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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mgcf1olem2 | Structured version Visualization version GIF version |
Description: Property of a Galois connection, lemma for mgcf1o 32725. (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) |
mgcf1olem2.1 | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
mgcf1olem2 | ⊢ (𝜑 → (𝐺‘(𝐹‘(𝐺‘𝑌))) = (𝐺‘𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mgcf1o.v | . 2 ⊢ (𝜑 → 𝑉 ∈ 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 | posprs 18302 | . . . . . . 7 ⊢ (𝑉 ∈ Poset → 𝑉 ∈ Proset ) | |
9 | 1, 8 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑉 ∈ Proset ) |
10 | mgcf1o.w | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Poset) | |
11 | posprs 18302 | . . . . . . 7 ⊢ (𝑊 ∈ Poset → 𝑊 ∈ Proset ) | |
12 | 10, 11 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Proset ) |
13 | 3, 4, 5, 6, 7, 9, 12 | dfmgc2 32718 | . . . . 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 | simplrd 769 | . . 3 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴) |
16 | 14 | simplld 767 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
17 | mgcf1olem2.1 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
18 | 15, 17 | ffvelcdmd 7090 | . . . 4 ⊢ (𝜑 → (𝐺‘𝑌) ∈ 𝐴) |
19 | 16, 18 | ffvelcdmd 7090 | . . 3 ⊢ (𝜑 → (𝐹‘(𝐺‘𝑌)) ∈ 𝐵) |
20 | 15, 19 | ffvelcdmd 7090 | . 2 ⊢ (𝜑 → (𝐺‘(𝐹‘(𝐺‘𝑌))) ∈ 𝐴) |
21 | 3, 4, 5, 6, 7, 9, 12, 2, 17 | mgccole2 32713 | . . 3 ⊢ (𝜑 → (𝐹‘(𝐺‘𝑌)) ≲ 𝑌) |
22 | 3, 4, 5, 6, 7, 9, 12, 2, 19, 17, 21 | mgcmnt2 32715 | . 2 ⊢ (𝜑 → (𝐺‘(𝐹‘(𝐺‘𝑌))) ≤ (𝐺‘𝑌)) |
23 | 3, 4, 5, 6, 7, 9, 12, 2, 18 | mgccole1 32712 | . 2 ⊢ (𝜑 → (𝐺‘𝑌) ≤ (𝐺‘(𝐹‘(𝐺‘𝑌)))) |
24 | 3, 5 | posasymb 18305 | . . 3 ⊢ ((𝑉 ∈ Poset ∧ (𝐺‘(𝐹‘(𝐺‘𝑌))) ∈ 𝐴 ∧ (𝐺‘𝑌) ∈ 𝐴) → (((𝐺‘(𝐹‘(𝐺‘𝑌))) ≤ (𝐺‘𝑌) ∧ (𝐺‘𝑌) ≤ (𝐺‘(𝐹‘(𝐺‘𝑌)))) ↔ (𝐺‘(𝐹‘(𝐺‘𝑌))) = (𝐺‘𝑌))) |
25 | 24 | biimpa 476 | . 2 ⊢ (((𝑉 ∈ Poset ∧ (𝐺‘(𝐹‘(𝐺‘𝑌))) ∈ 𝐴 ∧ (𝐺‘𝑌) ∈ 𝐴) ∧ ((𝐺‘(𝐹‘(𝐺‘𝑌))) ≤ (𝐺‘𝑌) ∧ (𝐺‘𝑌) ≤ (𝐺‘(𝐹‘(𝐺‘𝑌))))) → (𝐺‘(𝐹‘(𝐺‘𝑌))) = (𝐺‘𝑌)) |
26 | 1, 20, 18, 22, 23, 25 | syl32anc 1376 | 1 ⊢ (𝜑 → (𝐺‘(𝐹‘(𝐺‘𝑌))) = (𝐺‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∀wral 3057 class class class wbr 5143 ⟶wf 6539 ‘cfv 6543 (class class class)co 7415 Basecbs 17174 lecple 17234 Proset cproset 18279 Posetcpo 18293 MGalConncmgc 32701 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5144 df-opab 5206 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7418 df-oprab 7419 df-mpo 7420 df-map 8841 df-proset 18281 df-poset 18299 df-mgc 32703 |
This theorem is referenced by: mgcf1o 32725 |
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