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Theorem paddcom 38679
Description: Projective subspace sum commutes. (Contributed by NM, 3-Jan-2012.)
Hypotheses
Ref Expression
padd0.a 𝐴 = (Atomsβ€˜πΎ)
padd0.p + = (+π‘ƒβ€˜πΎ)
Assertion
Ref Expression
paddcom ((𝐾 ∈ Lat ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ (𝑋 + π‘Œ) = (π‘Œ + 𝑋))
Proof of Theorem paddcom
Dummy variables π‘ž 𝑝 π‘Ÿ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uncom 4153 . . . 4 (𝑋 βˆͺ π‘Œ) = (π‘Œ βˆͺ 𝑋)
21a1i 11 . . 3 ((𝐾 ∈ Lat ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ (𝑋 βˆͺ π‘Œ) = (π‘Œ βˆͺ 𝑋))
3 simpl1 1191 . . . . . . . 8 (((𝐾 ∈ Lat ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (π‘ž ∈ 𝑋 ∧ π‘Ÿ ∈ π‘Œ)) β†’ 𝐾 ∈ Lat)
4 simpl2 1192 . . . . . . . . . 10 (((𝐾 ∈ Lat ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (π‘ž ∈ 𝑋 ∧ π‘Ÿ ∈ π‘Œ)) β†’ 𝑋 βŠ† 𝐴)
5 simprl 769 . . . . . . . . . 10 (((𝐾 ∈ Lat ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (π‘ž ∈ 𝑋 ∧ π‘Ÿ ∈ π‘Œ)) β†’ π‘ž ∈ 𝑋)
64, 5sseldd 3983 . . . . . . . . 9 (((𝐾 ∈ Lat ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (π‘ž ∈ 𝑋 ∧ π‘Ÿ ∈ π‘Œ)) β†’ π‘ž ∈ 𝐴)
7 eqid 2732 . . . . . . . . . 10 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
8 padd0.a . . . . . . . . . 10 𝐴 = (Atomsβ€˜πΎ)
97, 8atbase 38154 . . . . . . . . 9 (π‘ž ∈ 𝐴 β†’ π‘ž ∈ (Baseβ€˜πΎ))
106, 9syl 17 . . . . . . . 8 (((𝐾 ∈ Lat ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (π‘ž ∈ 𝑋 ∧ π‘Ÿ ∈ π‘Œ)) β†’ π‘ž ∈ (Baseβ€˜πΎ))
11 simpl3 1193 . . . . . . . . . 10 (((𝐾 ∈ Lat ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (π‘ž ∈ 𝑋 ∧ π‘Ÿ ∈ π‘Œ)) β†’ π‘Œ βŠ† 𝐴)
12 simprr 771 . . . . . . . . . 10 (((𝐾 ∈ Lat ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (π‘ž ∈ 𝑋 ∧ π‘Ÿ ∈ π‘Œ)) β†’ π‘Ÿ ∈ π‘Œ)
1311, 12sseldd 3983 . . . . . . . . 9 (((𝐾 ∈ Lat ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (π‘ž ∈ 𝑋 ∧ π‘Ÿ ∈ π‘Œ)) β†’ π‘Ÿ ∈ 𝐴)
147, 8atbase 38154 . . . . . . . . 9 (π‘Ÿ ∈ 𝐴 β†’ π‘Ÿ ∈ (Baseβ€˜πΎ))
1513, 14syl 17 . . . . . . . 8 (((𝐾 ∈ Lat ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (π‘ž ∈ 𝑋 ∧ π‘Ÿ ∈ π‘Œ)) β†’ π‘Ÿ ∈ (Baseβ€˜πΎ))
16 eqid 2732 . . . . . . . . 9 (joinβ€˜πΎ) = (joinβ€˜πΎ)
177, 16latjcom 18399 . . . . . . . 8 ((𝐾 ∈ Lat ∧ π‘ž ∈ (Baseβ€˜πΎ) ∧ π‘Ÿ ∈ (Baseβ€˜πΎ)) β†’ (π‘ž(joinβ€˜πΎ)π‘Ÿ) = (π‘Ÿ(joinβ€˜πΎ)π‘ž))
183, 10, 15, 17syl3anc 1371 . . . . . . 7 (((𝐾 ∈ Lat ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (π‘ž ∈ 𝑋 ∧ π‘Ÿ ∈ π‘Œ)) β†’ (π‘ž(joinβ€˜πΎ)π‘Ÿ) = (π‘Ÿ(joinβ€˜πΎ)π‘ž))
1918breq2d 5160 . . . . . 6 (((𝐾 ∈ Lat ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (π‘ž ∈ 𝑋 ∧ π‘Ÿ ∈ π‘Œ)) β†’ (𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ) ↔ 𝑝(leβ€˜πΎ)(π‘Ÿ(joinβ€˜πΎ)π‘ž)))
20192rexbidva 3217 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ (βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ π‘Œ 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ) ↔ βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ π‘Œ 𝑝(leβ€˜πΎ)(π‘Ÿ(joinβ€˜πΎ)π‘ž)))
21 rexcom 3287 . . . . 5 (βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ π‘Œ 𝑝(leβ€˜πΎ)(π‘Ÿ(joinβ€˜πΎ)π‘ž) ↔ βˆƒπ‘Ÿ ∈ π‘Œ βˆƒπ‘ž ∈ 𝑋 𝑝(leβ€˜πΎ)(π‘Ÿ(joinβ€˜πΎ)π‘ž))
2220, 21bitrdi 286 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ (βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ π‘Œ 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ) ↔ βˆƒπ‘Ÿ ∈ π‘Œ βˆƒπ‘ž ∈ 𝑋 𝑝(leβ€˜πΎ)(π‘Ÿ(joinβ€˜πΎ)π‘ž)))
2322rabbidv 3440 . . 3 ((𝐾 ∈ Lat ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ {𝑝 ∈ 𝐴 ∣ βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ π‘Œ 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)} = {𝑝 ∈ 𝐴 ∣ βˆƒπ‘Ÿ ∈ π‘Œ βˆƒπ‘ž ∈ 𝑋 𝑝(leβ€˜πΎ)(π‘Ÿ(joinβ€˜πΎ)π‘ž)})
242, 23uneq12d 4164 . 2 ((𝐾 ∈ Lat ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ ((𝑋 βˆͺ π‘Œ) βˆͺ {𝑝 ∈ 𝐴 ∣ βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ π‘Œ 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)}) = ((π‘Œ βˆͺ 𝑋) βˆͺ {𝑝 ∈ 𝐴 ∣ βˆƒπ‘Ÿ ∈ π‘Œ βˆƒπ‘ž ∈ 𝑋 𝑝(leβ€˜πΎ)(π‘Ÿ(joinβ€˜πΎ)π‘ž)}))
25 eqid 2732 . . 3 (leβ€˜πΎ) = (leβ€˜πΎ)
26 padd0.p . . 3 + = (+π‘ƒβ€˜πΎ)
2725, 16, 8, 26paddval 38664 . 2 ((𝐾 ∈ Lat ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ (𝑋 + π‘Œ) = ((𝑋 βˆͺ π‘Œ) βˆͺ {𝑝 ∈ 𝐴 ∣ βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ π‘Œ 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)}))
2825, 16, 8, 26paddval 38664 . . 3 ((𝐾 ∈ Lat ∧ π‘Œ βŠ† 𝐴 ∧ 𝑋 βŠ† 𝐴) β†’ (π‘Œ + 𝑋) = ((π‘Œ βˆͺ 𝑋) βˆͺ {𝑝 ∈ 𝐴 ∣ βˆƒπ‘Ÿ ∈ π‘Œ βˆƒπ‘ž ∈ 𝑋 𝑝(leβ€˜πΎ)(π‘Ÿ(joinβ€˜πΎ)π‘ž)}))
29283com23 1126 . 2 ((𝐾 ∈ Lat ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ (π‘Œ + 𝑋) = ((π‘Œ βˆͺ 𝑋) βˆͺ {𝑝 ∈ 𝐴 ∣ βˆƒπ‘Ÿ ∈ π‘Œ βˆƒπ‘ž ∈ 𝑋 𝑝(leβ€˜πΎ)(π‘Ÿ(joinβ€˜πΎ)π‘ž)}))
3024, 27, 293eqtr4d 2782 1 ((𝐾 ∈ Lat ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ (𝑋 + π‘Œ) = (π‘Œ + 𝑋))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070  {crab 3432   βˆͺ cun 3946   βŠ† wss 3948   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7408  Basecbs 17143  lecple 17203  joincjn 18263  Latclat 18383  Atomscatm 38128  +𝑃cpadd 38661
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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
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-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  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-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975  df-lub 18298  df-join 18300  df-lat 18384  df-ats 38132  df-padd 38662
This theorem is referenced by:  paddass  38704  padd12N  38705  pmod2iN  38715  pmodN  38716  pmapjat2  38720
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