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Theorem paddidm 38700
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 483 . . . . 5 ((𝐾𝐵𝑋𝑆) → 𝐾𝐵)
2 eqid 2732 . . . . . 6 (Atoms‘𝐾) = (Atoms‘𝐾)
3 paddidm.s . . . . . 6 𝑆 = (PSubSp‘𝐾)
42, 3psubssat 38613 . . . . 5 ((𝐾𝐵𝑋𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
5 eqid 2732 . . . . . 6 (le‘𝐾) = (le‘𝐾)
6 eqid 2732 . . . . . 6 (join‘𝐾) = (join‘𝐾)
7 paddidm.p . . . . . 6 + = (+𝑃𝐾)
85, 6, 2, 7elpadd 38658 . . . . 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 1371 . . . 4 ((𝐾𝐵𝑋𝑆) → (𝑝 ∈ (𝑋 + 𝑋) ↔ ((𝑝𝑋𝑝𝑋) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞𝑋𝑟𝑋 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)))))
10 pm1.2 902 . . . . . 6 ((𝑝𝑋𝑝𝑋) → 𝑝𝑋)
1110a1i 11 . . . . 5 ((𝐾𝐵𝑋𝑆) → ((𝑝𝑋𝑝𝑋) → 𝑝𝑋))
125, 6, 2, 3psubspi 38606 . . . . . . 7 (((𝐾𝐵𝑋𝑆𝑝 ∈ (Atoms‘𝐾)) ∧ ∃𝑞𝑋𝑟𝑋 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)) → 𝑝𝑋)
13123exp1 1352 . . . . . 6 (𝐾𝐵 → (𝑋𝑆 → (𝑝 ∈ (Atoms‘𝐾) → (∃𝑞𝑋𝑟𝑋 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟) → 𝑝𝑋))))
1413imp4b 422 . . . . 5 ((𝐾𝐵𝑋𝑆) → ((𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞𝑋𝑟𝑋 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)) → 𝑝𝑋))
1511, 14jaod 857 . . . 4 ((𝐾𝐵𝑋𝑆) → (((𝑝𝑋𝑝𝑋) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞𝑋𝑟𝑋 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟))) → 𝑝𝑋))
169, 15sylbid 239 . . 3 ((𝐾𝐵𝑋𝑆) → (𝑝 ∈ (𝑋 + 𝑋) → 𝑝𝑋))
1716ssrdv 3987 . 2 ((𝐾𝐵𝑋𝑆) → (𝑋 + 𝑋) ⊆ 𝑋)
182, 7sspadd1 38674 . . 3 ((𝐾𝐵𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑋 ⊆ (Atoms‘𝐾)) → 𝑋 ⊆ (𝑋 + 𝑋))
191, 4, 4, 18syl3anc 1371 . 2 ((𝐾𝐵𝑋𝑆) → 𝑋 ⊆ (𝑋 + 𝑋))
2017, 19eqssd 3998 1 ((𝐾𝐵𝑋𝑆) → (𝑋 + 𝑋) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 845   = wceq 1541  wcel 2106  wrex 3070  wss 3947   class class class wbr 5147  cfv 6540  (class class class)co 7405  lecple 17200  joincjn 18260  Atomscatm 38121  PSubSpcpsubsp 38355  +𝑃cpadd 38654
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 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
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-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-psubsp 38362  df-padd 38655
This theorem is referenced by:  paddclN  38701  paddss  38704  pmod1i  38707
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