Theorem pclunN 39922

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 1136	. . 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 39832	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 ∪ 𝑌) ⊆ (𝑋 + 𝑌))
5	2, 3	paddssat 39838	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 + 𝑌) ⊆ 𝐴)
6		pclun.c	. . . 4 ⊢ 𝑈 = (PCl‘𝐾)
7	2, 6	pclssN 39918	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ (𝑋 ∪ 𝑌) ⊆ (𝑋 + 𝑌) ∧ (𝑋 + 𝑌) ⊆ 𝐴) → (𝑈‘(𝑋 ∪ 𝑌)) ⊆ (𝑈‘(𝑋 + 𝑌)))
8	1, 4, 5, 7	syl3anc 1373	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑈‘(𝑋 ∪ 𝑌)) ⊆ (𝑈‘(𝑋 + 𝑌)))
9		unss 4170	. . . . . . . . 9 ⊢ ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ↔ (𝑋 ∪ 𝑌) ⊆ 𝐴)
10	9	biimpi 216	. . . . . . . 8 ⊢ ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 ∪ 𝑌) ⊆ 𝐴)
11	10	3adant1 1130	. . . . . . 7 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 ∪ 𝑌) ⊆ 𝐴)
12	2, 6	pclssidN 39919	. . . . . . 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 4170	. . . . . 6 ⊢ ((𝑋 ⊆ (𝑈‘(𝑋 ∪ 𝑌)) ∧ 𝑌 ⊆ (𝑈‘(𝑋 ∪ 𝑌))) ↔ (𝑋 ∪ 𝑌) ⊆ (𝑈‘(𝑋 ∪ 𝑌)))
15	13, 14	sylibr 234	. . . . 5 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 ⊆ (𝑈‘(𝑋 ∪ 𝑌)) ∧ 𝑌 ⊆ (𝑈‘(𝑋 ∪ 𝑌))))
16		simp2 1137	. . . . . 6 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → 𝑋 ⊆ 𝐴)
17		simp3 1138	. . . . . 6 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → 𝑌 ⊆ 𝐴)
18		eqid 2736	. . . . . . . 8 ⊢ (PSubSp‘𝐾) = (PSubSp‘𝐾)
19	2, 18, 6	pclclN 39915	. . . . . . 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 39869	. . . . . 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 1374	. . . . 5 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → ((𝑋 ⊆ (𝑈‘(𝑋 ∪ 𝑌)) ∧ 𝑌 ⊆ (𝑈‘(𝑋 ∪ 𝑌))) ↔ (𝑋 + 𝑌) ⊆ (𝑈‘(𝑋 ∪ 𝑌))))
23	15, 22	mpbid 232	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 + 𝑌) ⊆ (𝑈‘(𝑋 ∪ 𝑌)))
24	2, 18	psubssat 39778	. . . . 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 39918	. . . 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 1373	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑈‘(𝑋 + 𝑌)) ⊆ (𝑈‘(𝑈‘(𝑋 ∪ 𝑌))))
28	18, 6	pclidN 39920	. . . 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 4000	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑈‘(𝑋 + 𝑌)) ⊆ (𝑈‘(𝑋 ∪ 𝑌)))
31	8, 30	eqssd 3981	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑈‘(𝑋 ∪ 𝑌)) = (𝑈‘(𝑋 + 𝑌)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ∪ cun 3929   ⊆ wss 3931  ‘cfv 6536  (class class class)co 7410  Atomscatm 39286  PSubSpcpsubsp 39520  +_𝑃cpadd 39819  PClcpclN 39911

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7993  df-2nd 7994  df-psubsp 39527  df-padd 39820  df-pclN 39912

This theorem is referenced by:  pclun2N  39923