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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ispsubclN | Structured version Visualization version GIF version |
Description: The predicate "is a closed projective subspace". (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
psubclset.a | ⊢ 𝐴 = (Atoms‘𝐾) |
psubclset.p | ⊢ ⊥ = (⊥𝑃‘𝐾) |
psubclset.c | ⊢ 𝐶 = (PSubCl‘𝐾) |
Ref | Expression |
---|---|
ispsubclN | ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝐶 ↔ (𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psubclset.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
2 | psubclset.p | . . . 4 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
3 | psubclset.c | . . . 4 ⊢ 𝐶 = (PSubCl‘𝐾) | |
4 | 1, 2, 3 | psubclsetN 35540 | . . 3 ⊢ (𝐾 ∈ 𝐷 → 𝐶 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)}) |
5 | 4 | eleq2d 2716 | . 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝐶 ↔ 𝑋 ∈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)})) |
6 | fvex 6239 | . . . . . 6 ⊢ (Atoms‘𝐾) ∈ V | |
7 | 1, 6 | eqeltri 2726 | . . . . 5 ⊢ 𝐴 ∈ V |
8 | 7 | ssex 4835 | . . . 4 ⊢ (𝑋 ⊆ 𝐴 → 𝑋 ∈ V) |
9 | 8 | adantr 480 | . . 3 ⊢ ((𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → 𝑋 ∈ V) |
10 | sseq1 3659 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ⊆ 𝐴 ↔ 𝑋 ⊆ 𝐴)) | |
11 | fveq2 6229 | . . . . . 6 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘𝑥) = ( ⊥ ‘𝑋)) | |
12 | 11 | fveq2d 6233 | . . . . 5 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘( ⊥ ‘𝑥)) = ( ⊥ ‘( ⊥ ‘𝑋))) |
13 | id 22 | . . . . 5 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋) | |
14 | 12, 13 | eqeq12d 2666 | . . . 4 ⊢ (𝑥 = 𝑋 → (( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥 ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)) |
15 | 10, 14 | anbi12d 747 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥) ↔ (𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋))) |
16 | 9, 15 | elab3 3390 | . 2 ⊢ (𝑋 ∈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)} ↔ (𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)) |
17 | 5, 16 | syl6bb 276 | 1 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝐶 ↔ (𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1523 ∈ wcel 2030 {cab 2637 Vcvv 3231 ⊆ wss 3607 ‘cfv 5926 Atomscatm 34868 ⊥𝑃cpolN 35506 PSubClcpscN 35538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-iota 5889 df-fun 5928 df-fv 5934 df-psubclN 35539 |
This theorem is referenced by: psubcliN 35542 psubcli2N 35543 0psubclN 35547 1psubclN 35548 atpsubclN 35549 pmapsubclN 35550 ispsubcl2N 35551 osumclN 35571 pexmidN 35573 pexmidlem6N 35579 |
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