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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 2738 | . . 3 ⊢ (Base‘𝑉) = (Base‘𝑉) | |
4 | eqid 2738 | . . 3 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
5 | eqid 2738 | . . 3 ⊢ (le‘𝑉) = (le‘𝑉) | |
6 | eqid 2738 | . . 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 30835 | . 2 ⊢ (𝜑 → 𝐹:(Base‘𝑉)⟶(Base‘𝑊)) |
10 | 3, 4, 5, 6, 7, 1, 2 | dfmgc2 30843 | . . . . 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 | 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 30830 | . . 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 838 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑉Monot𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2113 ∀wral 3053 class class class wbr 5027 ⟶wf 6329 ‘cfv 6333 (class class class)co 7164 Basecbs 16579 lecple 16668 Proset cproset 17645 Monotcmnt 30825 MGalConncmgc 30826 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-opab 5090 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-fv 6341 df-ov 7167 df-oprab 7168 df-mpo 7169 df-map 8432 df-proset 17647 df-mnt 30827 df-mgc 30828 |
This theorem is referenced by: (None) |
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