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Theorem paddss1 38991
Description: Subset law for projective subspace sum. (unss1 4179 analog.) (Contributed by NM, 7-Mar-2012.)
Hypotheses
Ref Expression
padd0.a 𝐴 = (Atomsβ€˜πΎ)
padd0.p + = (+π‘ƒβ€˜πΎ)
Assertion
Ref Expression
paddss1 ((𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) β†’ (𝑋 βŠ† π‘Œ β†’ (𝑋 + 𝑍) βŠ† (π‘Œ + 𝑍)))
Proof of Theorem paddss1
Dummy variables π‘ž 𝑝 π‘Ÿ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3975 . . . . . . 7 (𝑋 βŠ† π‘Œ β†’ (𝑝 ∈ 𝑋 β†’ 𝑝 ∈ π‘Œ))
21orim1d 964 . . . . . 6 (𝑋 βŠ† π‘Œ β†’ ((𝑝 ∈ 𝑋 ∨ 𝑝 ∈ 𝑍) β†’ (𝑝 ∈ π‘Œ ∨ 𝑝 ∈ 𝑍)))
3 ssrexv 4051 . . . . . . 7 (𝑋 βŠ† π‘Œ β†’ (βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ 𝑍 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ) β†’ βˆƒπ‘ž ∈ π‘Œ βˆƒπ‘Ÿ ∈ 𝑍 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)))
43anim2d 612 . . . . . 6 (𝑋 βŠ† π‘Œ β†’ ((𝑝 ∈ 𝐴 ∧ βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ 𝑍 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)) β†’ (𝑝 ∈ 𝐴 ∧ βˆƒπ‘ž ∈ π‘Œ βˆƒπ‘Ÿ ∈ 𝑍 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ))))
52, 4orim12d 963 . . . . 5 (𝑋 βŠ† π‘Œ β†’ (((𝑝 ∈ 𝑋 ∨ 𝑝 ∈ 𝑍) ∨ (𝑝 ∈ 𝐴 ∧ βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ 𝑍 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ))) β†’ ((𝑝 ∈ π‘Œ ∨ 𝑝 ∈ 𝑍) ∨ (𝑝 ∈ 𝐴 ∧ βˆƒπ‘ž ∈ π‘Œ βˆƒπ‘Ÿ ∈ 𝑍 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)))))
65adantl 482 . . . 4 (((𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ 𝑋 βŠ† π‘Œ) β†’ (((𝑝 ∈ 𝑋 ∨ 𝑝 ∈ 𝑍) ∨ (𝑝 ∈ 𝐴 ∧ βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ 𝑍 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ))) β†’ ((𝑝 ∈ π‘Œ ∨ 𝑝 ∈ 𝑍) ∨ (𝑝 ∈ 𝐴 ∧ βˆƒπ‘ž ∈ π‘Œ βˆƒπ‘Ÿ ∈ 𝑍 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)))))
7 simpl1 1191 . . . . 5 (((𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ 𝑋 βŠ† π‘Œ) β†’ 𝐾 ∈ 𝐡)
8 sstr 3990 . . . . . . 7 ((𝑋 βŠ† π‘Œ ∧ π‘Œ βŠ† 𝐴) β†’ 𝑋 βŠ† 𝐴)
983ad2antr2 1189 . . . . . 6 ((𝑋 βŠ† π‘Œ ∧ (𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ 𝑋 βŠ† 𝐴)
109ancoms 459 . . . . 5 (((𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ 𝑋 βŠ† π‘Œ) β†’ 𝑋 βŠ† 𝐴)
11 simpl3 1193 . . . . 5 (((𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ 𝑋 βŠ† π‘Œ) β†’ 𝑍 βŠ† 𝐴)
12 eqid 2732 . . . . . 6 (leβ€˜πΎ) = (leβ€˜πΎ)
13 eqid 2732 . . . . . 6 (joinβ€˜πΎ) = (joinβ€˜πΎ)
14 padd0.a . . . . . 6 𝐴 = (Atomsβ€˜πΎ)
15 padd0.p . . . . . 6 + = (+π‘ƒβ€˜πΎ)
1612, 13, 14, 15elpadd 38973 . . . . 5 ((𝐾 ∈ 𝐡 ∧ 𝑋 βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) β†’ (𝑝 ∈ (𝑋 + 𝑍) ↔ ((𝑝 ∈ 𝑋 ∨ 𝑝 ∈ 𝑍) ∨ (𝑝 ∈ 𝐴 ∧ βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ 𝑍 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)))))
177, 10, 11, 16syl3anc 1371 . . . 4 (((𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ 𝑋 βŠ† π‘Œ) β†’ (𝑝 ∈ (𝑋 + 𝑍) ↔ ((𝑝 ∈ 𝑋 ∨ 𝑝 ∈ 𝑍) ∨ (𝑝 ∈ 𝐴 ∧ βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ 𝑍 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)))))
1812, 13, 14, 15elpadd 38973 . . . . 5 ((𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) β†’ (𝑝 ∈ (π‘Œ + 𝑍) ↔ ((𝑝 ∈ π‘Œ ∨ 𝑝 ∈ 𝑍) ∨ (𝑝 ∈ 𝐴 ∧ βˆƒπ‘ž ∈ π‘Œ βˆƒπ‘Ÿ ∈ 𝑍 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)))))
1918adantr 481 . . . 4 (((𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ 𝑋 βŠ† π‘Œ) β†’ (𝑝 ∈ (π‘Œ + 𝑍) ↔ ((𝑝 ∈ π‘Œ ∨ 𝑝 ∈ 𝑍) ∨ (𝑝 ∈ 𝐴 ∧ βˆƒπ‘ž ∈ π‘Œ βˆƒπ‘Ÿ ∈ 𝑍 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)))))
206, 17, 193imtr4d 293 . . 3 (((𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ 𝑋 βŠ† π‘Œ) β†’ (𝑝 ∈ (𝑋 + 𝑍) β†’ 𝑝 ∈ (π‘Œ + 𝑍)))
2120ssrdv 3988 . 2 (((𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ 𝑋 βŠ† π‘Œ) β†’ (𝑋 + 𝑍) βŠ† (π‘Œ + 𝑍))
2221ex 413 1 ((𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) β†’ (𝑋 βŠ† π‘Œ β†’ (𝑋 + 𝑍) βŠ† (π‘Œ + 𝑍)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070   βŠ† wss 3948   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7411  lecple 17208  joincjn 18268  Atomscatm 38436  +𝑃cpadd 38969
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 7727
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-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-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-padd 38970
This theorem is referenced by:  paddss12  38993  paddasslem12  39005  pmod1i  39022  pl42lem3N  39155
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