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Theorem paddidm 35649
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 468 . . . . 5 ((𝐾𝐵𝑋𝑆) → 𝐾𝐵)
2 eqid 2771 . . . . . 6 (Atoms‘𝐾) = (Atoms‘𝐾)
3 paddidm.s . . . . . 6 𝑆 = (PSubSp‘𝐾)
42, 3psubssat 35562 . . . . 5 ((𝐾𝐵𝑋𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
5 eqid 2771 . . . . . 6 (le‘𝐾) = (le‘𝐾)
6 eqid 2771 . . . . . 6 (join‘𝐾) = (join‘𝐾)
7 paddidm.p . . . . . 6 + = (+𝑃𝐾)
85, 6, 2, 7elpadd 35607 . . . . 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 1476 . . . 4 ((𝐾𝐵𝑋𝑆) → (𝑝 ∈ (𝑋 + 𝑋) ↔ ((𝑝𝑋𝑝𝑋) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞𝑋𝑟𝑋 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)))))
10 pm1.2 889 . . . . . 6 ((𝑝𝑋𝑝𝑋) → 𝑝𝑋)
1110a1i 11 . . . . 5 ((𝐾𝐵𝑋𝑆) → ((𝑝𝑋𝑝𝑋) → 𝑝𝑋))
125, 6, 2, 3psubspi 35555 . . . . . . 7 (((𝐾𝐵𝑋𝑆𝑝 ∈ (Atoms‘𝐾)) ∧ ∃𝑞𝑋𝑟𝑋 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)) → 𝑝𝑋)
13123exp1 1445 . . . . . 6 (𝐾𝐵 → (𝑋𝑆 → (𝑝 ∈ (Atoms‘𝐾) → (∃𝑞𝑋𝑟𝑋 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟) → 𝑝𝑋))))
1413imp4b 408 . . . . 5 ((𝐾𝐵𝑋𝑆) → ((𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞𝑋𝑟𝑋 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)) → 𝑝𝑋))
1511, 14jaod 848 . . . 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 3758 . 2 ((𝐾𝐵𝑋𝑆) → (𝑋 + 𝑋) ⊆ 𝑋)
182, 7sspadd1 35623 . . 3 ((𝐾𝐵𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑋 ⊆ (Atoms‘𝐾)) → 𝑋 ⊆ (𝑋 + 𝑋))
191, 4, 4, 18syl3anc 1476 . 2 ((𝐾𝐵𝑋𝑆) → 𝑋 ⊆ (𝑋 + 𝑋))
2017, 19eqssd 3769 1 ((𝐾𝐵𝑋𝑆) → (𝑋 + 𝑋) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wo 836   = wceq 1631  wcel 2145  wrex 3062  wss 3723   class class class wbr 4786  cfv 6031  (class class class)co 6793  lecple 16156  joincjn 17152  Atomscatm 35072  PSubSpcpsubsp 35304  +𝑃cpadd 35603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-psubsp 35311  df-padd 35604
This theorem is referenced by:  paddclN  35650  paddss  35653  pmod1i  35656
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