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Mirrors > Home > MPE Home > Th. List > Mathboxes > mgcmnt1d | Structured version Visualization version GIF version |
Description: Galois connection implies monotonicity of the left 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 |
---|---|
mgcmnt1d | ⊢ (𝜑 → 𝐹 ∈ (𝑉Monot𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mgcmntd.2 | . 2 ⊢ (𝜑 → 𝑉 ∈ Proset ) | |
2 | mgcmntd.3 | . 2 ⊢ (𝜑 → 𝑊 ∈ Proset ) | |
3 | eqid 2758 | . . 3 ⊢ (Base‘𝑉) = (Base‘𝑉) | |
4 | eqid 2758 | . . 3 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
5 | eqid 2758 | . . 3 ⊢ (le‘𝑉) = (le‘𝑉) | |
6 | eqid 2758 | . . 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 30796 | . 2 ⊢ (𝜑 → 𝐹:(Base‘𝑉)⟶(Base‘𝑊)) |
10 | 3, 4, 5, 6, 7, 1, 2 | dfmgc2 30804 | . . . . 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 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 498 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝑉)∀𝑦 ∈ (Base‘𝑉)(𝑥(le‘𝑉)𝑦 → (𝐹‘𝑥)(le‘𝑊)(𝐹‘𝑦))) |
14 | 3, 4, 5, 6 | ismnt 30791 | . . 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 481 | . 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 837 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑉Monot𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3070 class class class wbr 5035 ⟶wf 6335 ‘cfv 6339 (class class class)co 7155 Basecbs 16546 lecple 16635 Proset cproset 17607 Monotcmnt 30786 MGalConncmgc 30787 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-sep 5172 ax-nul 5179 ax-pow 5237 ax-pr 5301 ax-un 7464 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ral 3075 df-rex 3076 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4802 df-br 5036 df-opab 5098 df-id 5433 df-xp 5533 df-rel 5534 df-cnv 5535 df-co 5536 df-dm 5537 df-rn 5538 df-iota 6298 df-fun 6341 df-fn 6342 df-f 6343 df-fv 6347 df-ov 7158 df-oprab 7159 df-mpo 7160 df-map 8423 df-proset 17609 df-mnt 30788 df-mgc 30789 |
This theorem is referenced by: (None) |
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