Theorem mgcmnt2d 33318

Description: Galois connection implies monotonicity of the right adjoint. (Contributed by Thierry Arnoux, 21-Jul-2024.)

Hypotheses

Ref	Expression
mgcmntd.1	⊢ 𝐻 = (𝑉MGalConn𝑊)
mgcmntd.2	⊢ (𝜑 → 𝑉 ∈ Proset )
mgcmntd.3	⊢ (𝜑 → 𝑊 ∈ Proset )
mgcmntd.4	⊢ (𝜑 → 𝐹𝐻𝐺)

Assertion

Ref	Expression
mgcmnt2d	⊢ (𝜑 → 𝐺 ∈ (𝑊Monot𝑉))

Proof of Theorem mgcmnt2d

Dummy variables 𝑢 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mgcmntd.3	. 2 ⊢ (𝜑 → 𝑊 ∈ Proset )
2		mgcmntd.2	. 2 ⊢ (𝜑 → 𝑉 ∈ Proset )
3		eqid 2763	. . 3 ⊢ (Base‘𝑉) = (Base‘𝑉)
4		eqid 2763	. . 3 ⊢ (Base‘𝑊) = (Base‘𝑊)
5		eqid 2763	. . 3 ⊢ (le‘𝑉) = (le‘𝑉)
6		eqid 2763	. . 3 ⊢ (le‘𝑊) = (le‘𝑊)
7		mgcmntd.1	. . 3 ⊢ 𝐻 = (𝑉MGalConn𝑊)
8		mgcmntd.4	. . 3 ⊢ (𝜑 → 𝐹𝐻𝐺)
9	3, 4, 5, 6, 7, 2, 1, 8	mgcf2 33309	. 2 ⊢ (𝜑 → 𝐺:(Base‘𝑊)⟶(Base‘𝑉))
10	3, 4, 5, 6, 7, 2, 1	dfmgc2 33316	. . . . 5 ⊢ (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:(Base‘𝑉)⟶(Base‘𝑊) ∧ 𝐺:(Base‘𝑊)⟶(Base‘𝑉)) ∧ ((∀𝑥 ∈ (Base‘𝑉)∀𝑦 ∈ (Base‘𝑉)(𝑥(le‘𝑉)𝑦 → (𝐹‘𝑥)(le‘𝑊)(𝐹‘𝑦)) ∧ ∀𝑢 ∈ (Base‘𝑊)∀𝑣 ∈ (Base‘𝑊)(𝑢(le‘𝑊)𝑣 → (𝐺‘𝑢)(le‘𝑉)(𝐺‘𝑣))) ∧ (∀𝑢 ∈ (Base‘𝑊)(𝐹‘(𝐺‘𝑢))(le‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑉)𝑥(le‘𝑉)(𝐺‘(𝐹‘𝑥)))))))
11	8, 10	mpbid 235	. . . 4 ⊢ (𝜑 → ((𝐹:(Base‘𝑉)⟶(Base‘𝑊) ∧ 𝐺:(Base‘𝑊)⟶(Base‘𝑉)) ∧ ((∀𝑥 ∈ (Base‘𝑉)∀𝑦 ∈ (Base‘𝑉)(𝑥(le‘𝑉)𝑦 → (𝐹‘𝑥)(le‘𝑊)(𝐹‘𝑦)) ∧ ∀𝑢 ∈ (Base‘𝑊)∀𝑣 ∈ (Base‘𝑊)(𝑢(le‘𝑊)𝑣 → (𝐺‘𝑢)(le‘𝑉)(𝐺‘𝑣))) ∧ (∀𝑢 ∈ (Base‘𝑊)(𝐹‘(𝐺‘𝑢))(le‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑉)𝑥(le‘𝑉)(𝐺‘(𝐹‘𝑥))))))
12	11	simprld 783	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ (Base‘𝑉)∀𝑦 ∈ (Base‘𝑉)(𝑥(le‘𝑉)𝑦 → (𝐹‘𝑥)(le‘𝑊)(𝐹‘𝑦)) ∧ ∀𝑢 ∈ (Base‘𝑊)∀𝑣 ∈ (Base‘𝑊)(𝑢(le‘𝑊)𝑣 → (𝐺‘𝑢)(le‘𝑉)(𝐺‘𝑣))))
13	12	simprd 500	. 2 ⊢ (𝜑 → ∀𝑢 ∈ (Base‘𝑊)∀𝑣 ∈ (Base‘𝑊)(𝑢(le‘𝑊)𝑣 → (𝐺‘𝑢)(le‘𝑉)(𝐺‘𝑣)))
14	4, 3, 6, 5	ismnt 33303	. . 3 ⊢ ((𝑊 ∈ Proset ∧ 𝑉 ∈ Proset ) → (𝐺 ∈ (𝑊Monot𝑉) ↔ (𝐺:(Base‘𝑊)⟶(Base‘𝑉) ∧ ∀𝑢 ∈ (Base‘𝑊)∀𝑣 ∈ (Base‘𝑊)(𝑢(le‘𝑊)𝑣 → (𝐺‘𝑢)(le‘𝑉)(𝐺‘𝑣)))))
15	14	biimpar 482	. 2 ⊢ (((𝑊 ∈ Proset ∧ 𝑉 ∈ Proset ) ∧ (𝐺:(Base‘𝑊)⟶(Base‘𝑉) ∧ ∀𝑢 ∈ (Base‘𝑊)∀𝑣 ∈ (Base‘𝑊)(𝑢(le‘𝑊)𝑣 → (𝐺‘𝑢)(le‘𝑉)(𝐺‘𝑣)))) → 𝐺 ∈ (𝑊Monot𝑉))
16	1, 2, 9, 13, 15	syl22anc 851	1 ⊢ (𝜑 → 𝐺 ∈ (𝑊Monot𝑉))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079   class class class wbr 5109  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  Basecbs 17264  lecple 17312   Proset cproset 18343  Monotcmnt 33298  MGalConncmgc 33299

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-map 8822  df-proset 18345  df-mnt 33300  df-mgc 33301

This theorem is referenced by:  mgcf1o  33323