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Mathbox for Thierry Arnoux |
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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 31672 | . . . . . 6 ⊢ (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦))))) |
10 | 2, 9 | mpbid 231 | . . . . 5 ⊢ (𝜑 → ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))) |
11 | 10 | simplld 767 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
12 | mgccole1.2 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
13 | 11, 12 | ffvelcdmd 7033 | . . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ 𝐵) |
14 | 4, 6 | prsref 18148 | . . 3 ⊢ ((𝑊 ∈ Proset ∧ (𝐹‘𝑋) ∈ 𝐵) → (𝐹‘𝑋) ≲ (𝐹‘𝑋)) |
15 | 1, 13, 14 | syl2anc 585 | . 2 ⊢ (𝜑 → (𝐹‘𝑋) ≲ (𝐹‘𝑋)) |
16 | fveq2 6840 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋)) | |
17 | 16 | breq1d 5114 | . . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) ≲ 𝑦 ↔ (𝐹‘𝑋) ≲ 𝑦)) |
18 | breq1 5107 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ (𝐺‘𝑦) ↔ 𝑋 ≤ (𝐺‘𝑦))) | |
19 | 17, 18 | bibi12d 346 | . . . . 5 ⊢ (𝑥 = 𝑋 → (((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)) ↔ ((𝐹‘𝑋) ≲ 𝑦 ↔ 𝑋 ≤ (𝐺‘𝑦)))) |
20 | 19 | ralbidv 3173 | . . . 4 ⊢ (𝑥 = 𝑋 → (∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)) ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑋) ≲ 𝑦 ↔ 𝑋 ≤ (𝐺‘𝑦)))) |
21 | 10 | simprd 497 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦))) |
22 | 20, 21, 12 | rspcdva 3581 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ((𝐹‘𝑋) ≲ 𝑦 ↔ 𝑋 ≤ (𝐺‘𝑦))) |
23 | simpr 486 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑋)) → 𝑦 = (𝐹‘𝑋)) | |
24 | 23 | breq2d 5116 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑋)) → ((𝐹‘𝑋) ≲ 𝑦 ↔ (𝐹‘𝑋) ≲ (𝐹‘𝑋))) |
25 | 23 | fveq2d 6844 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑋)) → (𝐺‘𝑦) = (𝐺‘(𝐹‘𝑋))) |
26 | 25 | breq2d 5116 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑋)) → (𝑋 ≤ (𝐺‘𝑦) ↔ 𝑋 ≤ (𝐺‘(𝐹‘𝑋)))) |
27 | 24, 26 | bibi12d 346 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑋)) → (((𝐹‘𝑋) ≲ 𝑦 ↔ 𝑋 ≤ (𝐺‘𝑦)) ↔ ((𝐹‘𝑋) ≲ (𝐹‘𝑋) ↔ 𝑋 ≤ (𝐺‘(𝐹‘𝑋))))) |
28 | 13, 27 | rspcdv 3572 | . . 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 231 | 1 ⊢ (𝜑 → 𝑋 ≤ (𝐺‘(𝐹‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3063 class class class wbr 5104 ⟶wf 6490 ‘cfv 6494 (class class class)co 7352 Basecbs 17043 lecple 17100 Proset cproset 18142 MGalConncmgc 31664 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-id 5530 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-fv 6502 df-ov 7355 df-oprab 7356 df-mpo 7357 df-map 8726 df-proset 18144 df-mgc 31666 |
This theorem is referenced by: mgcmnt1 31677 mgcmntco 31679 dfmgc2 31681 mgcf1olem1 31686 mgcf1olem2 31687 |
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