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Theorem paddss12 39332
Description: Subset law for projective subspace sum. (unss12 4184 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
paddss12 ((𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ π‘Š βŠ† 𝐴) β†’ ((𝑋 βŠ† π‘Œ ∧ 𝑍 βŠ† π‘Š) β†’ (𝑋 + 𝑍) βŠ† (π‘Œ + π‘Š)))
Proof of Theorem paddss12
StepHypRef Expression
1 simpl1 1188 . . . . 5 (((𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ π‘Š βŠ† 𝐴) ∧ (𝑋 βŠ† π‘Œ ∧ 𝑍 βŠ† π‘Š)) β†’ 𝐾 ∈ 𝐡)
2 simpl2 1189 . . . . 5 (((𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ π‘Š βŠ† 𝐴) ∧ (𝑋 βŠ† π‘Œ ∧ 𝑍 βŠ† π‘Š)) β†’ π‘Œ βŠ† 𝐴)
3 sstr 3990 . . . . . . . 8 ((𝑍 βŠ† π‘Š ∧ π‘Š βŠ† 𝐴) β†’ 𝑍 βŠ† 𝐴)
43ancoms 457 . . . . . . 7 ((π‘Š βŠ† 𝐴 ∧ 𝑍 βŠ† π‘Š) β†’ 𝑍 βŠ† 𝐴)
54ad2ant2l 744 . . . . . 6 (((π‘Œ βŠ† 𝐴 ∧ π‘Š βŠ† 𝐴) ∧ (𝑋 βŠ† π‘Œ ∧ 𝑍 βŠ† π‘Š)) β†’ 𝑍 βŠ† 𝐴)
653adantl1 1163 . . . . 5 (((𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ π‘Š βŠ† 𝐴) ∧ (𝑋 βŠ† π‘Œ ∧ 𝑍 βŠ† π‘Š)) β†’ 𝑍 βŠ† 𝐴)
71, 2, 63jca 1125 . . . 4 (((𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ π‘Š βŠ† 𝐴) ∧ (𝑋 βŠ† π‘Œ ∧ 𝑍 βŠ† π‘Š)) β†’ (𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴))
8 simprl 769 . . . 4 (((𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ π‘Š βŠ† 𝐴) ∧ (𝑋 βŠ† π‘Œ ∧ 𝑍 βŠ† π‘Š)) β†’ 𝑋 βŠ† π‘Œ)
9 padd0.a . . . . 5 𝐴 = (Atomsβ€˜πΎ)
10 padd0.p . . . . 5 + = (+π‘ƒβ€˜πΎ)
119, 10paddss1 39330 . . . 4 ((𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) β†’ (𝑋 βŠ† π‘Œ β†’ (𝑋 + 𝑍) βŠ† (π‘Œ + 𝑍)))
127, 8, 11sylc 65 . . 3 (((𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ π‘Š βŠ† 𝐴) ∧ (𝑋 βŠ† π‘Œ ∧ 𝑍 βŠ† π‘Š)) β†’ (𝑋 + 𝑍) βŠ† (π‘Œ + 𝑍))
139, 10paddss2 39331 . . . . . 6 ((𝐾 ∈ 𝐡 ∧ π‘Š βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ (𝑍 βŠ† π‘Š β†’ (π‘Œ + 𝑍) βŠ† (π‘Œ + π‘Š)))
14133com23 1123 . . . . 5 ((𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ π‘Š βŠ† 𝐴) β†’ (𝑍 βŠ† π‘Š β†’ (π‘Œ + 𝑍) βŠ† (π‘Œ + π‘Š)))
1514imp 405 . . . 4 (((𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ π‘Š βŠ† 𝐴) ∧ 𝑍 βŠ† π‘Š) β†’ (π‘Œ + 𝑍) βŠ† (π‘Œ + π‘Š))
1615adantrl 714 . . 3 (((𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ π‘Š βŠ† 𝐴) ∧ (𝑋 βŠ† π‘Œ ∧ 𝑍 βŠ† π‘Š)) β†’ (π‘Œ + 𝑍) βŠ† (π‘Œ + π‘Š))
1712, 16sstrd 3992 . 2 (((𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ π‘Š βŠ† 𝐴) ∧ (𝑋 βŠ† π‘Œ ∧ 𝑍 βŠ† π‘Š)) β†’ (𝑋 + 𝑍) βŠ† (π‘Œ + π‘Š))
1817ex 411 1 ((𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ π‘Š βŠ† 𝐴) β†’ ((𝑋 βŠ† π‘Œ ∧ 𝑍 βŠ† π‘Š) β†’ (𝑋 + 𝑍) βŠ† (π‘Œ + π‘Š)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   βŠ† wss 3949  β€˜cfv 6553  (class class class)co 7426  Atomscatm 38775  +𝑃cpadd 39308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 8001  df-2nd 8002  df-padd 39309
This theorem is referenced by:  paddssw1  39356  paddunN  39440  pl42lem2N  39493
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