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Theorem paddidm 35862
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 475 . . . . 5 ((𝐾𝐵𝑋𝑆) → 𝐾𝐵)
2 eqid 2799 . . . . . 6 (Atoms‘𝐾) = (Atoms‘𝐾)
3 paddidm.s . . . . . 6 𝑆 = (PSubSp‘𝐾)
42, 3psubssat 35775 . . . . 5 ((𝐾𝐵𝑋𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
5 eqid 2799 . . . . . 6 (le‘𝐾) = (le‘𝐾)
6 eqid 2799 . . . . . 6 (join‘𝐾) = (join‘𝐾)
7 paddidm.p . . . . . 6 + = (+𝑃𝐾)
85, 6, 2, 7elpadd 35820 . . . . 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 1491 . . . 4 ((𝐾𝐵𝑋𝑆) → (𝑝 ∈ (𝑋 + 𝑋) ↔ ((𝑝𝑋𝑝𝑋) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞𝑋𝑟𝑋 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)))))
10 pm1.2 928 . . . . . 6 ((𝑝𝑋𝑝𝑋) → 𝑝𝑋)
1110a1i 11 . . . . 5 ((𝐾𝐵𝑋𝑆) → ((𝑝𝑋𝑝𝑋) → 𝑝𝑋))
125, 6, 2, 3psubspi 35768 . . . . . . 7 (((𝐾𝐵𝑋𝑆𝑝 ∈ (Atoms‘𝐾)) ∧ ∃𝑞𝑋𝑟𝑋 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)) → 𝑝𝑋)
13123exp1 1462 . . . . . 6 (𝐾𝐵 → (𝑋𝑆 → (𝑝 ∈ (Atoms‘𝐾) → (∃𝑞𝑋𝑟𝑋 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟) → 𝑝𝑋))))
1413imp4b 413 . . . . 5 ((𝐾𝐵𝑋𝑆) → ((𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞𝑋𝑟𝑋 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)) → 𝑝𝑋))
1511, 14jaod 886 . . . 4 ((𝐾𝐵𝑋𝑆) → (((𝑝𝑋𝑝𝑋) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞𝑋𝑟𝑋 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟))) → 𝑝𝑋))
169, 15sylbid 232 . . 3 ((𝐾𝐵𝑋𝑆) → (𝑝 ∈ (𝑋 + 𝑋) → 𝑝𝑋))
1716ssrdv 3804 . 2 ((𝐾𝐵𝑋𝑆) → (𝑋 + 𝑋) ⊆ 𝑋)
182, 7sspadd1 35836 . . 3 ((𝐾𝐵𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑋 ⊆ (Atoms‘𝐾)) → 𝑋 ⊆ (𝑋 + 𝑋))
191, 4, 4, 18syl3anc 1491 . 2 ((𝐾𝐵𝑋𝑆) → 𝑋 ⊆ (𝑋 + 𝑋))
2017, 19eqssd 3815 1 ((𝐾𝐵𝑋𝑆) → (𝑋 + 𝑋) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  wo 874   = wceq 1653  wcel 2157  wrex 3090  wss 3769   class class class wbr 4843  cfv 6101  (class class class)co 6878  lecple 16274  joincjn 17259  Atomscatm 35284  PSubSpcpsubsp 35517  +𝑃cpadd 35816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-1st 7401  df-2nd 7402  df-psubsp 35524  df-padd 35817
This theorem is referenced by:  paddclN  35863  paddss  35866  pmod1i  35869
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