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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 2735 | . . 3 ⊢ (Base‘𝑉) = (Base‘𝑉) | |
4 | eqid 2735 | . . 3 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
5 | eqid 2735 | . . 3 ⊢ (le‘𝑉) = (le‘𝑉) | |
6 | eqid 2735 | . . 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 32963 | . 2 ⊢ (𝜑 → 𝐹:(Base‘𝑉)⟶(Base‘𝑊)) |
10 | 3, 4, 5, 6, 7, 1, 2 | dfmgc2 32971 | . . . . 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 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 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 32958 | . . 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 839 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑉Monot𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 class class class wbr 5148 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 lecple 17305 Proset cproset 18350 Monotcmnt 32953 MGalConncmgc 32954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8867 df-proset 18352 df-mnt 32955 df-mgc 32956 |
This theorem is referenced by: (None) |
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