Theorem pclunN 39881

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 39791	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 ∪ 𝑌) ⊆ (𝑋 + 𝑌))
5	2, 3	paddssat 39797	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 + 𝑌) ⊆ 𝐴)
6		pclun.c	. . . 4 ⊢ 𝑈 = (PCl‘𝐾)
7	2, 6	pclssN 39877	. . 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 4200	. . . . . . . . 9 ⊢ ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ↔ (𝑋 ∪ 𝑌) ⊆ 𝐴)
10	9	biimpi 216	. . . . . . . 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 39878	. . . . . . 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 4200	. . . . . 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 1136	. . . . . 6 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → 𝑋 ⊆ 𝐴)
17		simp3 1137	. . . . . 6 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → 𝑌 ⊆ 𝐴)
18		eqid 2735	. . . . . . . 8 ⊢ (PSubSp‘𝐾) = (PSubSp‘𝐾)
19	2, 18, 6	pclclN 39874	. . . . . . 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 39828	. . . . . 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 232	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 + 𝑌) ⊆ (𝑈‘(𝑋 ∪ 𝑌)))
24	2, 18	psubssat 39737	. . . . 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 39877	. . . 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 39879	. . . 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 4036	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑈‘(𝑋 + 𝑌)) ⊆ (𝑈‘(𝑋 ∪ 𝑌)))
31	8, 30	eqssd 4013	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑈‘(𝑋 ∪ 𝑌)) = (𝑈‘(𝑋 + 𝑌)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106   ∪ cun 3961   ⊆ wss 3963  ‘cfv 6563  (class class class)co 7431  Atomscatm 39245  PSubSpcpsubsp 39479  +_𝑃cpadd 39778  PClcpclN 39870

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-psubsp 39486  df-padd 39779  df-pclN 39871

This theorem is referenced by:  pclun2N  39882