Theorem mgcmnt1d 32969

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	⊢ (𝜑 → 𝑉 ∈ Proset )
mgcmntd.3	⊢ (𝜑 → 𝑊 ∈ Proset )
mgcmntd.4	⊢ (𝜑 → 𝐹𝐻𝐺)

Assertion

Ref	Expression
mgcmnt1d	⊢ (𝜑 → 𝐹 ∈ (𝑉Monot𝑊))

Proof of Theorem mgcmnt1d

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

Step	Hyp	Ref	Expression
1		mgcmntd.2	. 2 ⊢ (𝜑 → 𝑉 ∈ Proset )
2		mgcmntd.3	. 2 ⊢ (𝜑 → 𝑊 ∈ Proset )
3		eqid 2729	. . 3 ⊢ (Base‘𝑉) = (Base‘𝑉)
4		eqid 2729	. . 3 ⊢ (Base‘𝑊) = (Base‘𝑊)
5		eqid 2729	. . 3 ⊢ (le‘𝑉) = (le‘𝑉)
6		eqid 2729	. . 3 ⊢ (le‘𝑊) = (le‘𝑊)
7		mgcmntd.1	. . 3 ⊢ 𝐻 = (𝑉MGalConn𝑊)
8		mgcmntd.4	. . 3 ⊢ (𝜑 → 𝐹𝐻𝐺)
9	3, 4, 5, 6, 7, 1, 2, 8	mgcf1 32960	. 2 ⊢ (𝜑 → 𝐹:(Base‘𝑉)⟶(Base‘𝑊))
10	3, 4, 5, 6, 7, 1, 2	dfmgc2 32968	. . . . 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 232	. . . 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 771	. . 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	simpld 494	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝑉)∀𝑦 ∈ (Base‘𝑉)(𝑥(le‘𝑉)𝑦 → (𝐹‘𝑥)(le‘𝑊)(𝐹‘𝑦)))
14	3, 4, 5, 6	ismnt 32955	. . 3 ⊢ ((𝑉 ∈ Proset ∧ 𝑊 ∈ Proset ) → (𝐹 ∈ (𝑉Monot𝑊) ↔ (𝐹:(Base‘𝑉)⟶(Base‘𝑊) ∧ ∀𝑥 ∈ (Base‘𝑉)∀𝑦 ∈ (Base‘𝑉)(𝑥(le‘𝑉)𝑦 → (𝐹‘𝑥)(le‘𝑊)(𝐹‘𝑦)))))
15	14	biimpar 477	. 2 ⊢ (((𝑉 ∈ Proset ∧ 𝑊 ∈ Proset ) ∧ (𝐹:(Base‘𝑉)⟶(Base‘𝑊) ∧ ∀𝑥 ∈ (Base‘𝑉)∀𝑦 ∈ (Base‘𝑉)(𝑥(le‘𝑉)𝑦 → (𝐹‘𝑥)(le‘𝑊)(𝐹‘𝑦)))) → 𝐹 ∈ (𝑉Monot𝑊))
16	1, 2, 9, 13, 15	syl22anc 838	1 ⊢ (𝜑 → 𝐹 ∈ (𝑉Monot𝑊))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044   class class class wbr 5102  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369  Basecbs 17155  lecple 17203   Proset cproset 18233  Monotcmnt 32950  MGalConncmgc 32951

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-map 8778  df-proset 18235  df-mnt 32952  df-mgc 32953

This theorem is referenced by: (None)