Theorem paddclN 40541

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 1152	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝐾 ∈ HL)
2		eqid 2769	. . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
3		paddidm.s	. . . . 5 ⊢ 𝑆 = (PSubSp‘𝐾)
4	2, 3	psubssat 40453	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
5	4	3adant3 1148	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
6	2, 3	psubssat 40453	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝑆) → 𝑌 ⊆ (Atoms‘𝐾))
7	6	3adant2 1147	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝑌 ⊆ (Atoms‘𝐾))
8		paddidm.p	. . . 4 ⊢ + = (+𝑃‘𝐾)
9	2, 8	paddssat 40513	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾)) → (𝑋 + 𝑌) ⊆ (Atoms‘𝐾))
10	1, 5, 7, 9	syl3anc 1396	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) ⊆ (Atoms‘𝐾))
11		olc 881	. . . . 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 2769	. . . . . . . 8 ⊢ (le‘𝐾) = (le‘𝐾)
13		eqid 2769	. . . . . . . 8 ⊢ (join‘𝐾) = (join‘𝐾)
14	12, 13, 2, 8	elpadd 40498	. . . . . . 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 1396	. . . . . 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 40539	. . . . . . . . 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 1404	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → ((𝑋 + 𝑌) + (𝑋 + 𝑌)) = ((𝑋 + 𝑋) + (𝑌 + 𝑌)))
18	3, 8	paddidm 40540	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆) → (𝑋 + 𝑋) = 𝑋)
19	18	3adant3 1148	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑋) = 𝑋)
20	3, 8	paddidm 40540	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝑆) → (𝑌 + 𝑌) = 𝑌)
21	20	3adant2 1147	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑌 + 𝑌) = 𝑌)
22	19, 21	oveq12d 7429	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → ((𝑋 + 𝑋) + (𝑌 + 𝑌)) = (𝑋 + 𝑌))
23	17, 22	eqtrd 2804	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → ((𝑋 + 𝑌) + (𝑋 + 𝑌)) = (𝑋 + 𝑌))
24	23	eleq2d 2855	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑝 ∈ ((𝑋 + 𝑌) + (𝑋 + 𝑌)) ↔ 𝑝 ∈ (𝑋 + 𝑌)))
25	15, 24	bitr3d 284	. . . . 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 247	. . . 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 420	. . 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 3162	. 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 40445	. . 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 1149	. 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 725	1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095   ⊆ wss 3911   class class class wbr 5111  ‘cfv 6537  (class class class)co 7411  lecple 17317  joincjn 18367  Atomscatm 39962  HLchlt 40049  PSubSpcpsubsp 40195  +_𝑃cpadd 40494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3375  df-reu 3376  df-rab 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7986  df-2nd 7987  df-proset 18350  df-poset 18369  df-plt 18384  df-lub 18400  df-glb 18401  df-join 18402  df-meet 18403  df-p0 18479  df-lat 18488  df-clat 18555  df-oposet 39875  df-ol 39877  df-oml 39878  df-covers 39965  df-ats 39966  df-atl 39997  df-cvlat 40021  df-hlat 40050  df-psubsp 40202  df-padd 40495

This theorem is referenced by:  pmodl42N  40550  pclun2N  40598