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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 31266 | . 2 ⊢ (𝜑 → 𝐹:(Base‘𝑉)⟶(Base‘𝑊)) |
10 | 3, 4, 5, 6, 7, 1, 2 | dfmgc2 31274 | . . . . 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 231 | . . . 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 769 | . . 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 495 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝑉)∀𝑦 ∈ (Base‘𝑉)(𝑥(le‘𝑉)𝑦 → (𝐹‘𝑥)(le‘𝑊)(𝐹‘𝑦))) |
14 | 3, 4, 5, 6 | ismnt 31261 | . . 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 478 | . 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 836 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑉Monot𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 class class class wbr 5074 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 lecple 16969 Proset cproset 18011 Monotcmnt 31256 MGalConncmgc 31257 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-map 8617 df-proset 18013 df-mnt 31258 df-mgc 31259 |
This theorem is referenced by: (None) |
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