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Theorem mgcmnt1d 30844
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 2738 . . 3 (Base‘𝑉) = (Base‘𝑉)
4 eqid 2738 . . 3 (Base‘𝑊) = (Base‘𝑊)
5 eqid 2738 . . 3 (le‘𝑉) = (le‘𝑉)
6 eqid 2738 . . 3 (le‘𝑊) = (le‘𝑊)
7 mgcmntd.1 . . 3 𝐻 = (𝑉MGalConn𝑊)
8 mgcmntd.4 . . 3 (𝜑𝐹𝐻𝐺)
93, 4, 5, 6, 7, 1, 2, 8mgcf1 30835 . 2 (𝜑𝐹:(Base‘𝑉)⟶(Base‘𝑊))
103, 4, 5, 6, 7, 1, 2dfmgc2 30843 . . . . 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 235 . . . 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 498 . 2 (𝜑 → ∀𝑥 ∈ (Base‘𝑉)∀𝑦 ∈ (Base‘𝑉)(𝑥(le‘𝑉)𝑦 → (𝐹𝑥)(le‘𝑊)(𝐹𝑦)))
143, 4, 5, 6ismnt 30830 . . 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 481 . 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 838 1 (𝜑𝐹 ∈ (𝑉Monot𝑊))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2113  wral 3053   class class class wbr 5027  wf 6329  cfv 6333  (class class class)co 7164  Basecbs 16579  lecple 16668   Proset cproset 17645  Monotcmnt 30825  MGalConncmgc 30826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-br 5028  df-opab 5090  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-fv 6341  df-ov 7167  df-oprab 7168  df-mpo 7169  df-map 8432  df-proset 17647  df-mnt 30827  df-mgc 30828
This theorem is referenced by: (None)
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