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Theorem mgcmnt2d 31907
Description: Galois connection implies monotonicity of the right adjoint. (Contributed by Thierry Arnoux, 21-Jul-2024.)
Hypotheses
Ref Expression
mgcmntd.1 𝐻 = (𝑉MGalConnπ‘Š)
mgcmntd.2 (πœ‘ β†’ 𝑉 ∈ Proset )
mgcmntd.3 (πœ‘ β†’ π‘Š ∈ Proset )
mgcmntd.4 (πœ‘ β†’ 𝐹𝐻𝐺)
Assertion
Ref Expression
mgcmnt2d (πœ‘ β†’ 𝐺 ∈ (π‘ŠMonot𝑉))

Proof of Theorem mgcmnt2d
Dummy variables 𝑒 𝑣 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mgcmntd.3 . 2 (πœ‘ β†’ π‘Š ∈ Proset )
2 mgcmntd.2 . 2 (πœ‘ β†’ 𝑉 ∈ Proset )
3 eqid 2733 . . 3 (Baseβ€˜π‘‰) = (Baseβ€˜π‘‰)
4 eqid 2733 . . 3 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
5 eqid 2733 . . 3 (leβ€˜π‘‰) = (leβ€˜π‘‰)
6 eqid 2733 . . 3 (leβ€˜π‘Š) = (leβ€˜π‘Š)
7 mgcmntd.1 . . 3 𝐻 = (𝑉MGalConnπ‘Š)
8 mgcmntd.4 . . 3 (πœ‘ β†’ 𝐹𝐻𝐺)
93, 4, 5, 6, 7, 2, 1, 8mgcf2 31898 . 2 (πœ‘ β†’ 𝐺:(Baseβ€˜π‘Š)⟢(Baseβ€˜π‘‰))
103, 4, 5, 6, 7, 2, 1dfmgc2 31905 . . . . 5 (πœ‘ β†’ (𝐹𝐻𝐺 ↔ ((𝐹:(Baseβ€˜π‘‰)⟢(Baseβ€˜π‘Š) ∧ 𝐺:(Baseβ€˜π‘Š)⟢(Baseβ€˜π‘‰)) ∧ ((βˆ€π‘₯ ∈ (Baseβ€˜π‘‰)βˆ€π‘¦ ∈ (Baseβ€˜π‘‰)(π‘₯(leβ€˜π‘‰)𝑦 β†’ (πΉβ€˜π‘₯)(leβ€˜π‘Š)(πΉβ€˜π‘¦)) ∧ βˆ€π‘’ ∈ (Baseβ€˜π‘Š)βˆ€π‘£ ∈ (Baseβ€˜π‘Š)(𝑒(leβ€˜π‘Š)𝑣 β†’ (πΊβ€˜π‘’)(leβ€˜π‘‰)(πΊβ€˜π‘£))) ∧ (βˆ€π‘’ ∈ (Baseβ€˜π‘Š)(πΉβ€˜(πΊβ€˜π‘’))(leβ€˜π‘Š)𝑒 ∧ βˆ€π‘₯ ∈ (Baseβ€˜π‘‰)π‘₯(leβ€˜π‘‰)(πΊβ€˜(πΉβ€˜π‘₯)))))))
118, 10mpbid 231 . . . 4 (πœ‘ β†’ ((𝐹:(Baseβ€˜π‘‰)⟢(Baseβ€˜π‘Š) ∧ 𝐺:(Baseβ€˜π‘Š)⟢(Baseβ€˜π‘‰)) ∧ ((βˆ€π‘₯ ∈ (Baseβ€˜π‘‰)βˆ€π‘¦ ∈ (Baseβ€˜π‘‰)(π‘₯(leβ€˜π‘‰)𝑦 β†’ (πΉβ€˜π‘₯)(leβ€˜π‘Š)(πΉβ€˜π‘¦)) ∧ βˆ€π‘’ ∈ (Baseβ€˜π‘Š)βˆ€π‘£ ∈ (Baseβ€˜π‘Š)(𝑒(leβ€˜π‘Š)𝑣 β†’ (πΊβ€˜π‘’)(leβ€˜π‘‰)(πΊβ€˜π‘£))) ∧ (βˆ€π‘’ ∈ (Baseβ€˜π‘Š)(πΉβ€˜(πΊβ€˜π‘’))(leβ€˜π‘Š)𝑒 ∧ βˆ€π‘₯ ∈ (Baseβ€˜π‘‰)π‘₯(leβ€˜π‘‰)(πΊβ€˜(πΉβ€˜π‘₯))))))
1211simprld 771 . . 3 (πœ‘ β†’ (βˆ€π‘₯ ∈ (Baseβ€˜π‘‰)βˆ€π‘¦ ∈ (Baseβ€˜π‘‰)(π‘₯(leβ€˜π‘‰)𝑦 β†’ (πΉβ€˜π‘₯)(leβ€˜π‘Š)(πΉβ€˜π‘¦)) ∧ βˆ€π‘’ ∈ (Baseβ€˜π‘Š)βˆ€π‘£ ∈ (Baseβ€˜π‘Š)(𝑒(leβ€˜π‘Š)𝑣 β†’ (πΊβ€˜π‘’)(leβ€˜π‘‰)(πΊβ€˜π‘£))))
1312simprd 497 . 2 (πœ‘ β†’ βˆ€π‘’ ∈ (Baseβ€˜π‘Š)βˆ€π‘£ ∈ (Baseβ€˜π‘Š)(𝑒(leβ€˜π‘Š)𝑣 β†’ (πΊβ€˜π‘’)(leβ€˜π‘‰)(πΊβ€˜π‘£)))
144, 3, 6, 5ismnt 31892 . . 3 ((π‘Š ∈ Proset ∧ 𝑉 ∈ Proset ) β†’ (𝐺 ∈ (π‘ŠMonot𝑉) ↔ (𝐺:(Baseβ€˜π‘Š)⟢(Baseβ€˜π‘‰) ∧ βˆ€π‘’ ∈ (Baseβ€˜π‘Š)βˆ€π‘£ ∈ (Baseβ€˜π‘Š)(𝑒(leβ€˜π‘Š)𝑣 β†’ (πΊβ€˜π‘’)(leβ€˜π‘‰)(πΊβ€˜π‘£)))))
1514biimpar 479 . 2 (((π‘Š ∈ Proset ∧ 𝑉 ∈ Proset ) ∧ (𝐺:(Baseβ€˜π‘Š)⟢(Baseβ€˜π‘‰) ∧ βˆ€π‘’ ∈ (Baseβ€˜π‘Š)βˆ€π‘£ ∈ (Baseβ€˜π‘Š)(𝑒(leβ€˜π‘Š)𝑣 β†’ (πΊβ€˜π‘’)(leβ€˜π‘‰)(πΊβ€˜π‘£)))) β†’ 𝐺 ∈ (π‘ŠMonot𝑉))
161, 2, 9, 13, 15syl22anc 838 1 (πœ‘ β†’ 𝐺 ∈ (π‘ŠMonot𝑉))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061   class class class wbr 5106  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  lecple 17145   Proset cproset 18187  Monotcmnt 31887  MGalConncmgc 31888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8770  df-proset 18189  df-mnt 31889  df-mgc 31890
This theorem is referenced by:  mgcf1o  31912
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