Theorem pclunN 39899

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 39809	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 ∪ 𝑌) ⊆ (𝑋 + 𝑌))
5	2, 3	paddssat 39815	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 + 𝑌) ⊆ 𝐴)
6		pclun.c	. . . 4 ⊢ 𝑈 = (PCl‘𝐾)
7	2, 6	pclssN 39895	. . 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 4156	. . . . . . . . 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 39896	. . . . . . 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 4156	. . . . . 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 2730	. . . . . . . 8 ⊢ (PSubSp‘𝐾) = (PSubSp‘𝐾)
19	2, 18, 6	pclclN 39892	. . . . . . 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 39846	. . . . . 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 39755	. . . . 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 39895	. . . 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 39897	. . . 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 3986	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑈‘(𝑋 + 𝑌)) ⊆ (𝑈‘(𝑋 ∪ 𝑌)))
31	8, 30	eqssd 3967	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 3915   ⊆ wss 3917  ‘cfv 6514  (class class class)co 7390  Atomscatm 39263  PSubSpcpsubsp 39497  +_𝑃cpadd 39796  PClcpclN 39888

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-psubsp 39504  df-padd 39797  df-pclN 39889

This theorem is referenced by:  pclun2N  39900