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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 37074 | . . 3 ⊢ (𝐾 ∈ 𝐷 → 𝐶 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)}) |
5 | 4 | eleq2d 2900 | . 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝐶 ↔ 𝑋 ∈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)})) |
6 | 1 | fvexi 6686 | . . . . 5 ⊢ 𝐴 ∈ V |
7 | 6 | ssex 5227 | . . . 4 ⊢ (𝑋 ⊆ 𝐴 → 𝑋 ∈ V) |
8 | 7 | adantr 483 | . . 3 ⊢ ((𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → 𝑋 ∈ V) |
9 | sseq1 3994 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ⊆ 𝐴 ↔ 𝑋 ⊆ 𝐴)) | |
10 | 2fveq3 6677 | . . . . 5 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘( ⊥ ‘𝑥)) = ( ⊥ ‘( ⊥ ‘𝑋))) | |
11 | id 22 | . . . . 5 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋) | |
12 | 10, 11 | eqeq12d 2839 | . . . 4 ⊢ (𝑥 = 𝑋 → (( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥 ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)) |
13 | 9, 12 | anbi12d 632 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥) ↔ (𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋))) |
14 | 8, 13 | elab3 3676 | . 2 ⊢ (𝑋 ∈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)} ↔ (𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)) |
15 | 5, 14 | syl6bb 289 | 1 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝐶 ↔ (𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {cab 2801 Vcvv 3496 ⊆ wss 3938 ‘cfv 6357 Atomscatm 36401 ⊥𝑃cpolN 37040 PSubClcpscN 37072 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-psubclN 37073 |
This theorem is referenced by: psubcliN 37076 psubcli2N 37077 0psubclN 37081 1psubclN 37082 atpsubclN 37083 pmapsubclN 37084 ispsubcl2N 37085 osumclN 37105 pexmidN 37107 pexmidlem6N 37113 |
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