Theorem pclunN 37912

Description: The projective subspace closure of the union of two sets of atoms equals the closure of their projective sum. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
pclun.a	⊢ 𝐴 = (Atoms‘𝐾)
pclun.p	⊢ + = (+𝑃‘𝐾)
pclun.c	⊢ 𝑈 = (PCl‘𝐾)

Assertion

Ref	Expression
pclunN	⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑈‘(𝑋 ∪ 𝑌)) = (𝑈‘(𝑋 + 𝑌)))

Proof of Theorem pclunN

Step	Hyp	Ref	Expression
1		simp1 1135	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → 𝐾 ∈ 𝑉)
2		pclun.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
3		pclun.p	. . . 4 ⊢ + = (+𝑃‘𝐾)
4	2, 3	paddunssN 37822	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 ∪ 𝑌) ⊆ (𝑋 + 𝑌))
5	2, 3	paddssat 37828	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 + 𝑌) ⊆ 𝐴)
6		pclun.c	. . . 4 ⊢ 𝑈 = (PCl‘𝐾)
7	2, 6	pclssN 37908	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ (𝑋 ∪ 𝑌) ⊆ (𝑋 + 𝑌) ∧ (𝑋 + 𝑌) ⊆ 𝐴) → (𝑈‘(𝑋 ∪ 𝑌)) ⊆ (𝑈‘(𝑋 + 𝑌)))
8	1, 4, 5, 7	syl3anc 1370	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑈‘(𝑋 ∪ 𝑌)) ⊆ (𝑈‘(𝑋 + 𝑌)))
9		unss 4118	. . . . . . . . 9 ⊢ ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ↔ (𝑋 ∪ 𝑌) ⊆ 𝐴)
10	9	biimpi 215	. . . . . . . 8 ⊢ ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 ∪ 𝑌) ⊆ 𝐴)
11	10	3adant1 1129	. . . . . . 7 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 ∪ 𝑌) ⊆ 𝐴)
12	2, 6	pclssidN 37909	. . . . . . 7 ⊢ ((𝐾 ∈ 𝑉 ∧ (𝑋 ∪ 𝑌) ⊆ 𝐴) → (𝑋 ∪ 𝑌) ⊆ (𝑈‘(𝑋 ∪ 𝑌)))
13	1, 11, 12	syl2anc 584	. . . . . 6 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 ∪ 𝑌) ⊆ (𝑈‘(𝑋 ∪ 𝑌)))
14		unss 4118	. . . . . 6 ⊢ ((𝑋 ⊆ (𝑈‘(𝑋 ∪ 𝑌)) ∧ 𝑌 ⊆ (𝑈‘(𝑋 ∪ 𝑌))) ↔ (𝑋 ∪ 𝑌) ⊆ (𝑈‘(𝑋 ∪ 𝑌)))
15	13, 14	sylibr 233	. . . . 5 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 ⊆ (𝑈‘(𝑋 ∪ 𝑌)) ∧ 𝑌 ⊆ (𝑈‘(𝑋 ∪ 𝑌))))
16		simp2 1136	. . . . . 6 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → 𝑋 ⊆ 𝐴)
17		simp3 1137	. . . . . 6 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → 𝑌 ⊆ 𝐴)
18		eqid 2738	. . . . . . . 8 ⊢ (PSubSp‘𝐾) = (PSubSp‘𝐾)
19	2, 18, 6	pclclN 37905	. . . . . . 7 ⊢ ((𝐾 ∈ 𝑉 ∧ (𝑋 ∪ 𝑌) ⊆ 𝐴) → (𝑈‘(𝑋 ∪ 𝑌)) ∈ (PSubSp‘𝐾))
20	1, 11, 19	syl2anc 584	. . . . . 6 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑈‘(𝑋 ∪ 𝑌)) ∈ (PSubSp‘𝐾))
21	2, 18, 3	paddss 37859	. . . . . 6 ⊢ ((𝐾 ∈ 𝑉 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ (𝑈‘(𝑋 ∪ 𝑌)) ∈ (PSubSp‘𝐾))) → ((𝑋 ⊆ (𝑈‘(𝑋 ∪ 𝑌)) ∧ 𝑌 ⊆ (𝑈‘(𝑋 ∪ 𝑌))) ↔ (𝑋 + 𝑌) ⊆ (𝑈‘(𝑋 ∪ 𝑌))))
22	1, 16, 17, 20, 21	syl13anc 1371	. . . . 5 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → ((𝑋 ⊆ (𝑈‘(𝑋 ∪ 𝑌)) ∧ 𝑌 ⊆ (𝑈‘(𝑋 ∪ 𝑌))) ↔ (𝑋 + 𝑌) ⊆ (𝑈‘(𝑋 ∪ 𝑌))))
23	15, 22	mpbid 231	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 + 𝑌) ⊆ (𝑈‘(𝑋 ∪ 𝑌)))
24	2, 18	psubssat 37768	. . . . 5 ⊢ ((𝐾 ∈ 𝑉 ∧ (𝑈‘(𝑋 ∪ 𝑌)) ∈ (PSubSp‘𝐾)) → (𝑈‘(𝑋 ∪ 𝑌)) ⊆ 𝐴)
25	1, 20, 24	syl2anc 584	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑈‘(𝑋 ∪ 𝑌)) ⊆ 𝐴)
26	2, 6	pclssN 37908	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ (𝑋 + 𝑌) ⊆ (𝑈‘(𝑋 ∪ 𝑌)) ∧ (𝑈‘(𝑋 ∪ 𝑌)) ⊆ 𝐴) → (𝑈‘(𝑋 + 𝑌)) ⊆ (𝑈‘(𝑈‘(𝑋 ∪ 𝑌))))
27	1, 23, 25, 26	syl3anc 1370	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑈‘(𝑋 + 𝑌)) ⊆ (𝑈‘(𝑈‘(𝑋 ∪ 𝑌))))
28	18, 6	pclidN 37910	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ (𝑈‘(𝑋 ∪ 𝑌)) ∈ (PSubSp‘𝐾)) → (𝑈‘(𝑈‘(𝑋 ∪ 𝑌))) = (𝑈‘(𝑋 ∪ 𝑌)))
29	1, 20, 28	syl2anc 584	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑈‘(𝑈‘(𝑋 ∪ 𝑌))) = (𝑈‘(𝑋 ∪ 𝑌)))
30	27, 29	sseqtrd 3961	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑈‘(𝑋 + 𝑌)) ⊆ (𝑈‘(𝑋 ∪ 𝑌)))
31	8, 30	eqssd 3938	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑈‘(𝑋 ∪ 𝑌)) = (𝑈‘(𝑋 + 𝑌)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ∪ cun 3885   ⊆ wss 3887  ‘cfv 6433  (class class class)co 7275  Atomscatm 37277  PSubSpcpsubsp 37510  +_𝑃cpadd 37809  PClcpclN 37901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-psubsp 37517  df-padd 37810  df-pclN 37902

This theorem is referenced by:  pclun2N  37913