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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 32977 | . . . . . 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 18344 | . . 3 ⊢ ((𝑊 ∈ Proset ∧ (𝐹‘𝑋) ∈ 𝐵) → (𝐹‘𝑋) ≲ (𝐹‘𝑋)) | 
| 15 | 1, 13, 14 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐹‘𝑋) ≲ (𝐹‘𝑋)) | 
| 16 | fveq2 6906 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋)) | |
| 17 | 16 | breq1d 5153 | . . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) ≲ 𝑦 ↔ (𝐹‘𝑋) ≲ 𝑦)) | 
| 18 | breq1 5146 | . . . . . 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 3178 | . . . 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 5155 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑋)) → ((𝐹‘𝑋) ≲ 𝑦 ↔ (𝐹‘𝑋) ≲ (𝐹‘𝑋))) | 
| 25 | 23 | fveq2d 6910 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑋)) → (𝐺‘𝑦) = (𝐺‘(𝐹‘𝑋))) | 
| 26 | 25 | breq2d 5155 | . . . . 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 1540 ∈ wcel 2108 ∀wral 3061 class class class wbr 5143 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 lecple 17304 Proset cproset 18338 MGalConncmgc 32969 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 df-proset 18340 df-mgc 32971 | 
| This theorem is referenced by: mgcmnt1 32982 mgcmntco 32984 dfmgc2 32986 mgcf1olem1 32991 mgcf1olem2 32992 | 
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