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Theorem paddidm 34953
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 473 . . . . 5 ((𝐾𝐵𝑋𝑆) → 𝐾𝐵)
2 eqid 2621 . . . . . 6 (Atoms‘𝐾) = (Atoms‘𝐾)
3 paddidm.s . . . . . 6 𝑆 = (PSubSp‘𝐾)
42, 3psubssat 34866 . . . . 5 ((𝐾𝐵𝑋𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
5 eqid 2621 . . . . . 6 (le‘𝐾) = (le‘𝐾)
6 eqid 2621 . . . . . 6 (join‘𝐾) = (join‘𝐾)
7 paddidm.p . . . . . 6 + = (+𝑃𝐾)
85, 6, 2, 7elpadd 34911 . . . . 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 1325 . . . 4 ((𝐾𝐵𝑋𝑆) → (𝑝 ∈ (𝑋 + 𝑋) ↔ ((𝑝𝑋𝑝𝑋) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞𝑋𝑟𝑋 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)))))
10 pm1.2 535 . . . . . 6 ((𝑝𝑋𝑝𝑋) → 𝑝𝑋)
1110a1i 11 . . . . 5 ((𝐾𝐵𝑋𝑆) → ((𝑝𝑋𝑝𝑋) → 𝑝𝑋))
125, 6, 2, 3psubspi 34859 . . . . . . 7 (((𝐾𝐵𝑋𝑆𝑝 ∈ (Atoms‘𝐾)) ∧ ∃𝑞𝑋𝑟𝑋 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)) → 𝑝𝑋)
13123exp1 1282 . . . . . 6 (𝐾𝐵 → (𝑋𝑆 → (𝑝 ∈ (Atoms‘𝐾) → (∃𝑞𝑋𝑟𝑋 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟) → 𝑝𝑋))))
1413imp4b 613 . . . . 5 ((𝐾𝐵𝑋𝑆) → ((𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞𝑋𝑟𝑋 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)) → 𝑝𝑋))
1511, 14jaod 395 . . . 4 ((𝐾𝐵𝑋𝑆) → (((𝑝𝑋𝑝𝑋) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞𝑋𝑟𝑋 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟))) → 𝑝𝑋))
169, 15sylbid 230 . . 3 ((𝐾𝐵𝑋𝑆) → (𝑝 ∈ (𝑋 + 𝑋) → 𝑝𝑋))
1716ssrdv 3607 . 2 ((𝐾𝐵𝑋𝑆) → (𝑋 + 𝑋) ⊆ 𝑋)
182, 7sspadd1 34927 . . 3 ((𝐾𝐵𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑋 ⊆ (Atoms‘𝐾)) → 𝑋 ⊆ (𝑋 + 𝑋))
191, 4, 4, 18syl3anc 1325 . 2 ((𝐾𝐵𝑋𝑆) → 𝑋 ⊆ (𝑋 + 𝑋))
2017, 19eqssd 3618 1 ((𝐾𝐵𝑋𝑆) → (𝑋 + 𝑋) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384   = wceq 1482  wcel 1989  wrex 2912  wss 3572   class class class wbr 4651  cfv 5886  (class class class)co 6647  lecple 15942  joincjn 16938  Atomscatm 34376  PSubSpcpsubsp 34608  +𝑃cpadd 34907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-1st 7165  df-2nd 7166  df-psubsp 34615  df-padd 34908
This theorem is referenced by:  paddclN  34954  paddss  34957  pmod1i  34960
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