Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pmod1i Structured version   Visualization version   GIF version
Theorem pmod1i 39325
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 2727 . . . . 5 (leβ€˜πΎ) = (leβ€˜πΎ)
2 eqid 2727 . . . . 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 39324 . . . 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 1115 . . 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 4229 . . . . 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 1213 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ π‘Œ βŠ† 𝐴)
11 simplr1 1212 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ 𝑋 βŠ† 𝐴)
123, 5paddss2 39295 . . . . . 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 1368 . . . . 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 481 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) β†’ 𝐾 ∈ HL)
163, 4psubssat 39231 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑍 ∈ 𝑆) β†’ 𝑍 βŠ† 𝐴)
17163ad2antr3 1187 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) β†’ 𝑍 βŠ† 𝐴)
18 simpr2 1192 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) β†’ π‘Œ βŠ† 𝐴)
19 ssinss1 4238 . . . . . . . 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 39294 . . . . . . 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 1368 . . . . . 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 405 . . . . 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 1214 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ 𝑍 ∈ 𝑆)
259, 24, 16syl2anc 582 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ 𝑍 βŠ† 𝐴)
26 inss2 4230 . . . . . . . 8 (π‘Œ ∩ 𝑍) βŠ† 𝑍
273, 5paddss2 39295 . . . . . . . 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 1368 . . . . . 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 39318 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑍 ∈ 𝑆) β†’ (𝑍 + 𝑍) = 𝑍)
319, 24, 30syl2anc 582 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ (𝑍 + 𝑍) = 𝑍)
3229, 31sseqtrd 4020 . . . . 5 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ (𝑍 + (π‘Œ ∩ 𝑍)) βŠ† 𝑍)
3323, 32sstrd 3990 . . . 4 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ (𝑋 + (π‘Œ ∩ 𝑍)) βŠ† 𝑍)
3414, 33ssind 4233 . . 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 3997 . 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 411 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ∩ cin 3946   βŠ† wss 3947  β€˜cfv 6551  (class class class)co 7424  lecple 17245  joincjn 18308  Atomscatm 38739  HLchlt 38826  PSubSpcpsubsp 38973  +𝑃cpadd 39272
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 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744
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 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 7997  df-2nd 7998  df-proset 18292  df-poset 18310  df-plt 18327  df-lub 18343  df-glb 18344  df-join 18345  df-meet 18346  df-p0 18422  df-lat 18429  df-covers 38742  df-ats 38743  df-atl 38774  df-cvlat 38798  df-hlat 38827  df-psubsp 38980  df-padd 39273
This theorem is referenced by:  pmod2iN  39326  pmodN  39327  pmodl42N  39328  hlmod1i  39333
  Copyright terms: Public domain W3C validator