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Theorem pmod1i 38707
Description: The modular law holds in a projective subspace. (Contributed by NM, 10-Mar-2012.)
Hypotheses
Ref Expression
pmod.a 𝐴 = (Atomsβ€˜πΎ)
pmod.s 𝑆 = (PSubSpβ€˜πΎ)
pmod.p + = (+π‘ƒβ€˜πΎ)
Assertion
Ref Expression
pmod1i ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) β†’ (𝑋 βŠ† 𝑍 β†’ ((𝑋 + π‘Œ) ∩ 𝑍) = (𝑋 + (π‘Œ ∩ 𝑍))))
Proof of Theorem pmod1i
StepHypRef Expression
1 eqid 2732 . . . . 5 (leβ€˜πΎ) = (leβ€˜πΎ)
2 eqid 2732 . . . . 5 (joinβ€˜πΎ) = (joinβ€˜πΎ)
3 pmod.a . . . . 5 𝐴 = (Atomsβ€˜πΎ)
4 pmod.s . . . . 5 𝑆 = (PSubSpβ€˜πΎ)
5 pmod.p . . . . 5 + = (+π‘ƒβ€˜πΎ)
61, 2, 3, 4, 5pmodlem2 38706 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 βŠ† 𝑍) β†’ ((𝑋 + π‘Œ) ∩ 𝑍) βŠ† (𝑋 + (π‘Œ ∩ 𝑍)))
763expa 1118 . . 3 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ ((𝑋 + π‘Œ) ∩ 𝑍) βŠ† (𝑋 + (π‘Œ ∩ 𝑍)))
8 inss1 4227 . . . . 5 (π‘Œ ∩ 𝑍) βŠ† π‘Œ
9 simpll 765 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ 𝐾 ∈ HL)
10 simplr2 1216 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ π‘Œ βŠ† 𝐴)
11 simplr1 1215 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ 𝑋 βŠ† 𝐴)
123, 5paddss2 38677 . . . . . 6 ((𝐾 ∈ HL ∧ π‘Œ βŠ† 𝐴 ∧ 𝑋 βŠ† 𝐴) β†’ ((π‘Œ ∩ 𝑍) βŠ† π‘Œ β†’ (𝑋 + (π‘Œ ∩ 𝑍)) βŠ† (𝑋 + π‘Œ)))
139, 10, 11, 12syl3anc 1371 . . . . 5 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ ((π‘Œ ∩ 𝑍) βŠ† π‘Œ β†’ (𝑋 + (π‘Œ ∩ 𝑍)) βŠ† (𝑋 + π‘Œ)))
148, 13mpi 20 . . . 4 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ (𝑋 + (π‘Œ ∩ 𝑍)) βŠ† (𝑋 + π‘Œ))
15 simpl 483 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) β†’ 𝐾 ∈ HL)
163, 4psubssat 38613 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑍 ∈ 𝑆) β†’ 𝑍 βŠ† 𝐴)
17163ad2antr3 1190 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) β†’ 𝑍 βŠ† 𝐴)
18 simpr2 1195 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) β†’ π‘Œ βŠ† 𝐴)
19 ssinss1 4236 . . . . . . . 8 (π‘Œ βŠ† 𝐴 β†’ (π‘Œ ∩ 𝑍) βŠ† 𝐴)
2018, 19syl 17 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) β†’ (π‘Œ ∩ 𝑍) βŠ† 𝐴)
213, 5paddss1 38676 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑍 βŠ† 𝐴 ∧ (π‘Œ ∩ 𝑍) βŠ† 𝐴) β†’ (𝑋 βŠ† 𝑍 β†’ (𝑋 + (π‘Œ ∩ 𝑍)) βŠ† (𝑍 + (π‘Œ ∩ 𝑍))))
2215, 17, 20, 21syl3anc 1371 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) β†’ (𝑋 βŠ† 𝑍 β†’ (𝑋 + (π‘Œ ∩ 𝑍)) βŠ† (𝑍 + (π‘Œ ∩ 𝑍))))
2322imp 407 . . . . 5 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ (𝑋 + (π‘Œ ∩ 𝑍)) βŠ† (𝑍 + (π‘Œ ∩ 𝑍)))
24 simplr3 1217 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ 𝑍 ∈ 𝑆)
259, 24, 16syl2anc 584 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ 𝑍 βŠ† 𝐴)
26 inss2 4228 . . . . . . . 8 (π‘Œ ∩ 𝑍) βŠ† 𝑍
273, 5paddss2 38677 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑍 βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) β†’ ((π‘Œ ∩ 𝑍) βŠ† 𝑍 β†’ (𝑍 + (π‘Œ ∩ 𝑍)) βŠ† (𝑍 + 𝑍)))
2826, 27mpi 20 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑍 βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) β†’ (𝑍 + (π‘Œ ∩ 𝑍)) βŠ† (𝑍 + 𝑍))
299, 25, 25, 28syl3anc 1371 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ (𝑍 + (π‘Œ ∩ 𝑍)) βŠ† (𝑍 + 𝑍))
304, 5paddidm 38700 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑍 ∈ 𝑆) β†’ (𝑍 + 𝑍) = 𝑍)
319, 24, 30syl2anc 584 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ (𝑍 + 𝑍) = 𝑍)
3229, 31sseqtrd 4021 . . . . 5 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ (𝑍 + (π‘Œ ∩ 𝑍)) βŠ† 𝑍)
3323, 32sstrd 3991 . . . 4 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ (𝑋 + (π‘Œ ∩ 𝑍)) βŠ† 𝑍)
3414, 33ssind 4231 . . 3 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ (𝑋 + (π‘Œ ∩ 𝑍)) βŠ† ((𝑋 + π‘Œ) ∩ 𝑍))
357, 34eqssd 3998 . 2 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ ((𝑋 + π‘Œ) ∩ 𝑍) = (𝑋 + (π‘Œ ∩ 𝑍)))
3635ex 413 1 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) β†’ (𝑋 βŠ† 𝑍 β†’ ((𝑋 + π‘Œ) ∩ 𝑍) = (𝑋 + (π‘Œ ∩ 𝑍))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ∩ cin 3946   βŠ† wss 3947  β€˜cfv 6540  (class class class)co 7405  lecple 17200  joincjn 18260  Atomscatm 38121  HLchlt 38208  PSubSpcpsubsp 38355  +𝑃cpadd 38654
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 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
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 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 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-proset 18244  df-poset 18262  df-plt 18279  df-lub 18295  df-glb 18296  df-join 18297  df-meet 18298  df-p0 18374  df-lat 18381  df-covers 38124  df-ats 38125  df-atl 38156  df-cvlat 38180  df-hlat 38209  df-psubsp 38362  df-padd 38655
This theorem is referenced by:  pmod2iN  38708  pmodN  38709  pmodl42N  38710  hlmod1i  38715
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