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Theorem paddidm 39843
Description: Projective subspace sum is idempotent. Part of Lemma 16.2 of [MaedaMaeda] p. 68. (Contributed by NM, 13-Jan-2012.)
Hypotheses
Ref Expression
paddidm.s 𝑆 = (PSubSp‘𝐾)
paddidm.p + = (+𝑃𝐾)
Assertion
Ref Expression
paddidm ((𝐾𝐵𝑋𝑆) → (𝑋 + 𝑋) = 𝑋)

Proof of Theorem paddidm
Dummy variables 𝑝 𝑞 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 482 . . . . 5 ((𝐾𝐵𝑋𝑆) → 𝐾𝐵)
2 eqid 2737 . . . . . 6 (Atoms‘𝐾) = (Atoms‘𝐾)
3 paddidm.s . . . . . 6 𝑆 = (PSubSp‘𝐾)
42, 3psubssat 39756 . . . . 5 ((𝐾𝐵𝑋𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
5 eqid 2737 . . . . . 6 (le‘𝐾) = (le‘𝐾)
6 eqid 2737 . . . . . 6 (join‘𝐾) = (join‘𝐾)
7 paddidm.p . . . . . 6 + = (+𝑃𝐾)
85, 6, 2, 7elpadd 39801 . . . . 5 ((𝐾𝐵𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑋 ⊆ (Atoms‘𝐾)) → (𝑝 ∈ (𝑋 + 𝑋) ↔ ((𝑝𝑋𝑝𝑋) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞𝑋𝑟𝑋 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)))))
91, 4, 4, 8syl3anc 1373 . . . 4 ((𝐾𝐵𝑋𝑆) → (𝑝 ∈ (𝑋 + 𝑋) ↔ ((𝑝𝑋𝑝𝑋) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞𝑋𝑟𝑋 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)))))
10 pm1.2 904 . . . . . 6 ((𝑝𝑋𝑝𝑋) → 𝑝𝑋)
1110a1i 11 . . . . 5 ((𝐾𝐵𝑋𝑆) → ((𝑝𝑋𝑝𝑋) → 𝑝𝑋))
125, 6, 2, 3psubspi 39749 . . . . . . 7 (((𝐾𝐵𝑋𝑆𝑝 ∈ (Atoms‘𝐾)) ∧ ∃𝑞𝑋𝑟𝑋 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)) → 𝑝𝑋)
13123exp1 1353 . . . . . 6 (𝐾𝐵 → (𝑋𝑆 → (𝑝 ∈ (Atoms‘𝐾) → (∃𝑞𝑋𝑟𝑋 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟) → 𝑝𝑋))))
1413imp4b 421 . . . . 5 ((𝐾𝐵𝑋𝑆) → ((𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞𝑋𝑟𝑋 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)) → 𝑝𝑋))
1511, 14jaod 860 . . . 4 ((𝐾𝐵𝑋𝑆) → (((𝑝𝑋𝑝𝑋) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞𝑋𝑟𝑋 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟))) → 𝑝𝑋))
169, 15sylbid 240 . . 3 ((𝐾𝐵𝑋𝑆) → (𝑝 ∈ (𝑋 + 𝑋) → 𝑝𝑋))
1716ssrdv 3989 . 2 ((𝐾𝐵𝑋𝑆) → (𝑋 + 𝑋) ⊆ 𝑋)
182, 7sspadd1 39817 . . 3 ((𝐾𝐵𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑋 ⊆ (Atoms‘𝐾)) → 𝑋 ⊆ (𝑋 + 𝑋))
191, 4, 4, 18syl3anc 1373 . 2 ((𝐾𝐵𝑋𝑆) → 𝑋 ⊆ (𝑋 + 𝑋))
2017, 19eqssd 4001 1 ((𝐾𝐵𝑋𝑆) → (𝑋 + 𝑋) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848   = wceq 1540  wcel 2108  wrex 3070  wss 3951   class class class wbr 5143  cfv 6561  (class class class)co 7431  lecple 17304  joincjn 18357  Atomscatm 39264  PSubSpcpsubsp 39498  +𝑃cpadd 39797
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-psubsp 39505  df-padd 39798
This theorem is referenced by:  paddclN  39844  paddss  39847  pmod1i  39850
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