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Theorem paddidm 39016
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 = (PSubSpβ€˜πΎ)
paddidm.p + = (+π‘ƒβ€˜πΎ)
Assertion
Ref Expression
paddidm ((𝐾 ∈ 𝐡 ∧ 𝑋 ∈ 𝑆) β†’ (𝑋 + 𝑋) = 𝑋)
Proof of Theorem paddidm
Dummy variables 𝑝 π‘ž π‘Ÿ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 482 . . . . 5 ((𝐾 ∈ 𝐡 ∧ 𝑋 ∈ 𝑆) β†’ 𝐾 ∈ 𝐡)
2 eqid 2731 . . . . . 6 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
3 paddidm.s . . . . . 6 𝑆 = (PSubSpβ€˜πΎ)
42, 3psubssat 38929 . . . . 5 ((𝐾 ∈ 𝐡 ∧ 𝑋 ∈ 𝑆) β†’ 𝑋 βŠ† (Atomsβ€˜πΎ))
5 eqid 2731 . . . . . 6 (leβ€˜πΎ) = (leβ€˜πΎ)
6 eqid 2731 . . . . . 6 (joinβ€˜πΎ) = (joinβ€˜πΎ)
7 paddidm.p . . . . . 6 + = (+π‘ƒβ€˜πΎ)
85, 6, 2, 7elpadd 38974 . . . . 5 ((𝐾 ∈ 𝐡 ∧ 𝑋 βŠ† (Atomsβ€˜πΎ) ∧ 𝑋 βŠ† (Atomsβ€˜πΎ)) β†’ (𝑝 ∈ (𝑋 + 𝑋) ↔ ((𝑝 ∈ 𝑋 ∨ 𝑝 ∈ 𝑋) ∨ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ 𝑋 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)))))
91, 4, 4, 8syl3anc 1370 . . . 4 ((𝐾 ∈ 𝐡 ∧ 𝑋 ∈ 𝑆) β†’ (𝑝 ∈ (𝑋 + 𝑋) ↔ ((𝑝 ∈ 𝑋 ∨ 𝑝 ∈ 𝑋) ∨ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ 𝑋 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)))))
10 pm1.2 901 . . . . . 6 ((𝑝 ∈ 𝑋 ∨ 𝑝 ∈ 𝑋) β†’ 𝑝 ∈ 𝑋)
1110a1i 11 . . . . 5 ((𝐾 ∈ 𝐡 ∧ 𝑋 ∈ 𝑆) β†’ ((𝑝 ∈ 𝑋 ∨ 𝑝 ∈ 𝑋) β†’ 𝑝 ∈ 𝑋))
125, 6, 2, 3psubspi 38922 . . . . . . 7 (((𝐾 ∈ 𝐡 ∧ 𝑋 ∈ 𝑆 ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) ∧ βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ 𝑋 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)) β†’ 𝑝 ∈ 𝑋)
13123exp1 1351 . . . . . 6 (𝐾 ∈ 𝐡 β†’ (𝑋 ∈ 𝑆 β†’ (𝑝 ∈ (Atomsβ€˜πΎ) β†’ (βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ 𝑋 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ) β†’ 𝑝 ∈ 𝑋))))
1413imp4b 421 . . . . 5 ((𝐾 ∈ 𝐡 ∧ 𝑋 ∈ 𝑆) β†’ ((𝑝 ∈ (Atomsβ€˜πΎ) ∧ βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ 𝑋 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)) β†’ 𝑝 ∈ 𝑋))
1511, 14jaod 856 . . . 4 ((𝐾 ∈ 𝐡 ∧ 𝑋 ∈ 𝑆) β†’ (((𝑝 ∈ 𝑋 ∨ 𝑝 ∈ 𝑋) ∨ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ 𝑋 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ))) β†’ 𝑝 ∈ 𝑋))
169, 15sylbid 239 . . 3 ((𝐾 ∈ 𝐡 ∧ 𝑋 ∈ 𝑆) β†’ (𝑝 ∈ (𝑋 + 𝑋) β†’ 𝑝 ∈ 𝑋))
1716ssrdv 3988 . 2 ((𝐾 ∈ 𝐡 ∧ 𝑋 ∈ 𝑆) β†’ (𝑋 + 𝑋) βŠ† 𝑋)
182, 7sspadd1 38990 . . 3 ((𝐾 ∈ 𝐡 ∧ 𝑋 βŠ† (Atomsβ€˜πΎ) ∧ 𝑋 βŠ† (Atomsβ€˜πΎ)) β†’ 𝑋 βŠ† (𝑋 + 𝑋))
191, 4, 4, 18syl3anc 1370 . 2 ((𝐾 ∈ 𝐡 ∧ 𝑋 ∈ 𝑆) β†’ 𝑋 βŠ† (𝑋 + 𝑋))
2017, 19eqssd 3999 1 ((𝐾 ∈ 𝐡 ∧ 𝑋 ∈ 𝑆) β†’ (𝑋 + 𝑋) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 844   = wceq 1540   ∈ wcel 2105  βˆƒwrex 3069   βŠ† wss 3948   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7412  lecple 17209  joincjn 18269  Atomscatm 38437  PSubSpcpsubsp 38671  +𝑃cpadd 38970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7978  df-2nd 7979  df-psubsp 38678  df-padd 38971
This theorem is referenced by:  paddclN  39017  paddss  39020  pmod1i  39023
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