Theorem paddclN 38582

Description: The projective sum of two subspaces is a subspace. Part of Lemma 16.2 of [MaedaMaeda] p. 68. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
paddidm.s	⊢ 𝑆 = (PSubSp‘𝐾)
paddidm.p	⊢ + = (+𝑃‘𝐾)

Assertion

Ref	Expression
paddclN	⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆)

Proof of Theorem paddclN

Dummy variables 𝑝 𝑞 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1136	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝐾 ∈ HL)
2		eqid 2732	. . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
3		paddidm.s	. . . . 5 ⊢ 𝑆 = (PSubSp‘𝐾)
4	2, 3	psubssat 38494	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
5	4	3adant3 1132	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
6	2, 3	psubssat 38494	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝑆) → 𝑌 ⊆ (Atoms‘𝐾))
7	6	3adant2 1131	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝑌 ⊆ (Atoms‘𝐾))
8		paddidm.p	. . . 4 ⊢ + = (+𝑃‘𝐾)
9	2, 8	paddssat 38554	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾)) → (𝑋 + 𝑌) ⊆ (Atoms‘𝐾))
10	1, 5, 7, 9	syl3anc 1371	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) ⊆ (Atoms‘𝐾))
11		olc 866	. . . . 5 ⊢ ((𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞 ∈ (𝑋 + 𝑌)∃𝑟 ∈ (𝑋 + 𝑌)𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)) → ((𝑝 ∈ (𝑋 + 𝑌) ∨ 𝑝 ∈ (𝑋 + 𝑌)) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞 ∈ (𝑋 + 𝑌)∃𝑟 ∈ (𝑋 + 𝑌)𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟))))
12		eqid 2732	. . . . . . . 8 ⊢ (le‘𝐾) = (le‘𝐾)
13		eqid 2732	. . . . . . . 8 ⊢ (join‘𝐾) = (join‘𝐾)
14	12, 13, 2, 8	elpadd 38539	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 + 𝑌) ⊆ (Atoms‘𝐾) ∧ (𝑋 + 𝑌) ⊆ (Atoms‘𝐾)) → (𝑝 ∈ ((𝑋 + 𝑌) + (𝑋 + 𝑌)) ↔ ((𝑝 ∈ (𝑋 + 𝑌) ∨ 𝑝 ∈ (𝑋 + 𝑌)) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞 ∈ (𝑋 + 𝑌)∃𝑟 ∈ (𝑋 + 𝑌)𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)))))
15	1, 10, 10, 14	syl3anc 1371	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑝 ∈ ((𝑋 + 𝑌) + (𝑋 + 𝑌)) ↔ ((𝑝 ∈ (𝑋 + 𝑌) ∨ 𝑝 ∈ (𝑋 + 𝑌)) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞 ∈ (𝑋 + 𝑌)∃𝑟 ∈ (𝑋 + 𝑌)𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)))))
16	2, 8	padd4N 38580	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾)) ∧ (𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾))) → ((𝑋 + 𝑌) + (𝑋 + 𝑌)) = ((𝑋 + 𝑋) + (𝑌 + 𝑌)))
17	1, 5, 7, 5, 7, 16	syl122anc 1379	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → ((𝑋 + 𝑌) + (𝑋 + 𝑌)) = ((𝑋 + 𝑋) + (𝑌 + 𝑌)))
18	3, 8	paddidm 38581	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆) → (𝑋 + 𝑋) = 𝑋)
19	18	3adant3 1132	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑋) = 𝑋)
20	3, 8	paddidm 38581	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝑆) → (𝑌 + 𝑌) = 𝑌)
21	20	3adant2 1131	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑌 + 𝑌) = 𝑌)
22	19, 21	oveq12d 7412	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → ((𝑋 + 𝑋) + (𝑌 + 𝑌)) = (𝑋 + 𝑌))
23	17, 22	eqtrd 2772	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → ((𝑋 + 𝑌) + (𝑋 + 𝑌)) = (𝑋 + 𝑌))
24	23	eleq2d 2819	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑝 ∈ ((𝑋 + 𝑌) + (𝑋 + 𝑌)) ↔ 𝑝 ∈ (𝑋 + 𝑌)))
25	15, 24	bitr3d 280	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (((𝑝 ∈ (𝑋 + 𝑌) ∨ 𝑝 ∈ (𝑋 + 𝑌)) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞 ∈ (𝑋 + 𝑌)∃𝑟 ∈ (𝑋 + 𝑌)𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟))) ↔ 𝑝 ∈ (𝑋 + 𝑌)))
26	11, 25	imbitrid 243	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → ((𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞 ∈ (𝑋 + 𝑌)∃𝑟 ∈ (𝑋 + 𝑌)𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)) → 𝑝 ∈ (𝑋 + 𝑌)))
27	26	expd 416	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑝 ∈ (Atoms‘𝐾) → (∃𝑞 ∈ (𝑋 + 𝑌)∃𝑟 ∈ (𝑋 + 𝑌)𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟) → 𝑝 ∈ (𝑋 + 𝑌))))
28	27	ralrimiv 3145	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → ∀𝑝 ∈ (Atoms‘𝐾)(∃𝑞 ∈ (𝑋 + 𝑌)∃𝑟 ∈ (𝑋 + 𝑌)𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟) → 𝑝 ∈ (𝑋 + 𝑌)))
29	12, 13, 2, 3	ispsubsp2 38486	. . 3 ⊢ (𝐾 ∈ HL → ((𝑋 + 𝑌) ∈ 𝑆 ↔ ((𝑋 + 𝑌) ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ (Atoms‘𝐾)(∃𝑞 ∈ (𝑋 + 𝑌)∃𝑟 ∈ (𝑋 + 𝑌)𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟) → 𝑝 ∈ (𝑋 + 𝑌)))))
30	29	3ad2ant1 1133	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → ((𝑋 + 𝑌) ∈ 𝑆 ↔ ((𝑋 + 𝑌) ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ (Atoms‘𝐾)(∃𝑞 ∈ (𝑋 + 𝑌)∃𝑟 ∈ (𝑋 + 𝑌)𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟) → 𝑝 ∈ (𝑋 + 𝑌)))))
31	10, 28, 30	mpbir2and 711	1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3061  ∃wrex 3070   ⊆ wss 3945   class class class wbr 5142  ‘cfv 6533  (class class class)co 7394  lecple 17188  joincjn 18248  Atomscatm 38002  HLchlt 38089  PSubSpcpsubsp 38236  +_𝑃cpadd 38535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7350  df-ov 7397  df-oprab 7398  df-mpo 7399  df-1st 7959  df-2nd 7960  df-proset 18232  df-poset 18250  df-plt 18267  df-lub 18283  df-glb 18284  df-join 18285  df-meet 18286  df-p0 18362  df-lat 18369  df-clat 18436  df-oposet 37915  df-ol 37917  df-oml 37918  df-covers 38005  df-ats 38006  df-atl 38037  df-cvlat 38061  df-hlat 38090  df-psubsp 38243  df-padd 38536

This theorem is referenced by:  pmodl42N  38591  pclun2N  38639