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Mirrors > Home > MPE Home > Th. List > Mathboxes > elpcliN | Structured version Visualization version GIF version |
Description: Implication of membership in the projective subspace closure function. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
elpcli.s | ⊢ 𝑆 = (PSubSp‘𝐾) |
elpcli.c | ⊢ 𝑈 = (PCl‘𝐾) |
Ref | Expression |
---|---|
elpcliN | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) ∧ 𝑄 ∈ (𝑈‘𝑋)) → 𝑄 ∈ 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1128 | . . . . . 6 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) → 𝐾 ∈ 𝑉) | |
2 | simp2 1129 | . . . . . . 7 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) → 𝑋 ⊆ 𝑌) | |
3 | eqid 2818 | . . . . . . . . 9 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
4 | elpcli.s | . . . . . . . . 9 ⊢ 𝑆 = (PSubSp‘𝐾) | |
5 | 3, 4 | psubssat 36770 | . . . . . . . 8 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑌 ∈ 𝑆) → 𝑌 ⊆ (Atoms‘𝐾)) |
6 | 5 | 3adant2 1123 | . . . . . . 7 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) → 𝑌 ⊆ (Atoms‘𝐾)) |
7 | 2, 6 | sstrd 3974 | . . . . . 6 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) → 𝑋 ⊆ (Atoms‘𝐾)) |
8 | elpcli.c | . . . . . . 7 ⊢ 𝑈 = (PCl‘𝐾) | |
9 | 3, 4, 8 | pclvalN 36906 | . . . . . 6 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ (Atoms‘𝐾)) → (𝑈‘𝑋) = ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑧}) |
10 | 1, 7, 9 | syl2anc 584 | . . . . 5 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) → (𝑈‘𝑋) = ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑧}) |
11 | 10 | eleq2d 2895 | . . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) → (𝑄 ∈ (𝑈‘𝑋) ↔ 𝑄 ∈ ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑧})) |
12 | elintrabg 4880 | . . . . 5 ⊢ (𝑄 ∈ ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑧} → (𝑄 ∈ ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑧} ↔ ∀𝑧 ∈ 𝑆 (𝑋 ⊆ 𝑧 → 𝑄 ∈ 𝑧))) | |
13 | 12 | ibi 268 | . . . 4 ⊢ (𝑄 ∈ ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑧} → ∀𝑧 ∈ 𝑆 (𝑋 ⊆ 𝑧 → 𝑄 ∈ 𝑧)) |
14 | 11, 13 | syl6bi 254 | . . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) → (𝑄 ∈ (𝑈‘𝑋) → ∀𝑧 ∈ 𝑆 (𝑋 ⊆ 𝑧 → 𝑄 ∈ 𝑧))) |
15 | sseq2 3990 | . . . . . . . 8 ⊢ (𝑧 = 𝑌 → (𝑋 ⊆ 𝑧 ↔ 𝑋 ⊆ 𝑌)) | |
16 | eleq2 2898 | . . . . . . . 8 ⊢ (𝑧 = 𝑌 → (𝑄 ∈ 𝑧 ↔ 𝑄 ∈ 𝑌)) | |
17 | 15, 16 | imbi12d 346 | . . . . . . 7 ⊢ (𝑧 = 𝑌 → ((𝑋 ⊆ 𝑧 → 𝑄 ∈ 𝑧) ↔ (𝑋 ⊆ 𝑌 → 𝑄 ∈ 𝑌))) |
18 | 17 | rspccv 3617 | . . . . . 6 ⊢ (∀𝑧 ∈ 𝑆 (𝑋 ⊆ 𝑧 → 𝑄 ∈ 𝑧) → (𝑌 ∈ 𝑆 → (𝑋 ⊆ 𝑌 → 𝑄 ∈ 𝑌))) |
19 | 18 | com13 88 | . . . . 5 ⊢ (𝑋 ⊆ 𝑌 → (𝑌 ∈ 𝑆 → (∀𝑧 ∈ 𝑆 (𝑋 ⊆ 𝑧 → 𝑄 ∈ 𝑧) → 𝑄 ∈ 𝑌))) |
20 | 19 | imp 407 | . . . 4 ⊢ ((𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) → (∀𝑧 ∈ 𝑆 (𝑋 ⊆ 𝑧 → 𝑄 ∈ 𝑧) → 𝑄 ∈ 𝑌)) |
21 | 20 | 3adant1 1122 | . . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) → (∀𝑧 ∈ 𝑆 (𝑋 ⊆ 𝑧 → 𝑄 ∈ 𝑧) → 𝑄 ∈ 𝑌)) |
22 | 14, 21 | syld 47 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) → (𝑄 ∈ (𝑈‘𝑋) → 𝑄 ∈ 𝑌)) |
23 | 22 | imp 407 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) ∧ 𝑄 ∈ (𝑈‘𝑋)) → 𝑄 ∈ 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∀wral 3135 {crab 3139 ⊆ wss 3933 ∩ cint 4867 ‘cfv 6348 Atomscatm 36279 PSubSpcpsubsp 36512 PClcpclN 36903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-psubsp 36519 df-pclN 36904 |
This theorem is referenced by: pclfinclN 36966 |
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