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Theorem paddss2 38992
Description: Subset law for projective subspace sum. (unss2 4180 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
paddss2 ((𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) β†’ (𝑋 βŠ† π‘Œ β†’ (𝑍 + 𝑋) βŠ† (𝑍 + π‘Œ)))
Proof of Theorem paddss2
Dummy variables π‘ž 𝑝 π‘Ÿ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3974 . . . . . . 7 (𝑋 βŠ† π‘Œ β†’ (𝑝 ∈ 𝑋 β†’ 𝑝 ∈ π‘Œ))
21orim2d 963 . . . . . 6 (𝑋 βŠ† π‘Œ β†’ ((𝑝 ∈ 𝑍 ∨ 𝑝 ∈ 𝑋) β†’ (𝑝 ∈ 𝑍 ∨ 𝑝 ∈ π‘Œ)))
3 ssrexv 4050 . . . . . . . 8 (𝑋 βŠ† π‘Œ β†’ (βˆƒπ‘Ÿ ∈ 𝑋 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ) β†’ βˆƒπ‘Ÿ ∈ π‘Œ 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)))
43reximdv 3168 . . . . . . 7 (𝑋 βŠ† π‘Œ β†’ (βˆƒπ‘ž ∈ 𝑍 βˆƒπ‘Ÿ ∈ 𝑋 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ) β†’ βˆƒπ‘ž ∈ 𝑍 βˆƒπ‘Ÿ ∈ π‘Œ 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)))
54anim2d 610 . . . . . 6 (𝑋 βŠ† π‘Œ β†’ ((𝑝 ∈ 𝐴 ∧ βˆƒπ‘ž ∈ 𝑍 βˆƒπ‘Ÿ ∈ 𝑋 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)) β†’ (𝑝 ∈ 𝐴 ∧ βˆƒπ‘ž ∈ 𝑍 βˆƒπ‘Ÿ ∈ π‘Œ 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ))))
62, 5orim12d 961 . . . . 5 (𝑋 βŠ† π‘Œ β†’ (((𝑝 ∈ 𝑍 ∨ 𝑝 ∈ 𝑋) ∨ (𝑝 ∈ 𝐴 ∧ βˆƒπ‘ž ∈ 𝑍 βˆƒπ‘Ÿ ∈ 𝑋 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ))) β†’ ((𝑝 ∈ 𝑍 ∨ 𝑝 ∈ π‘Œ) ∨ (𝑝 ∈ 𝐴 ∧ βˆƒπ‘ž ∈ 𝑍 βˆƒπ‘Ÿ ∈ π‘Œ 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)))))
76adantl 480 . . . 4 (((𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ 𝑋 βŠ† π‘Œ) β†’ (((𝑝 ∈ 𝑍 ∨ 𝑝 ∈ 𝑋) ∨ (𝑝 ∈ 𝐴 ∧ βˆƒπ‘ž ∈ 𝑍 βˆƒπ‘Ÿ ∈ 𝑋 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ))) β†’ ((𝑝 ∈ 𝑍 ∨ 𝑝 ∈ π‘Œ) ∨ (𝑝 ∈ 𝐴 ∧ βˆƒπ‘ž ∈ 𝑍 βˆƒπ‘Ÿ ∈ π‘Œ 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)))))
8 simpl1 1189 . . . . 5 (((𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ 𝑋 βŠ† π‘Œ) β†’ 𝐾 ∈ 𝐡)
9 simpl3 1191 . . . . 5 (((𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ 𝑋 βŠ† π‘Œ) β†’ 𝑍 βŠ† 𝐴)
10 sstr 3989 . . . . . . 7 ((𝑋 βŠ† π‘Œ ∧ π‘Œ βŠ† 𝐴) β†’ 𝑋 βŠ† 𝐴)
11103ad2antr2 1187 . . . . . 6 ((𝑋 βŠ† π‘Œ ∧ (𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ 𝑋 βŠ† 𝐴)
1211ancoms 457 . . . . 5 (((𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ 𝑋 βŠ† π‘Œ) β†’ 𝑋 βŠ† 𝐴)
13 eqid 2730 . . . . . 6 (leβ€˜πΎ) = (leβ€˜πΎ)
14 eqid 2730 . . . . . 6 (joinβ€˜πΎ) = (joinβ€˜πΎ)
15 padd0.a . . . . . 6 𝐴 = (Atomsβ€˜πΎ)
16 padd0.p . . . . . 6 + = (+π‘ƒβ€˜πΎ)
1713, 14, 15, 16elpadd 38973 . . . . 5 ((𝐾 ∈ 𝐡 ∧ 𝑍 βŠ† 𝐴 ∧ 𝑋 βŠ† 𝐴) β†’ (𝑝 ∈ (𝑍 + 𝑋) ↔ ((𝑝 ∈ 𝑍 ∨ 𝑝 ∈ 𝑋) ∨ (𝑝 ∈ 𝐴 ∧ βˆƒπ‘ž ∈ 𝑍 βˆƒπ‘Ÿ ∈ 𝑋 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)))))
188, 9, 12, 17syl3anc 1369 . . . 4 (((𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ 𝑋 βŠ† π‘Œ) β†’ (𝑝 ∈ (𝑍 + 𝑋) ↔ ((𝑝 ∈ 𝑍 ∨ 𝑝 ∈ 𝑋) ∨ (𝑝 ∈ 𝐴 ∧ βˆƒπ‘ž ∈ 𝑍 βˆƒπ‘Ÿ ∈ 𝑋 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)))))
19 simpl2 1190 . . . . 5 (((𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ 𝑋 βŠ† π‘Œ) β†’ π‘Œ βŠ† 𝐴)
2013, 14, 15, 16elpadd 38973 . . . . 5 ((𝐾 ∈ 𝐡 ∧ 𝑍 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ (𝑝 ∈ (𝑍 + π‘Œ) ↔ ((𝑝 ∈ 𝑍 ∨ 𝑝 ∈ π‘Œ) ∨ (𝑝 ∈ 𝐴 ∧ βˆƒπ‘ž ∈ 𝑍 βˆƒπ‘Ÿ ∈ π‘Œ 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)))))
218, 9, 19, 20syl3anc 1369 . . . 4 (((𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ 𝑋 βŠ† π‘Œ) β†’ (𝑝 ∈ (𝑍 + π‘Œ) ↔ ((𝑝 ∈ 𝑍 ∨ 𝑝 ∈ π‘Œ) ∨ (𝑝 ∈ 𝐴 ∧ βˆƒπ‘ž ∈ 𝑍 βˆƒπ‘Ÿ ∈ π‘Œ 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)))))
227, 18, 213imtr4d 293 . . 3 (((𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ 𝑋 βŠ† π‘Œ) β†’ (𝑝 ∈ (𝑍 + 𝑋) β†’ 𝑝 ∈ (𝑍 + π‘Œ)))
2322ssrdv 3987 . 2 (((𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ 𝑋 βŠ† π‘Œ) β†’ (𝑍 + 𝑋) βŠ† (𝑍 + π‘Œ))
2423ex 411 1 ((𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) β†’ (𝑋 βŠ† π‘Œ β†’ (𝑍 + 𝑋) βŠ† (𝑍 + π‘Œ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  βˆƒwrex 3068   βŠ† wss 3947   class class class wbr 5147  β€˜cfv 6542  (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 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  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  pmod1i  39022
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