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Mirrors > Home > MPE Home > Th. List > Mathboxes > mgcmnt2d | Structured version Visualization version GIF version |
Description: Galois connection implies monotonicity of the right adjoint. (Contributed by Thierry Arnoux, 21-Jul-2024.) |
Ref | Expression |
---|---|
mgcmntd.1 | ⊢ 𝐻 = (𝑉MGalConn𝑊) |
mgcmntd.2 | ⊢ (𝜑 → 𝑉 ∈ Proset ) |
mgcmntd.3 | ⊢ (𝜑 → 𝑊 ∈ Proset ) |
mgcmntd.4 | ⊢ (𝜑 → 𝐹𝐻𝐺) |
Ref | Expression |
---|---|
mgcmnt2d | ⊢ (𝜑 → 𝐺 ∈ (𝑊Monot𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mgcmntd.3 | . 2 ⊢ (𝜑 → 𝑊 ∈ Proset ) | |
2 | mgcmntd.2 | . 2 ⊢ (𝜑 → 𝑉 ∈ Proset ) | |
3 | eqid 2739 | . . 3 ⊢ (Base‘𝑉) = (Base‘𝑉) | |
4 | eqid 2739 | . . 3 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
5 | eqid 2739 | . . 3 ⊢ (le‘𝑉) = (le‘𝑉) | |
6 | eqid 2739 | . . 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 31017 | . 2 ⊢ (𝜑 → 𝐺:(Base‘𝑊)⟶(Base‘𝑉)) |
10 | 3, 4, 5, 6, 7, 2, 1 | dfmgc2 31024 | . . . . 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 772 | . . 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 499 | . 2 ⊢ (𝜑 → ∀𝑢 ∈ (Base‘𝑊)∀𝑣 ∈ (Base‘𝑊)(𝑢(le‘𝑊)𝑣 → (𝐺‘𝑢)(le‘𝑉)(𝐺‘𝑣))) |
14 | 4, 3, 6, 5 | ismnt 31011 | . . 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 481 | . 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 839 | 1 ⊢ (𝜑 → 𝐺 ∈ (𝑊Monot𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∀wral 3064 class class class wbr 5069 ⟶wf 6396 ‘cfv 6400 (class class class)co 7234 Basecbs 16792 lecple 16841 Proset cproset 17832 Monotcmnt 31006 MGalConncmgc 31007 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-sep 5208 ax-nul 5215 ax-pow 5274 ax-pr 5338 ax-un 7544 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3425 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4456 df-pw 4531 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4836 df-br 5070 df-opab 5132 df-id 5471 df-xp 5574 df-rel 5575 df-cnv 5576 df-co 5577 df-dm 5578 df-rn 5579 df-iota 6358 df-fun 6402 df-fn 6403 df-f 6404 df-fv 6408 df-ov 7237 df-oprab 7238 df-mpo 7239 df-map 8533 df-proset 17834 df-mnt 31008 df-mgc 31009 |
This theorem is referenced by: mgcf1o 31031 |
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