7, 1, 2dfmgc2 32986 . . . . 5 (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:(Base‘𝑉)⟶(Base‘𝑊) ∧ 𝐺:(Base‘𝑊)⟶(Base‘𝑉)) ∧ ((∀𝑥 ∈ (Base‘𝑉)∀𝑦 ∈ (Base‘𝑉)(𝑥(le‘𝑉)𝑦 → (𝐹𝑥)(le‘𝑊)(𝐹𝑦)) ∧ ∀𝑢 ∈ (Base‘𝑊)∀𝑣 ∈ (Base‘𝑊)(𝑢(le‘𝑊)𝑣 → (𝐺𝑢)(le‘𝑉)(𝐺𝑣))) ∧ (∀𝑢 ∈ (Base‘𝑊)(𝐹‘(𝐺𝑢))(le‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑉)𝑥(le‘𝑉)(𝐺‘(𝐹𝑥)))))))
118, 10mpbid 232 . . . 4 (𝜑 → ((𝐹:(Base‘𝑉)⟶(Base‘𝑊) ∧ 𝐺:(Base‘𝑊)⟶(Base‘𝑉)) ∧ ((∀𝑥 ∈ (Base‘𝑉)∀𝑦 ∈ (Base‘𝑉)(𝑥(le‘𝑉)𝑦 → (𝐹𝑥)(le‘𝑊)(𝐹𝑦)) ∧ ∀𝑢 ∈ (Base‘𝑊)∀𝑣 ∈ (Base‘𝑊)(𝑢(le‘𝑊)𝑣 → (𝐺𝑢)(le‘𝑉)(𝐺𝑣))) ∧ (∀𝑢 ∈ (Base‘𝑊)(𝐹‘(𝐺𝑢))(le‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑉)𝑥(le‘𝑉)(𝐺‘(𝐹𝑥))))))
1211simprld 772 . . 3 (𝜑 → (∀𝑥 ∈ (Base‘𝑉)∀𝑦 ∈ (Base‘𝑉)(𝑥(le‘𝑉)𝑦 → (𝐹𝑥)(le‘𝑊)(𝐹𝑦)) ∧ ∀𝑢 ∈ (Base‘𝑊)∀𝑣 ∈ (Base‘𝑊)(𝑢(le‘𝑊)𝑣 → (𝐺𝑢)(le‘𝑉)(𝐺𝑣))))
1312simpld 494 . 2 (𝜑 → ∀𝑥 ∈ (Base‘𝑉)∀𝑦 ∈ (Base‘𝑉)(𝑥(le‘𝑉)𝑦 → (𝐹𝑥)(le‘𝑊)(𝐹𝑦)))
143, 4, 5, 6ismnt 32973 . . 3 ((𝑉 ∈ Proset ∧ 𝑊 ∈ Proset ) → (𝐹 ∈ (𝑉Monot𝑊) ↔ (𝐹:(Base‘𝑉)⟶(Base‘𝑊) ∧ ∀𝑥 ∈ (Base‘𝑉)∀𝑦 ∈ (Base‘𝑉)(𝑥(le‘𝑉)𝑦 → (𝐹𝑥)(le‘𝑊)(𝐹𝑦)))))
1514biimpar 477 . 2 (((𝑉 ∈ Proset ∧ 𝑊 ∈ Proset ) ∧ (𝐹:(Base‘𝑉)⟶(Base‘𝑊) ∧ ∀𝑥 ∈ (Base‘𝑉)∀𝑦 ∈ (Base‘𝑉)(𝑥(le‘𝑉)𝑦 → (𝐹𝑥)(le‘𝑊)(𝐹𝑦)))) → 𝐹 ∈ (𝑉Monot𝑊))
161, 2, 9, 13, 15syl22anc 839 1 (𝜑𝐹 ∈ (𝑉Monot𝑊))
Colors of variables: wff setvar class
Syntax hints:  wi 4  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  Monotcmnt 32968  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-mnt 32970  df-mgc 32971
This theorem is referenced by: (None)
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