![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > mgccole1 | Structured version Visualization version GIF version |
Description: An inequality for the kernel operator 𝐺 ∘ 𝐹. (Contributed by Thierry Arnoux, 26-Apr-2024.) |
Ref | Expression |
---|---|
mgcoval.1 | ⊢ 𝐴 = (Base‘𝑉) |
mgcoval.2 | ⊢ 𝐵 = (Base‘𝑊) |
mgcoval.3 | ⊢ ≤ = (le‘𝑉) |
mgcoval.4 | ⊢ ≲ = (le‘𝑊) |
mgcval.1 | ⊢ 𝐻 = (𝑉MGalConn𝑊) |
mgcval.2 | ⊢ (𝜑 → 𝑉 ∈ Proset ) |
mgcval.3 | ⊢ (𝜑 → 𝑊 ∈ Proset ) |
mgccole.1 | ⊢ (𝜑 → 𝐹𝐻𝐺) |
mgccole1.2 | ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
Ref | Expression |
---|---|
mgccole1 | ⊢ (𝜑 → 𝑋 ≤ (𝐺‘(𝐹‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mgcval.3 | . . 3 ⊢ (𝜑 → 𝑊 ∈ Proset ) | |
2 | mgccole.1 | . . . . . 6 ⊢ (𝜑 → 𝐹𝐻𝐺) | |
3 | mgcoval.1 | . . . . . . 7 ⊢ 𝐴 = (Base‘𝑉) | |
4 | mgcoval.2 | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑊) | |
5 | mgcoval.3 | . . . . . . 7 ⊢ ≤ = (le‘𝑉) | |
6 | mgcoval.4 | . . . . . . 7 ⊢ ≲ = (le‘𝑊) | |
7 | mgcval.1 | . . . . . . 7 ⊢ 𝐻 = (𝑉MGalConn𝑊) | |
8 | mgcval.2 | . . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ Proset ) | |
9 | 3, 4, 5, 6, 7, 8, 1 | mgcval 32962 | . . . . . 6 ⊢ (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦))))) |
10 | 2, 9 | mpbid 232 | . . . . 5 ⊢ (𝜑 → ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))) |
11 | 10 | simplld 768 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
12 | mgccole1.2 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
13 | 11, 12 | ffvelcdmd 7105 | . . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ 𝐵) |
14 | 4, 6 | prsref 18356 | . . 3 ⊢ ((𝑊 ∈ Proset ∧ (𝐹‘𝑋) ∈ 𝐵) → (𝐹‘𝑋) ≲ (𝐹‘𝑋)) |
15 | 1, 13, 14 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐹‘𝑋) ≲ (𝐹‘𝑋)) |
16 | fveq2 6907 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋)) | |
17 | 16 | breq1d 5158 | . . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) ≲ 𝑦 ↔ (𝐹‘𝑋) ≲ 𝑦)) |
18 | breq1 5151 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ (𝐺‘𝑦) ↔ 𝑋 ≤ (𝐺‘𝑦))) | |
19 | 17, 18 | bibi12d 345 | . . . . 5 ⊢ (𝑥 = 𝑋 → (((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)) ↔ ((𝐹‘𝑋) ≲ 𝑦 ↔ 𝑋 ≤ (𝐺‘𝑦)))) |
20 | 19 | ralbidv 3176 | . . . 4 ⊢ (𝑥 = 𝑋 → (∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)) ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑋) ≲ 𝑦 ↔ 𝑋 ≤ (𝐺‘𝑦)))) |
21 | 10 | simprd 495 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦))) |
22 | 20, 21, 12 | rspcdva 3623 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ((𝐹‘𝑋) ≲ 𝑦 ↔ 𝑋 ≤ (𝐺‘𝑦))) |
23 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑋)) → 𝑦 = (𝐹‘𝑋)) | |
24 | 23 | breq2d 5160 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑋)) → ((𝐹‘𝑋) ≲ 𝑦 ↔ (𝐹‘𝑋) ≲ (𝐹‘𝑋))) |
25 | 23 | fveq2d 6911 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑋)) → (𝐺‘𝑦) = (𝐺‘(𝐹‘𝑋))) |
26 | 25 | breq2d 5160 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑋)) → (𝑋 ≤ (𝐺‘𝑦) ↔ 𝑋 ≤ (𝐺‘(𝐹‘𝑋)))) |
27 | 24, 26 | bibi12d 345 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑋)) → (((𝐹‘𝑋) ≲ 𝑦 ↔ 𝑋 ≤ (𝐺‘𝑦)) ↔ ((𝐹‘𝑋) ≲ (𝐹‘𝑋) ↔ 𝑋 ≤ (𝐺‘(𝐹‘𝑋))))) |
28 | 13, 27 | rspcdv 3614 | . . 3 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 ((𝐹‘𝑋) ≲ 𝑦 ↔ 𝑋 ≤ (𝐺‘𝑦)) → ((𝐹‘𝑋) ≲ (𝐹‘𝑋) ↔ 𝑋 ≤ (𝐺‘(𝐹‘𝑋))))) |
29 | 22, 28 | mpd 15 | . 2 ⊢ (𝜑 → ((𝐹‘𝑋) ≲ (𝐹‘𝑋) ↔ 𝑋 ≤ (𝐺‘(𝐹‘𝑋)))) |
30 | 15, 29 | mpbid 232 | 1 ⊢ (𝜑 → 𝑋 ≤ (𝐺‘(𝐹‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 class class class wbr 5148 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 lecple 17305 Proset cproset 18350 MGalConncmgc 32954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8867 df-proset 18352 df-mgc 32956 |
This theorem is referenced by: mgcmnt1 32967 mgcmntco 32969 dfmgc2 32971 mgcf1olem1 32976 mgcf1olem2 32977 |
Copyright terms: Public domain | W3C validator |