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Theorem paddss12 39203
Description: Subset law for projective subspace sum. (unss12 4177 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 3985 . . . . . . . 8 ((𝑍 βŠ† π‘Š ∧ π‘Š βŠ† 𝐴) β†’ 𝑍 βŠ† 𝐴)
43ancoms 458 . . . . . . 7 ((π‘Š βŠ† 𝐴 ∧ 𝑍 βŠ† π‘Š) β†’ 𝑍 βŠ† 𝐴)
54ad2ant2l 743 . . . . . 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 768 . . . 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 39201 . . . 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 39202 . . . . . 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 406 . . . 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 713 . . 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 3987 . 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 412 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   βŠ† wss 3943  β€˜cfv 6537  (class class class)co 7405  Atomscatm 38646  +𝑃cpadd 39179
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-padd 39180
This theorem is referenced by:  paddssw1  39227  paddunN  39311  pl42lem2N  39364
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