| Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > pclun2N | Structured version Visualization version GIF version | ||
| Description: The projective subspace closure of the union of two subspaces equals their projective sum. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| pclun2.s | ⊢ 𝑆 = (PSubSp‘𝐾) |
| pclun2.p | ⊢ + = (+𝑃‘𝐾) |
| pclun2.c | ⊢ 𝑈 = (PCl‘𝐾) |
| Ref | Expression |
|---|---|
| pclun2N | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑈‘(𝑋 ∪ 𝑌)) = (𝑋 + 𝑌)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simp1 1137 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝐾 ∈ HL) | |
| 2 | eqid 2737 | . . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
| 3 | pclun2.s | . . . . 5 ⊢ 𝑆 = (PSubSp‘𝐾) | |
| 4 | 2, 3 | psubssat 39756 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆) → 𝑋 ⊆ (Atoms‘𝐾)) |
| 5 | 4 | 3adant3 1133 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝑋 ⊆ (Atoms‘𝐾)) |
| 6 | 2, 3 | psubssat 39756 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝑆) → 𝑌 ⊆ (Atoms‘𝐾)) |
| 7 | 6 | 3adant2 1132 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝑌 ⊆ (Atoms‘𝐾)) |
| 8 | pclun2.p | . . . 4 ⊢ + = (+𝑃‘𝐾) | |
| 9 | pclun2.c | . . . 4 ⊢ 𝑈 = (PCl‘𝐾) | |
| 10 | 2, 8, 9 | pclunN 39900 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾)) → (𝑈‘(𝑋 ∪ 𝑌)) = (𝑈‘(𝑋 + 𝑌))) |
| 11 | 1, 5, 7, 10 | syl3anc 1373 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑈‘(𝑋 ∪ 𝑌)) = (𝑈‘(𝑋 + 𝑌))) |
| 12 | 3, 8 | paddclN 39844 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆) |
| 13 | 3, 9 | pclidN 39898 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 + 𝑌) ∈ 𝑆) → (𝑈‘(𝑋 + 𝑌)) = (𝑋 + 𝑌)) |
| 14 | 1, 12, 13 | syl2anc 584 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑈‘(𝑋 + 𝑌)) = (𝑋 + 𝑌)) |
| 15 | 11, 14 | eqtrd 2777 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑈‘(𝑋 ∪ 𝑌)) = (𝑋 + 𝑌)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∪ cun 3949 ⊆ wss 3951 ‘cfv 6561 (class class class)co 7431 Atomscatm 39264 HLchlt 39351 PSubSpcpsubsp 39498 +𝑃cpadd 39797 PClcpclN 39889 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-proset 18340 df-poset 18359 df-plt 18375 df-lub 18391 df-glb 18392 df-join 18393 df-meet 18394 df-p0 18470 df-lat 18477 df-clat 18544 df-oposet 39177 df-ol 39179 df-oml 39180 df-covers 39267 df-ats 39268 df-atl 39299 df-cvlat 39323 df-hlat 39352 df-psubsp 39505 df-padd 39798 df-pclN 39890 |
| This theorem is referenced by: (None) |
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