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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 33069 | . . . . . 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 767 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
| 12 | mgccole1.2 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
| 13 | 11, 12 | ffvelcdmd 7030 | . . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ 𝐵) |
| 14 | 4, 6 | prsref 18221 | . . 3 ⊢ ((𝑊 ∈ Proset ∧ (𝐹‘𝑋) ∈ 𝐵) → (𝐹‘𝑋) ≲ (𝐹‘𝑋)) |
| 15 | 1, 13, 14 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐹‘𝑋) ≲ (𝐹‘𝑋)) |
| 16 | fveq2 6834 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋)) | |
| 17 | 16 | breq1d 5108 | . . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) ≲ 𝑦 ↔ (𝐹‘𝑋) ≲ 𝑦)) |
| 18 | breq1 5101 | . . . . . 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 3159 | . . . 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 3577 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ((𝐹‘𝑋) ≲ 𝑦 ↔ 𝑋 ≤ (𝐺‘𝑦))) |
| 23 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑋)) → 𝑦 = (𝐹‘𝑋)) | |
| 24 | 23 | breq2d 5110 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑋)) → ((𝐹‘𝑋) ≲ 𝑦 ↔ (𝐹‘𝑋) ≲ (𝐹‘𝑋))) |
| 25 | 23 | fveq2d 6838 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑋)) → (𝐺‘𝑦) = (𝐺‘(𝐹‘𝑋))) |
| 26 | 25 | breq2d 5110 | . . . . 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 3568 | . . 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 1541 ∈ wcel 2113 ∀wral 3051 class class class wbr 5098 ⟶wf 6488 ‘cfv 6492 (class class class)co 7358 Basecbs 17136 lecple 17184 Proset cproset 18215 MGalConncmgc 33061 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-br 5099 df-opab 5161 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-fv 6500 df-ov 7361 df-oprab 7362 df-mpo 7363 df-map 8765 df-proset 18217 df-mgc 33063 |
| This theorem is referenced by: mgcmnt1 33074 mgcmntco 33076 dfmgc2 33078 mgcf1olem1 33083 mgcf1olem2 33084 |
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