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 31281. (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 18034 | . . . . . . 7 ⊢ (𝑉 ∈ Poset → 𝑉 ∈ Proset ) | |
10 | 8, 9 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑉 ∈ Proset ) |
11 | posprs 18034 | . . . . . . 7 ⊢ (𝑊 ∈ Poset → 𝑊 ∈ Proset ) | |
12 | 1, 11 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Proset ) |
13 | 3, 4, 5, 6, 7, 10, 12 | dfmgc2 31274 | . . . . 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 231 | . . . 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 765 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
16 | 14 | simplrd 767 | . . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴) |
17 | mgcf1olem1.1 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
18 | 15, 17 | ffvelrnd 6962 | . . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ 𝐵) |
19 | 16, 18 | ffvelrnd 6962 | . . 3 ⊢ (𝜑 → (𝐺‘(𝐹‘𝑋)) ∈ 𝐴) |
20 | 15, 19 | ffvelrnd 6962 | . 2 ⊢ (𝜑 → (𝐹‘(𝐺‘(𝐹‘𝑋))) ∈ 𝐵) |
21 | 3, 4, 5, 6, 7, 10, 12, 2, 18 | mgccole2 31269 | . 2 ⊢ (𝜑 → (𝐹‘(𝐺‘(𝐹‘𝑋))) ≲ (𝐹‘𝑋)) |
22 | 3, 4, 5, 6, 7, 10, 12, 2, 17 | mgccole1 31268 | . . 3 ⊢ (𝜑 → 𝑋 ≤ (𝐺‘(𝐹‘𝑋))) |
23 | 3, 4, 5, 6, 7, 10, 12, 2, 17, 19, 22 | mgcmnt1 31270 | . 2 ⊢ (𝜑 → (𝐹‘𝑋) ≲ (𝐹‘(𝐺‘(𝐹‘𝑋)))) |
24 | 4, 6 | posasymb 18037 | . . 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 477 | . 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 1377 | 1 ⊢ (𝜑 → (𝐹‘(𝐺‘(𝐹‘𝑋))) = (𝐹‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = 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 Posetcpo 18025 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-poset 18031 df-mgc 31259 |
This theorem is referenced by: mgcf1o 31281 |
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