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 30858. (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 17675 | . . . . . . 7 ⊢ (𝑉 ∈ Poset → 𝑉 ∈ Proset ) | |
10 | 8, 9 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑉 ∈ Proset ) |
11 | posprs 17675 | . . . . . . 7 ⊢ (𝑊 ∈ Poset → 𝑊 ∈ Proset ) | |
12 | 1, 11 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Proset ) |
13 | 3, 4, 5, 6, 7, 10, 12 | dfmgc2 30851 | . . . . 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 768 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
16 | 14 | simplrd 770 | . . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴) |
17 | mgcf1olem1.1 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
18 | 15, 17 | ffvelrnd 6862 | . . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ 𝐵) |
19 | 16, 18 | ffvelrnd 6862 | . . 3 ⊢ (𝜑 → (𝐺‘(𝐹‘𝑋)) ∈ 𝐴) |
20 | 15, 19 | ffvelrnd 6862 | . 2 ⊢ (𝜑 → (𝐹‘(𝐺‘(𝐹‘𝑋))) ∈ 𝐵) |
21 | 3, 4, 5, 6, 7, 10, 12, 2, 18 | mgccole2 30846 | . 2 ⊢ (𝜑 → (𝐹‘(𝐺‘(𝐹‘𝑋))) ≲ (𝐹‘𝑋)) |
22 | 3, 4, 5, 6, 7, 10, 12, 2, 17 | mgccole1 30845 | . . 3 ⊢ (𝜑 → 𝑋 ≤ (𝐺‘(𝐹‘𝑋))) |
23 | 3, 4, 5, 6, 7, 10, 12, 2, 17, 19, 22 | mgcmnt1 30847 | . 2 ⊢ (𝜑 → (𝐹‘𝑋) ≲ (𝐹‘(𝐺‘(𝐹‘𝑋)))) |
24 | 4, 6 | posasymb 17678 | . . 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 1379 | 1 ⊢ (𝜑 → (𝐹‘(𝐺‘(𝐹‘𝑋))) = (𝐹‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ∀wral 3053 class class class wbr 5030 ⟶wf 6335 ‘cfv 6339 (class class class)co 7170 Basecbs 16586 lecple 16675 Proset cproset 17652 Posetcpo 17666 MGalConncmgc 30834 |
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 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 |
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 2075 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 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-fv 6347 df-ov 7173 df-oprab 7174 df-mpo 7175 df-map 8439 df-proset 17654 df-poset 17672 df-mgc 30836 |
This theorem is referenced by: mgcf1o 30858 |
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