Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mgcf1olem1 | Structured version Visualization version GIF version |
Description: Property of a Galois connection, lemma for mgcf1o 30811. (Contributed by Thierry Arnoux, 26-Jul-2024.) |
Ref | Expression |
---|---|
mgcf1o.h | ⊢ 𝐻 = (𝑉MGalConn𝑊) |
mgcf1o.a | ⊢ 𝐴 = (Base‘𝑉) |
mgcf1o.b | ⊢ 𝐵 = (Base‘𝑊) |
mgcf1o.1 | ⊢ ≤ = (le‘𝑉) |
mgcf1o.2 | ⊢ ≲ = (le‘𝑊) |
mgcf1o.v | ⊢ (𝜑 → 𝑉 ∈ Poset) |
mgcf1o.w | ⊢ (𝜑 → 𝑊 ∈ Poset) |
mgcf1o.f | ⊢ (𝜑 → 𝐹𝐻𝐺) |
mgcf1olem1.1 | ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
Ref | Expression |
---|---|
mgcf1olem1 | ⊢ (𝜑 → (𝐹‘(𝐺‘(𝐹‘𝑋))) = (𝐹‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mgcf1o.w | . 2 ⊢ (𝜑 → 𝑊 ∈ Poset) | |
2 | mgcf1o.f | . . . . 5 ⊢ (𝜑 → 𝐹𝐻𝐺) | |
3 | mgcf1o.a | . . . . . 6 ⊢ 𝐴 = (Base‘𝑉) | |
4 | mgcf1o.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑊) | |
5 | mgcf1o.1 | . . . . . 6 ⊢ ≤ = (le‘𝑉) | |
6 | mgcf1o.2 | . . . . . 6 ⊢ ≲ = (le‘𝑊) | |
7 | mgcf1o.h | . . . . . 6 ⊢ 𝐻 = (𝑉MGalConn𝑊) | |
8 | mgcf1o.v | . . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ Poset) | |
9 | posprs 17630 | . . . . . . 7 ⊢ (𝑉 ∈ Poset → 𝑉 ∈ Proset ) | |
10 | 8, 9 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑉 ∈ Proset ) |
11 | posprs 17630 | . . . . . . 7 ⊢ (𝑊 ∈ Poset → 𝑊 ∈ Proset ) | |
12 | 1, 11 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Proset ) |
13 | 3, 4, 5, 6, 7, 10, 12 | dfmgc2 30804 | . . . . 5 ⊢ (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ∧ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣))) ∧ (∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢 ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥))))))) |
14 | 2, 13 | mpbid 235 | . . . 4 ⊢ (𝜑 → ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ∧ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣))) ∧ (∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢 ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥)))))) |
15 | 14 | simplld 767 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
16 | 14 | simplrd 769 | . . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴) |
17 | mgcf1olem1.1 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
18 | 15, 17 | ffvelrnd 6848 | . . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ 𝐵) |
19 | 16, 18 | ffvelrnd 6848 | . . 3 ⊢ (𝜑 → (𝐺‘(𝐹‘𝑋)) ∈ 𝐴) |
20 | 15, 19 | ffvelrnd 6848 | . 2 ⊢ (𝜑 → (𝐹‘(𝐺‘(𝐹‘𝑋))) ∈ 𝐵) |
21 | 3, 4, 5, 6, 7, 10, 12, 2, 18 | mgccole2 30799 | . 2 ⊢ (𝜑 → (𝐹‘(𝐺‘(𝐹‘𝑋))) ≲ (𝐹‘𝑋)) |
22 | 3, 4, 5, 6, 7, 10, 12, 2, 17 | mgccole1 30798 | . . 3 ⊢ (𝜑 → 𝑋 ≤ (𝐺‘(𝐹‘𝑋))) |
23 | 3, 4, 5, 6, 7, 10, 12, 2, 17, 19, 22 | mgcmnt1 30800 | . 2 ⊢ (𝜑 → (𝐹‘𝑋) ≲ (𝐹‘(𝐺‘(𝐹‘𝑋)))) |
24 | 4, 6 | posasymb 17633 | . . 3 ⊢ ((𝑊 ∈ Poset ∧ (𝐹‘(𝐺‘(𝐹‘𝑋))) ∈ 𝐵 ∧ (𝐹‘𝑋) ∈ 𝐵) → (((𝐹‘(𝐺‘(𝐹‘𝑋))) ≲ (𝐹‘𝑋) ∧ (𝐹‘𝑋) ≲ (𝐹‘(𝐺‘(𝐹‘𝑋)))) ↔ (𝐹‘(𝐺‘(𝐹‘𝑋))) = (𝐹‘𝑋))) |
25 | 24 | biimpa 480 | . 2 ⊢ (((𝑊 ∈ Poset ∧ (𝐹‘(𝐺‘(𝐹‘𝑋))) ∈ 𝐵 ∧ (𝐹‘𝑋) ∈ 𝐵) ∧ ((𝐹‘(𝐺‘(𝐹‘𝑋))) ≲ (𝐹‘𝑋) ∧ (𝐹‘𝑋) ≲ (𝐹‘(𝐺‘(𝐹‘𝑋))))) → (𝐹‘(𝐺‘(𝐹‘𝑋))) = (𝐹‘𝑋)) |
26 | 1, 20, 18, 21, 23, 25 | syl32anc 1375 | 1 ⊢ (𝜑 → (𝐹‘(𝐺‘(𝐹‘𝑋))) = (𝐹‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = 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 Posetcpo 17621 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-poset 17627 df-mgc 30789 |
This theorem is referenced by: mgcf1o 30811 |
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