![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > ispsubsp | Structured version Visualization version GIF version |
Description: The predicate "is a projective subspace". (Contributed by NM, 2-Oct-2011.) |
Ref | Expression |
---|---|
psubspset.l | ⊢ ≤ = (le‘𝐾) |
psubspset.j | ⊢ ∨ = (join‘𝐾) |
psubspset.a | ⊢ 𝐴 = (Atoms‘𝐾) |
psubspset.s | ⊢ 𝑆 = (PSubSp‘𝐾) |
Ref | Expression |
---|---|
ispsubsp | ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑆 ↔ (𝑋 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑋 ∀𝑞 ∈ 𝑋 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑋)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psubspset.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
2 | psubspset.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
3 | psubspset.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | psubspset.s | . . . 4 ⊢ 𝑆 = (PSubSp‘𝐾) | |
5 | 1, 2, 3, 4 | psubspset 39082 | . . 3 ⊢ (𝐾 ∈ 𝐷 → 𝑆 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑥 ∀𝑞 ∈ 𝑥 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑥))}) |
6 | 5 | eleq2d 2818 | . 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑆 ↔ 𝑋 ∈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑥 ∀𝑞 ∈ 𝑥 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑥))})) |
7 | 3 | fvexi 6905 | . . . . 5 ⊢ 𝐴 ∈ V |
8 | 7 | ssex 5321 | . . . 4 ⊢ (𝑋 ⊆ 𝐴 → 𝑋 ∈ V) |
9 | 8 | adantr 480 | . . 3 ⊢ ((𝑋 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑋 ∀𝑞 ∈ 𝑋 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑋)) → 𝑋 ∈ V) |
10 | sseq1 4007 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ⊆ 𝐴 ↔ 𝑋 ⊆ 𝐴)) | |
11 | eleq2 2821 | . . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑟 ∈ 𝑥 ↔ 𝑟 ∈ 𝑋)) | |
12 | 11 | imbi2d 340 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → ((𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑥) ↔ (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑋))) |
13 | 12 | ralbidv 3176 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑥) ↔ ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑋))) |
14 | 13 | raleqbi1dv 3332 | . . . . 5 ⊢ (𝑥 = 𝑋 → (∀𝑞 ∈ 𝑥 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑥) ↔ ∀𝑞 ∈ 𝑋 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑋))) |
15 | 14 | raleqbi1dv 3332 | . . . 4 ⊢ (𝑥 = 𝑋 → (∀𝑝 ∈ 𝑥 ∀𝑞 ∈ 𝑥 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑥) ↔ ∀𝑝 ∈ 𝑋 ∀𝑞 ∈ 𝑋 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑋))) |
16 | 10, 15 | anbi12d 630 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑥 ∀𝑞 ∈ 𝑥 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑥)) ↔ (𝑋 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑋 ∀𝑞 ∈ 𝑋 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑋)))) |
17 | 9, 16 | elab3 3676 | . 2 ⊢ (𝑋 ∈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑥 ∀𝑞 ∈ 𝑥 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑥))} ↔ (𝑋 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑋 ∀𝑞 ∈ 𝑋 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑋))) |
18 | 6, 17 | bitrdi 287 | 1 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑆 ↔ (𝑋 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑋 ∀𝑞 ∈ 𝑋 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑋)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 {cab 2708 ∀wral 3060 Vcvv 3473 ⊆ wss 3948 class class class wbr 5148 ‘cfv 6543 (class class class)co 7412 lecple 17211 joincjn 18274 Atomscatm 38600 PSubSpcpsubsp 38834 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7415 df-psubsp 38841 |
This theorem is referenced by: ispsubsp2 39084 0psubN 39087 snatpsubN 39088 linepsubN 39090 atpsubN 39091 psubssat 39092 pmapsub 39106 pclclN 39229 pclfinN 39238 |
Copyright terms: Public domain | W3C validator |